Modules and Proportions

Posted on August 2, 2023 by Graham Shawcross

Introduction

The primary reason why systems of proportion have been and continue to be important for architecture is that they enable our buildings to embody a mathematical order that we either distil out of or impose upon nature. Besides this, the other reasons commonly put forward — that these systems bring about a pleasing visual harmony, or that through modular coordination they enable components to fit together neatly and without waste – are relatively trivial.

Richard Padovan, 2001, p.10

I think that Padovan seriously overestimates the influence of proportion in current architectural practice, particularly as he denigrates the role of proportion in generating pleasing visual harmonies. I also think that he underestimates the importance to architecture of modularity; as a means of exhibiting order as well as allowing components to fit together neatly and without excessive waste.

Modularity

The numerals of a number system exemplify modularity with a module of the number one and all the other numerals being multiples of that number. Place value number systems, like our base 10 system, also give rise to a hierarchy of larger sized units: decades, hundreds etc.

Figure 1: Cuisenaire Counting Rods

Cuisenaire counting rods are a direct physical representation of this relationship, with one size and colour of rod for each of 1 to 10 numbers with length and colour encoding number.

Figure 2: The Evolving House. Vol. III. Rational Design (from Bemis A. F. 1937, p71)

The much-referenced images above, by Albert F. Bemis, merely show that with a small enough module it is possible to describe all the structural elements of a house. Prefabrication and standardisation efforts assume that it is desirable to reduce the variety and number of necessary components in a building programme. Repetition of recognisable physical units of similar size or proportion might also be considered aesthetically desirable. The use of multiples of module sizes can reduce component variety and, by using a modular grid, control their location both horizontally and vertically. However, the horizontal and vertical modules do not need to be the same and as shown later often reflect quite different functional requirements.

Figure 3: Modular Fixing and Location Grids

House Design Program

The Scottish Special Housing Association (SSHA) and the Edinburgh University Architectural Research Unit (ARU) developed a Computer Aided Design program, called House Design. (Bijl et al., 1971) Whilst working at Edinburgh University and SSHA, I was mainly  responsible for the practical implementation of this program particularly simplifying the interactive positioning of components

House Design allowed experienced designers to interactively design house types within Circular 36/69 restraints, which had already been incorporated, with some minor variations, into the Scottish Building Regulations.

All location, component and assembly drawings and bills of quantities were then automatically produced without further interaction, by reference to a complete set of standard component and assembly details. That is all possible component and assembly details had been identified, detailed and quantified in advance of their being required.

External walls, windows and external doors were located by placing pairs of 300mm square symbols constrained to be within a 300 mm grid. Structural partitions, non-loadbearing partitions and internal doors were located by placing pairs of 100mm square symbols constrained to be within a 100mm sub-grid. External walls, windows and external doors were therefore notionally 300mm wide whilst structural partitions, non-loadbearing partitions and internal doors were notionally 100mm wide.

Figure 4: House Design: Grid Based Component Location

Extra information such as window height or door swing direction was supplied by pulldown menus or extra pointing. At the time this was described as being a 2½D model; most information being derived from the 2D plan with some default height information and a little extra input to supply information about the 3rd dimension.

Figure 5: House Design: Vertical Dimensions

Of particular importance was the adoption of a standard 2600mm floor to floor height. This allowed stair symbols and floor openings to be accurately known. Thus far, this all conformed to Design Bulletin 16: Dimensional Coordination in Housing (MoHLG, 1969). Problems arose though when actual component widths differed from their 300 or 100mm notional widths.

Figure 6: House Design: Symmetrically Located Components

Without finishes, external walls were actually 240mm wide rather than 300mm, structural partitions 74mm wide rather than 100mm and non-loadbearing partitions 50mm wide rather than 100mm. Design Bulletin 16 recommended (probably required) that internal components were located with their finished faces on a 100mm grid line. Somewhat surprisingly it was thought that this would aid the location of components on site, perhaps assuming that they would all be pre-finished and in modular lengths.

Locating components symmetrically within their grid space with 30, 13 and 25mm offsets as illustrated above, meant that location plans could be automatically dimensioned, with dimensions to the structural face of the components, that is before plasterboard etc. was fixed and exactly as site operatives found them. This meant that as shown below grids could be ignored on site and not shown on location drawings; their work having been done in organising the system, they were no longer needed and could be discarded.

The use of 300 and 100mm grids did however usefully reduce the number of component sizes especially for windows, external doors (300mm increments) and internal doors (100mm increments). This was at the expense of non-modular partition lengths which in any case were less likely to be manufactured off site.

It is probably worth saying that modern production methods mean that components, like doors, can be produced to almost any dimension. Apart from the need to have perceivable visual order, this transfers site assembly problems to identifying slightly varying components and the difficult logistics associated with ensuring that individual components are delivered as and when required.

Figure 7: House Design: Accurately Dimensioned Location Drawing

Detail Complexity

The requirement to locate components with their finished faces on grid greatly increases the number of assembly details that are required. The enumerative analysis that follows was used to justify locating components symmetrically within their grid zones rather than face on grid.

The diagram below shows all the possible ways internal components can meet when they are located symmetrically within a grid. The numbers in brackets indicate the number of ways that components in that configuration can meet if they are located face on grid.

Figure 8: Possible Symmetrical Partition Locations

The following diagram, not part of the original analysis, shows the 16 ways components in the circled arrangement above, can meet when they are located with their faces on grid.

Figure 9: Possible Face On Grid Partition Locations for Circled Configuration

In summary, locating internal components symmetrically within their grid space reduces the number of assembly details required by a factor of almost 8 and thereby avoids the need for lots of otherwise very similar and potentially confusing details.

Combinations of Numbers in Building

As another means of reducing the number of required components, Combinations of Numbers in Building (Dunstone, P. H. 1965) demonstrates the way small numbers of modular components can be used in combination to satisfy the requirement of filling a continuous series of modular sizes. For instance, in the example below what modular spaces can be filled with two components that are 3 and 5 modules long? With these lengths it is not possible to fill spaces of 4 or 7 modules, but it is possible to fill all modular spaces of 8 modules or more. Making 8 what Dunstone calls the Critical Number (CN) for these sizes.

Figure 10: Combinations of Numbers in Building (after Dunstone, P. H. 1965, p.20)

In the days before computers or pocket calculators, Dunstone produced a number of graphical tools to aid in selecting appropriate component sizes to give particular Critical Numbers. However, the Critical Number for two integer modular sizes a and b can easily be calculated as follows, CN = (a – 1) x (b – 1). In the example given above CN = (3 – 1) x (5 – 1) = 8. With 3 or more sizes the calculations can just be done in pairs and the lowest CN used.

Dunstone gives a number of heuristics for choosing component sizes but surprisingly given what follows correctly warns that if two component sizes of a trio, combine to make a third in Fibonacci series fashion (e.g. 3, 5 and 8) then the CN of the trio is not reduced by the third member but the number of ways individual sizes can be made up is increased.

In my early professional career, I used this procedure to design a prefabricated foundation system for the MoHLG Research and Development Group’s Finchampstead Project. A small number of modular beam lengths were selected that could be arranged in different ways so that all modular block lengths and depths could be accommodated, and foundation pads always placed so as to avoid the position of service entry plugs and soil stacks, which were the only things allowed to pierce the ground floor slab.

Figure 11: Finchampstead Project: Foundation Design

Proportion

Proportion can have practical as well as aesthetic uses. In particular, a number of rectangles with specific aspect ratios can be recursively divided into rectangles with the same aspect ratio. This implies that these rectangles can be inflated as well as divided, that is become part of a larger rectangle with the same aspect ratio. The diagram below illustrates the three rectangles that can initially be divided into two such rectangles. In each series, equal aspect ratios are indicated by diagonal lines and squares by inscribed circles.

Figure 12: Recursively Dividing Rectangles in Two

The rectangle with a root 2 aspect ratio, on the left above, allows itself to be split in half across its longer edge to give 2 rectangles with the same aspect ratio, a process that can be repeated ad infinitum. This is the ratio underlying the A series of paper sizes, A1 is half A0 etc. and all sizes usefully have the same aspect ratio. Something that seems satisfying in both aesthetic terms and the parsimonious use of materials.

The double square 2:1 ratio allows each square to be divided into two rectangles with the same 2:1 aspect ratio, again a process that can be repeated indefinitely. This is the ratio that prosaically underlies standard brick sizes, with coordinating dimensions, including one 10mm mortar joint in each direction, of 225 x 112.5 x 75mm (length x width x height).

The division of the Golden Section rectangle is a little more difficult to describe. This rectangle can be split into a square and a smaller rectangle that has the same aspect ratio, a process that can then continue to be applied repeatedly to the smaller rectangle.

The Golden Section

The value of the Golden Section derives directly from the following relationship, that the ratio of L to S is the same as the ratio of (S + L) to L.

Figure 13: Golden Section Derivation

Although formally the Golden Section Ø is an irrational number and therefore cannot be represented as the ratio of two integers, in fact with any Fibonacci type sequence, where n = (n – 1) + (n – 2), as n increases the ratio n / (n – 1) progressively approaches Ø.

Figure 14: Fibonacci Approximation to Golden Ratio (1)

Figure 15: Fibonacci Approximation to Golden Ratio (2)

(8 / 5) is a fair approximation of Ø but (13 / 8), (21 / 13), (55 / 34) and (89 / 55) etc. are progressively better approximations. Moreover, rectangles drawn with these ratios cannot be visually discriminated from ones drawn more accurately and there would seem to be no point in trying to be more accurate than these integer approximations. The rectangles below were drawn with the same shorter edge and the longer edge scaled to give the stated long to short ratio.

Figure 16: Discriminating the Golden Rectangle

When trying to fit a Golden Rectangle to a classical building like the Parthenon, it always seems difficult to decide what should be included. I think that all that can reasonably be concluded is that the Parthenon has a pleasing proportion that is similar to that of the Golden Rectangle.

Figure 17: The Parthenon and the Golden Rectangle

Tiling the Plane with Aperiodic Tiles

As shown in Shadows of Reality : The Fourth Dimension in Relativity , Cubism , and Modern Thought (Robbin, 2002), the Golden Section (and other irrational numbers like root 2 and the Plastic Number) can play a part in proving that aperiodic tiles tile the plane, see Aperiodic Tiling. These irrational ratios can be used to generate the long and short ‘musical sequences’ that describe large sets of aperiodic tiling.

Figure 18: Musical Sequence from the Golden Section (after Robbin 2002, p.65)

In musical sequences a long interval can follow another long interval, but a short interval must follow two long intervals, and a short interval must be followed by a long one.

Figure 19: Short and Long Relationship

In inflation, or composition, the new short interval is equal to the old long interval and the new long interval is equal to the old short interval plus the old long interval. The substitutions S’ = L and L’ = L + S, mandate the Golden Section L  /  S = (L + S)  /  L = Ø.

Figure 20: Inflation of Musical Sequences (after Grünbaum and Sheppard, p.574)

As described in Tilings and Patterns (Grünbaum and Sheppard, 1986), the fact that such inflation (composition) can continue for ever, proves the ability of aperiodic tiling to tile the plane.

Figure 21: Composition of Penrose P2 Tiling (after Grünbaum and Sheppard, 1986, p.573)

In partial explanation as shown below, the Golden Section and Penrose P2 Tiling both have an underlying relationship to the pentagon.

Figure 22: Relation of Pentagon to Ø and Penrose P2 Tiles

V+A Spiral Project and Aperiodic Tiling

Aperiodic tiles were used by Daniel Libeskind and Cecil Balmond in their Victoria and Albert Museum ‘Spiral’ proposal. Ammann A2 tiling was used in a hierarchical fractal manner at different scales for physical surface treatments, tiles and panels. Ammann bars, enforcing aperiodicity, also provided a linear support structure.

 Figure 23: Ammann A2 Tiling (after Grünbaum and Sheppard 1986, p.550)

Figure 24: V+A Spiral Project (from Cecil Balmond Studio and Daniel Libeskind Studio)

The use of aperiodic tiles in this project is well documented in Informal (Balmond and Smith, 2007) including this quote from Cecil Balmond (as formatted by him).

Balmond and Smith 2007, p.245.

The Modulor

Richard Padovan in Proportion Science, Philosophy, Architecture (Padovan, 2001) describes the precursors and early development of Le Corbusier’s Modulor. In its later versions the Modulor attempted to do two things, to reconcile the supposedly more anthropometric imperial units with the supposedly more mechanical metric units, and to link human dimensions to the Golden Section, in red and blue doubled scales.

Figure 25: Le Corbusier, The Modulor: 1951, cover

Basically, the Modulor is a vertical scale, relating the height of a rather tall man to the Golden Section, and therefore does not relate particularly to modular planning in the horizontal plane.

Figure 26: Le Corbusier, The Modulor: 1951, p.67

Interestingly Le Corbusier himself carried around a customised tape measure marked with imperial and metric scales on one side and the divisions of the Modulor marked on the other. This looks like an attempt to use the Modulor to find order in the existing world as well as impose order on a future world.

Figure 27: Modulor Tape Measure: Modern Facsimile

BRS Modular Number Pattern

Ezra Ehrenkrantz in The Modular Number Pattern, Flexibility Through Standardisation (Ehrenkrantz, 1956) was concerned about dimensional coordination in the British building industry as a whole. He was aware of Corbusier’s Modulor and its relation to the Golden Section and the Fibonacci Series. Produced before the advent of the personal computer or electronic calculators, he presented his preferred numbers on a set of three separated transparent sheets.

Figure 28: BRS Modular Number Pattern 1 (from Ehrenkrantz, E. 1956, insert)

Figure 29: BRS Modular Number Pattern 2 (after Ehrenkrantz, E. 1956, insert sheets)

The top sheet has the Fibonacci series down the left-hand edge and a doubling series across the top (both in red) with their products in the remaining squares (ignoring anything above 1000). The second sheet has the values on the first sheet multiplied by 3 and the third sheet multiplied again by 3. Whilst the Fibonacci and doubling series might provide useful combinations, it is not clear why the multiplications by three are useful but still only 82 black and 23 grey sizes are suggested.

Based on his experience of the Hertfordshire schools building programme whilst in England and working with the Building Research Station, Ehrenkrantz went on to develop the much lauded Southern California Schools Development program. (Ehrenkrantz, Ezra, 1989) Using a 60” (5’) module all components except the building envelope, but including the structure, mechanical services, lighting and interior partitions were realised as a performance specified, modularly coordinated kit of parts, where parts could be easily assembled, disassembled or reconfigured. The 60” (5’) module giving rise to organised spaces with 10’, 20’ and 30’ dimensions.

Figure 30: SCSD Components (from Ehrenkrantz, Ezra, 1989, p.141)

New Proportional Systems

In a recent Guardian article The golden ratio has spawned a beautiful new curve: the Harris curve (https://tinyurl.com/yxf98lez), Alex Bellos wrote about the American mathematician, Edmund Harris, who has identified a series of proportional systems where rectangles can be split into squares and same aspect ratio rectangles. These include the root 2, Ø and double square rectangles described previously, where the given rectangle is initially divided into two parts. But Harris shows that there is also a further family of proportional systems based on rectangles where the original rectangle is initially split into three parts, that again are either squares or rectangles with the same aspect ratio.

The Golden Rectangle and Double Square examples greyed out below can be thought of as second-generation divisions of their two division parents. The root 2 three-part division, however, is different from the previously described root 2 version in that the rectangle is divided into two different sized squares and a rectangle with the same aspect ratio rather than two rectangles with the same aspect ratio.

The two-division ratios given previously and the ratios below are all solutions to simple equations, sometimes known as algebraic numbers. These equations are given above the top-left corner of each rectangle. For example, the equation for the golden rectangle is x2 = x + 1 and for the Plastic Number x3 = x +1 etc.

Figure 31: Other Proportional Systems

The Plastic Number

We are overwhelmed by nature and we are looking for artificial principles to dissolve this contrast, to again get a grip on space, to again understand and control it. Architecture in this sense becomes a necessary instrument for our intellect as well as an expression of a regained authority, of an understood space.

Van der Laan, 1941
quoted in Caroline Voet, 2016, p.4

The architect and Benedictine monk, Dom Hans van der Laan (1904 – 1991) discovered, or rediscovered, the Plastic Number in 1928 although until 1955 he called it the ‘ground ratio’. Like the Golden Section, the Plastic Number has a related Fibonacci like sequence where n = (n – 2) + (n – 3). This is now called the Padovan sequence (OEIS A90031), named for van der Laan’s biographer, Richard Padovan.

Figure 32: Padovan Series Approximation to the Plastic Number 1

Figure 33: Padovan Series Approximation to the Plastic Number 2

With the Padovan sequence, the (4 / 3) ratio is extremely close to the Plastic Number (being a little more than half a percent off its true value). Consequently, the convergence of the succeeding ratios is not so pronounced as with the Fibonacci series and the Golden Ratio. (4 / 3) is a better approximation to the Plastic Number than either (9 / 7) or (21 / 16) but is not as good as (49 / 37).

The closeness of the (4 / 3) ratio to the true irrational value has led people to describe the Plastic Number theory as an additive system rather than a proportional one or at least an additive system that facilitates pleasant proportions.

Van der Laan himself believed in the philosophical notion of there being a perceptible order in space, that could be revealed in architecture through a commensurable proportional system. He assumed that the minimum ratio between two sizes that could be differentiated was 4:3. By differentiating he meant recognising a difference that can be counted or named. He does not mention Weber’s Law or the Distance Effect but in accounting for the 4:3 preference does refer to their just ‘discernible difference’.

Figure 34: Differentiating Ratios

Van der Laan thought that the Plastic Number was superior to the Golden Section because it produced a continuous series of same proportion divisions (on the right below), whilst the Golden Section when applied repetitively leads to two equal length divisions (on the left below).

He also, according to Padovan erroneously, thought that the Plastic Number was inherently more 3-dimensional than the Golden Section because it was the solution to the equation x3 = x + 1 rather than x2 = x + 1 for the Golden Section.

Figure 35: Golden Section and Plastic Number Ratios

The application of van der Laan’s design philosophy produced a fairly austere, spiritual, although according to him not minimalist, architecture.

Figure 36: St. Benedictusberg Abbey, Vaals, (photo by Coen van der Heiden 2008).

Despite the austere nature of his architecture, van der Laan displayed a playful nature when promoting his theories. For instance, leaving ordered piles of pebbles for his students to find and designing Cuisenaire like sets of proportional wooden bricks.

In an experiment that van der Laan conducted with his students, as shown on the left below, he carefully selected 36 pebbles each of which was 1/25th less in diameter than the previous one, considering this a difference that could be perceived without measurement. He then asked his students to sort the mixed pebbles into sets that appeared to be of the same size, a process shown in the middle column below.

Figuire 37: Hans van der Laan: Number and Ratio (from C. Voet 2016, p.5)

Van der Laan claimed that all his students selected 7 groups of ‘equal’ sized pebbles and that the ratio between the group sizes was 4:3 as shown in the right-hand column above. In Between Looking and Making: Unravelling Dom Hans van der Laan’s Plastic Number,Caroline Voet describes repeating this with her students and not getting quite such clear results. (Voet, 2016)

Van der Laan was particularly interested in the number 7 and the ratio 1:7 which he thought was the maximum difference between two sizes that could still be related, what he called their ‘nearness’.

Figure 38: Nearness and the Ratio 1:7

Using the ratios 3:4 and 1:7 van der Laan produced a series of 8 ‘authentic’ sizes, where each size starting at 4:3 closely approximates to a power of Ψ, the Plastic Number, shown in red below. Van der Laan also introduced a derived set of sizes where each authentic size is coupled with a derived size that is 6/7th of its size.

Figure 39: Plastic Number: Van der Laan’s Authentic and Derived Sizes

In van der Laan’s toy like, Form Banks or Morphotheeks, he uses the authentic range of sizes to create arrays of wooden shapes and volumes, which he classifies as blocks (brown below), bars (yellow), slabs (blue) and blanks (white). Each type also has a representative form, shown in a lighter colour. This resulted in 36 preferred shapes and 122 preferred volumes which designers could play with and come to understand their relationships.

Figure 40: Van der Laan Form Banks

Figure 41: Form Bank (photo Jeroen Verrecht)

Being Commensurate

As with the House Design example above, when communicating sizes on drawings or in instructions it is desirable to use values that are easily understood, that is whole numbers or if necessary, simple fractions like 1/2. The adoption of millimetres as the UK official unit of measure for the building industry greatly facilitates this, the unit being small enough to almost entirely remove the need for fractions.

Padovan’s analysis of the Parthenon below also illustrates the need for dimensions to be commensurate. (Padovan, 2001) He suggests a column spacing of 5 units, a stylobate that is 3 units wide wrapped round a core of 15 x 6 column spacings. Assuming that column edges are aligned with the outer edge of the stylobate and that the corner column spacings are reduced, this gives rise to overall dimensions of 81 x 36 units. A core ratio of 15:6 (5:2) becomes an overall ratio of 81:36 (9:4) but all the important dimensions are small whole numbers.

Figure 42: Parthenon: Whole Number Interpretation (after Pardovan, 2001, p.95)

Another historical example is Palladio’s use of whole numbers of Vicentine feet and occasional simple fractions in his drawings of the Villa Foscari and other villas in his Four Books of Architecture. (Palladio, trans Tavenor and Schofield, 1997)

Figure 43: VillaFoscari (from Palladio, trans Tavenor and Schofield, 1997, p.128)

Conclusions

Repetition can be desirable for both aesthetic and practical reasons. Modularity facilitates repetition of object sizes whilst proportion facilitates repetition of the aspect ratio of shapes in a potentially fractal manner but essentially without size repetition.

Number systems are a natural model for modular sizes, where the module is the number one and the natural numers are its multiples. If the number system is a place value system, see Place Value, then the individual places can provide a hierarchy of larger modules sizes.

Planning grids encourage modularity by regulating the placing and size of building elements, but horizontal and vertical modules do not need to be the same. The way elements are located within their grid space can facilitate appropriate dimensioning and remove the need for grids to be shown on locations drawing. Placing components symmetrically within their grid space also significantly reduces the number of assembly details that are required.

Component variety can also be reduced by carefully choosing modular sizes that can combine together. However, choosing sizes that are related in a Fibonacci like manner does not increase the possible range of sizes but does increase the number of ways that they can be combined.

Whilst proportional systems are all defined by irrational numbers, in practice it is impossible to tell if rational integer approximations have been used in their presentation rather than an approximation to several or many decimal places.

Proportional systems have practical as well as aesthetic uses. For instance, the root 2 rectangle giving rise to A paper sizes and the Golden Section (Ø) being part of the proof that aperiodic tiles tile the plane. Ø and its underlying Fibonacci series also form the basis of Corbusier’s Modular and Ehrenkrantz’s BRS Modular Number Pattern, neither of which seem to have had any lasting effect or legacy.

The fact that the ratio 4:3 is extremely close to the Plastic Number (Ψ) means that the proportional system proposed by Dom Hans van der Laan can usefully bridge the gap between modular and proportional systems.

As shown in the Round and Sharp Numbers post some numbers are round and a slightly different set, relating to the base 10 of our number system, are favourites. These are the numbers that we would naturally prefer to use or find when measuring. Like the House Design example this relates to dimensions needing to be commensurate that is wherever possible whole numbers or simple fractions.

Figure 44: Component Sizes, Favourite Numbers and Greedy Algorithm

Bibliography

Bijl, A. et al. (1971) ARU research project A25/SSHA-DOE: house design ; application of computer graphics to architectural practice, ARU CAAD Studies. Edinburgh Unniversity, Architecture Research Unit, 1971.

Ehrenkrantz, Ezra, D. (1989) Architectural Systems: A Needs, Resources and Design Approach. McGraw-Hill.

Ehrenkrantz, E. (1956) The Modular Number Pattern, Flexibility Through Standardisation. HMSO.

Grünbaum, B. and Sheppard, G. C. (1986) Tilings and Patterns. W. H. Freeman and Company.

MoHLG (1969) Design Bulletin 16: Dimensional Coordination in Housing.

Padovan, R. (2001) Proportion, Science, Philosophy, Architecture.

Palladio, A., trans Tavenor, R. and Schofield, R. (1997) The Four Books of Architecture. The MIT Press.

Robbin, T. (2002) ‘Shadows of Reality : The Fourth Dimension in Relativity , Cubism , and Modern Thought’, Dimension Contemporary German Arts And Letters, pp. 74–76.

Voet, C. (2016) ‘Between Looking and Making: Unravelling Dom Hans van der Laan’s Plastic Number’, Architectural Histories. Ubiquity Press, Ltd., 4(1), pp. 1–24. doi: 10.5334/ah.119.

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Size and Distance Effect

Posted on November 19, 2022 by Graham Shawcross

The Distance Effect

In whatever way they are presented, it is easier to compare two magnitudes that are quantitatively further apart, than it is to compare two magnitudes that are quantitatively closer together. This is the distance effect.

Figure 1: Distance Effect: 2 versus 8 and 8 versus 9

It is easier to differentiate 2 and 8 dots (on the left) to decide which is smaller or larger, than it is to differentiate 8 and 9 dots (on the right). This is because these two comparisons have different separation distances of 6 (8 – 2) and 1 (9 – 8) respectively and it is easier to recognise a difference of 6 than a difference of 1.

The Size Effect

If the numerical distance between two magnitudes remains the same, then it is easier to compare two small magnitudes than it is to compare two larger magnitudes. This is the size effect.

Figure 2: Size Effect: 3 versus 4 and 8 versus 9

It is easier to differentiate 3 and 4 dots (on the left) than it is to differentiate 8 and 9 dots (on the right), where both comparisons have a separation distance of 1 (4 – 3) and (9 – 8) respectively.

The distance and size effects follow Weber’s Law, are present in different modalities and are present with two-digit numbers. For instance, when comparing two-digit numbers against a fixed reference such as 65, comparing 79 with 65 (difference 14) takes less time than comparing 71 and 65 (difference 6) an effect that is not subject to decade boundaries.

Figure 3: Numerical Distance Effect (from Dehaene 2011, p.64)

Origin of Size and Distance Effect

According to Gelman and Gallistel, in Language and the Origin of Numerical Concepts (Gelman and Gallistel, 2004) one can imagine that repeated perceptions of a given numerosity give rise to normal signal distributions like those shown below. The wider spread the distribution is, the less precise and noisy the representation. The extent to which two signal distributions overlap, determines the likelihood of confusion as to which distribution an individual signal belongs to and the more processing time is required to resolve the situation. So, no matter how the numerosities are presented, it is much easier (takes less time) to discriminate 10 from 2 and is much harder (takes longer) to discriminate 2 from 3.

Figure 4: Mental Magnitude Signal Distributions (after Gelman and Galliestal 2004, p.442 A)

This is the generally accepted explanation of the size and distance effects. It ties basic arithmetic reasoning with numerical symbols (order judgements) to an imprecise non-verbal representation of number.

Gelman and Galliestal 2004, p.441
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Numerical Separators

Posted on November 1, 2022 by Graham Shawcross

With a base 10 place value number system, Britain and America use the period as the radix symbol, to separate integers and decimals, and use a comma to separate groups of digits; for example, they would write

3,200,100.56

Other Europeans such as the French, Italians, Spanish and Norwegians use the comma as a radix symbol and a period as a separator, so would write

3.200.100,56

Yet others like the Swedes and Finns use the comma as a radix symbol and use a small space as a separator, so would write

3 200 100,56

The Germans however would write

3 200.100,56

This can represent a problem for translation systems. For instance, internationally the following number is ambiguous

100.523

However, in all cases the underlying principle is that breaking long numbers, into groups of 3 or less digits, facilitates speed and accuracy when reading numbers.

The following earlier posts are relevant

Subitising
Place Value
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Round and Sharp Numbers

Posted on October 30, 2022 by Graham Shawcross

“Don’t interrupt,” Bruno said as we came in. “I’m counting the Pigs in the field!”

“How many are there?” I enquired.

“About a thousand and four,” said Bruno.

“You mean ‘about a thousand,’” Sylvie corrected him. “There’s no good saying ‘and four’: you can’t be sure about the four!”

“And you’re as wrong as ever!” Bruno exclaimed triumphantly. “It’s just the four I can be sure about; ‘cause they’re here, grubbing under the window! It is the thousand I isn’t pruffickly sure about

Lewis Carroll (Sylvie and Bruno Concluded)
quoted in Dehaene 2011, p. 95

According to Dehaene in The Number Sense: How The Mind Creates Mathematics (Dehaene, 2011), numbers are either round or sharp. Round numbers represent approximate quantity whilst all other numbers are sharp and have a precise meaning. The conversation above sounds strange because Bruno is using a sharp number, a thousand and four, as if it were a round number like a thousand. Sylvie as ever is correct.

If you have to describe a population of 5,424,000 individuals and you are unsure about the exact number or do not wish to be precise, you might say that the population is 5 million, implying that it is 5 million plus or minus a million. If you are a little more sure about the number, you might say that the population is 5 million 400 thousand, implying that your estimate is accurate to the nearest hundred thousand. Obvious problems arise if the actual number is the same as what might otherwise be regarded as a round number. This can be overcome by using a locution such as, the population is exactly 5,424,000.

All the languages of the world seem to have selected a set of round numbers. Why this universality? Probably because all humans share the same mental apparatus and are, therefore, confronted with the difficulty of conceptualizing large quantities. The larger a number, the less accurate is our mental representation of it. Language, if it wants to be a faithful vehicle for thought, must incorporate devices that express this increasing uncertainty.

Dehaene 2011, p.96

Round numbers are therefore a device that lets language express inaccuracy and uncertainty much as it is presented by the Approximate Number System.

Approximate Number Pairs

All languages also have a large vocabulary of words for expressing various degrees of numerical uncertainty. For instance, the English words, about, almost, approximately, around, barely, circa, close to, just about, more or less, nearly, roughly etc. Some of these words such as around, close to and nearly are expressly spatial.

In many languages one can also indicate an approximate quantity by saying a pair of numbers, in English for example, one might say “four or five toys”. Unfortunately, in English this can have a disjunctive as well as an approximate meaning. It can mean either four or five which just means one or the other, as well as approximately four or five which might mean three four, five or six etc.

In approximate expressions like these, not all combinations of numbers are possible or acceptable, Following work by Channell and Sigurd, Pollmann and Jansen in The Language User as an Arithmetician suggest that acceptable pairings follow a number of simple rules.(Pollmann and Jansen, 1996)

  1. The Roundness Rule. At least one of the pair of numbers needs to be round, that is in one of the following sets of numbers: the integers 1-20 and multiples of 5, 10, 20, 50 or 100.
  2. The Ordering Rule. The smallest number of an approximate pair expression comes first, so one can say two or three toys but not three or two toys. An exception is twelve or one, probably because one-o-clock comes immediately after twelve-o-clock.
  3. The Difference Rule. Both numbers must be of the same order, so one can say twenty or thirty people but not in this sense ten or one thousand dollars.
  4. The Sequence Rule. The numbers in a two number approximate expression have to be part of the same arithmetic sequence, and have to follow each other in that sequence.

Pollmann and Jansen 1996, p.223

Pollmann and Jansen then expand the sequence rule as follows

  1. The starting point of any sequence has to be equal to the difference between the numbers in the sequence; so when the difference is 1, the starting point also has to be 1 and the sequence is 1, 2, 3, 4, 5, etc. When the difference and starting point is 2, the sequence is 2, 4, 6, 8, 10, 12, etc. And when the difference and starting point is 10, the sequence is 10, 20, 30, 40, 50, etc. Expressed like this, the sequence rule correctly predicts that expressions like 3 or 4, 4 or 5 and 20 or 30 are acceptable, while expressions like 1 or 3 and 2 or 5 are not.
  2. Only sequences where the starting point and difference is 10n, 5 x 10n, 2 x 10n or 2.5 x 10n can be used as sources for approximate expressions. Therefore, expression pairs like 3 or 6, 40 or 80 and 600 or 1200 that would form part of sequences with differences equaling 3 x 10n, 4 x 10n or 6 x 10n are not acceptable.
  3. The sequence involved can only be used up to its 20th number at most. 12 or 13 and 90 or 95 are therefore acceptable, but 32 or 33 and 150 or 155 are not. If the starting point and difference is 2 or 20 etc., or 2.5 or 25 etc. the sequence might be even shorter than 20. Meaning that large numbers with small differences are not acceptable members of an estimation pair.

Pollmann and Jansen 1996, p.224

Favourite Numbers

With regard to round numbers, Pollmann and Jansen suggest that estimations made using the Approximate Number System can be thought of as being mapped onto a set of favourite numbers. So that with any particular base n there is a set of favourite numbers that consists of:

  • any integer power of the base
  • half, double, and half of half of any integer power of the base
Pollmann and Jansen 1996, p.225

Giving a set of favourite numbers for our base 10 number system of something like:

n-powers of base 10           0.01, 0.1,   1,      10,    100, 1000 etc.

doubles                                       0.2,   2,      20,    200, 2000 etc.

halves                                                 0.5,     5,      50,   500 etc.

halves of halves                                 0.25,   2.5,   25,   250 etc.

Pollmann and Jansen then wanted to see if favourite numbers like these occurred in everyday usage and examined the denominations used for different currencies. They examined the denominations used by 84 countries with a total of over 1000 separate denominations and found that they all used base 10 favourite numbers as defined above, with just 13 exceptions.

Sterling             1p, 2p, 5p, 10p, 20p, 50p, £1, £2, £5, £10, £20, £50

Euro                 1, 2, 5, 10, 20, 50 cents €1, €2, €5, €10, €20, €50, €100, €200, €500

Dollar               1, 5, 10, 25, 50 cents $1, $2, $5, $10, $50, $100

The few exceptions mainly included countries that have a 3-unit coin or banknote. Before metrication in 1971, Great Britain also had threepence (3d) and sixpence (6d) coins although these would have been members of a base 12 favourite number list, there being 12d in a shilling and 6d would half of that and 3d a half of a half of a shilling.

When thinking about satisfying a particular value, for instance when proffering money or change, a so-called ‘greedy algorithm’ is sufficient. That is first proffering the largest denomination less than the required value, then subtracting that denomination from the value to create a new value and continuing the process until the value becomes zero.

If you are asked for 75p you would, if possible, first proffer 50p leaving 75 – 50 = 25p, then proffer 20p leaving 25 – 20 = 5p and finally proffer 5p leaving 5 – 5 = 0p. Greedy algorithms like this are not necessarily globally optimal but with the currencies and denominations listed above they are. This means that these denominations are good models for sets of coordinated modular sizes.

Component Sizes

If favourite numbers are used for component sizes a greedy algorithm guarantees that any overall size can be fulfilled with the minimum number of components.

Figure 2: Favourite Numbers and Greedy Algorithm

Saliency of Favourite Numbers

Round numbers are a device that lets language express the inaccuracy and uncertainty of our perception of larger numbers as presented by the Approximate Number System. These numbers and especially their Favourite Number subset have great saliency and are the numbers that we prefer to see when measuring and prefer to use when designing.

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Less Chaos More Order

Posted on September 10, 2023 by Graham Shawcross

This is a quick posting adding to a previous post Cafetières, Disorder, Chaos and Anarchy. This morning whist sitting in the sun on our breakfast seat at Scoughall, I noticed that the coffee pot we always use both here and at home answers all the problems, real or imaginary, posed by Bodum cafetières.

Proudly boasting that it is ‘Made in England’, it just has one opening because it is ceramic, but has a mark on the lid to tell you where that opening is.

And it makes a great cup of coffee.

IKEA have nearly got the right idea

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Finchampstead

Posted on May 12, 2023 by Graham Shawcross

BACKGROUND

This post recreates the report issued for the opening of the Gorse Ride Housing Estate Finchamptead by Mr Peter Walker Minister of Housing and Local Government on 17th July 1970. The estate was designed by the Ministry’s Research and Development Team. The reports authors were Pat Tindale, Rosemary Stjernstedt and Graham Shawcross.

INTRODUCTION

This brochure briefly describes a housing scheme designed by the Ministry of Housing and Local Government Research and Development Group. The R and D Group is a team of architects, sociologists, quantity surveyors and administrators. Ove Arup and Partners are structural consultants to the Group;

Development on constructional techniques was carried out in collaboration with the Production Division of the Building Research Station and on the coordination of underground services with the Directorate of Research and Development of the Ministry of Public Buildings and Works.

THE SCHEME

The Research and Development Group were appointed by the Rural District Council of Wokinham to design a housing scheme for a site of 25 acres in the village of Finchamstead at a density of 60-70 persons per acre.

An analysis of the Council’s waiting list and a survey of local residents was undertaken by the Group’s sociologists in order to determine the sizes of households to be accommodated and the facilities to be provided.

The residents’ main concerns about the area were the infrequency and inflexibility of public transport, the lack of local shops and the danger of fast moving cars to their children.

Car ownership among local authority tenants in the area has risen already to 60%. The layout adopted allows for a high proportion of garages attached to houses. The Radburn principle of layout was rejected in favour of a controlled, mixed side for vehicles and pedestrians where vehicles are few and slow while the rear sides of the houses are kept pedestrian and give access to children’s play areas.

Direct pedestrian routes across the site were arranged to connect the existing shopping centre, which the planners did not wish to expand, to a new centre which will contain a primary school, library, shops and public house.

It was estimated that best value for money would be obtained by medium frontage terrace houses, two and three-person accommodation being one-storey in height and four, five and six-person two storeys. The site is virtually flat but well wooded and steps and staggers were therefore unnecessary.

CONTRACT PROCEDURE

The contract for Phase 1 of the scheme is for 172 houses and ancillary works. The roads were constructed under a separate contract before the building contract commenced. The contractor was appointed by a two-stage tender procedure.

The builder was selected at the stage when the design of the layout and house types was almost complete but the constructional details were still in generalised form. The selection was based, following an open invitation to be considered, on the builder’s experience of house building, on the capacity of his management organisation to participate in pre-contract decisions on the form and details of the components, and on his general level of pricing as determined by a notional bill of quantities for the scheme. During a six month period following selection of the builder, fortnightly meetings between architects, quantity surveyors, BRS and from a director, contracts manager and site agent from the building firm took place and production information prepared.

Tenders for components were invited from a selected list of manufactures. These were invited to quote for external wall panels, party wall panels, partitions, floor panels and/ or joists and deck, and roof panels and / or deck, all to be delivered in house sets loaded in erection sequence and delivered to site as programmed by the main contractor at the average rate of one house shell per day.

The selected manufacturer did not quote for floor deck or roof deck and these were supplied by the main contractor.

The Council requested that there should be no excess over the cost yardstick allowance for the scheme and the tender submitted was 3.5% below this figure.

HOUSE PLANS

Space and equipment follow the standards laid down in Circular 36/67 and Design Bulletin 6 “Space in the Home” and the arrangement of rooms is based on data provided by user surveys of earlier schemes designed by the Development Group. The four, five and six-person houses have two living spaces; one a dining area attached to the kitchen and the other, a separate living room. Four-person houses have either two double bedrooms or one double and two single bedrooms. The five-person have alternative ground floor plans, one with living room at the front and one with living room at the back in order that the living room may always have south or west orientation.

The plans are based on a 1 ft planning grid and follow in imperial measurements the principles of the dimensional framework laid down in Design Bulletin 16.

All houses are rectangular and two-storey house have the same depth.

BUILDING SEQUENCE

DETAILS

LOCATIONS

Not part of original paper

Corner of Dart Close Showing 6 person dwelling

Back of Firs Close across the public open space

Old persons bungalows at Firs Close

Posted in Architecture, House Building Productivity, Housing | Tagged | 1 Comment

House Design

Posted on May 5, 2023 by Graham Shawcross

This post adds extra information to part of an earlier post “Severely Constrained Design”.

The Scottish Special Housing Association (SSHA) and the Edinburgh University Architectural Research Unit (ARU) developed a Computer Aided Design program, called House Design. (Bijl et al., 1971) Whilst working at Edinburgh University and SSHA, I was partly responsible for the practical implementation of this program particularly simplifying the interactive positioning of components.

House Design allowed experienced designers to interactively design house types within Circular 36/69 restraints, which had already been incorporated, with some minor variations, into the Scottish Building Regulations.

All location, component and assembly drawings and bills of quantities were then automatically produced without further interaction, by reference to a complete set of standard component and assembly details. That is all possible component and assembly details had been identified, detailed and quantified in advance of their being required.

External walls, windows and external doors were located by placing pairs of 300mm square symbols constrained to be within a 300 mm grid. Structural partitions, non-loadbearing partitions and internal doors were located by placing pairs of 100mm square symbols constrained to be within a 100mm sub-grid. External walls, windows and external doors were therefore notionally 300mm wide whilst structural partitions, non-loadbearing partitions and internal doors were notionally 100mm wide.

Grid Based Component Location

Extra information such as window height or door swing direction was supplied by pulldown menus or extra pointing. At the time this was described as being a 2½D model; most information being derived from the 2D plan with some default height information and a little extra input to supply information about the 3rd dimension.

House Design Dimensional Constraints

Of particular importance was the adoption of a standard 2600mm floor to floor height. This allowed stair symbols and floor openings to be accurately known. Thus far, this all conformed to Design Bulletin 16: Dimensional Coordination in Housing (MoHLG, 1969). Problems arose though when actual component widths differed from their 300 or 100mm notional widths.

Symmetrically Located Components

Without finishes, external walls were actually 240mm wide rather than 300mm, structural partitions 74mm wide rather than 100mm and non-loadbearing partitions 50mm wide rather than 100mm. Design Bulletin 16 recommended (probably required) that internal components were located with their finished faces on a 100mm grid line. Somewhat surprisingly it was thought that this would aid the location of components on site, perhaps assuming that they would all be pre-finished and in modular lengths.

Locating components symmetrically within their grid space with 30, 13 and 25mm offsets as illustrated above, meant that location plans could be automatically dimensioned, with dimensions to the structural face of the components, that is before plasterboard etc. was fixed and exactly as site operatives found them. This meant that as shown below grids could be ignored on site and not shown on location drawings; their work having been done in organising the system, they were no longer needed and could be discarded.

House Design: Accurately Dimensioned Plan

The use of 300 and 100mm grids did however usefully reduce the number of component sizes especially for windows, external doors (300mm increments) and internal doors (100mm increments). This was at the expense of non-modular partition lengths, which in any case were less likely to be manufactured off site.

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Graham’s Maze Game: TestFlight Help

Posted on October 8, 2021 by Graham Shawcross
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Königsberg Bridges

Posted on March 7, 2016 by Graham Shawcross

Background

The great Swiss mathematician Leonhard Euler, who had been asked by the Mayor of Danzig to provide a solution to the Königsberg Bridge problem, sent him this disdainful reply:

“. . .  Thus you see, most noble Sir, how this type of solution bears little relationship to mathematics, and I do not understand why you expect a mathematician to produce it, rather than anyone else, for the solution is based on reason alone, and its discovery does not depend on any mathematical principle.  Because of this, I do not know why even questions which bear so little relationship to mathematics are solved more quickly by mathematicians than by others.”

However in the same year he wrote this, in a letter to the Italian mathematician and engineer, Giovanni Marinoni.

“This question is so banal, but seemed to me worthy of attention in that [neither] geometry, nor algebra, nor even the art of counting was sufficient to solve it.”

On August 26, 1735, Euler presented a paper containing the solution to the Königsberg bridge problem, in which he addresses both the specific problem, and gives a general solution with any number of land masses and any number of bridges.  This paper, titled  ‘Solutio problematis ad geometriam situs pertinentis,’ was  published later in 1741. What follows is a verbatim version of an edited version of this paper translated by James R.Newman, that appeared  in my 1978 set of reprints from Scientific American entitled, ‘Mathematics: An Introduction to its Spirit and Use’. I have just reinstated the paragraph numbering from the original paper.

Euler’s Paper

1. The branch of geometry that deals with magnitudes has been zealously studied throughout the past, but there is another branch that was then almost unknown up to now; Leibnitz spoke of this first, calling it the “geometry of position” (geometry situs). This branch of geometry deals with relations dependent on position alone, and investigates the properties of position; it does not take magnitude into consideration, nor does it involve calculation with quantities. But as yet no satisfactory definition has been given of the problems that belong to this geometry of position or of the method to be used in solving them. Recently there was announced a problem which, whilst it certainly seemed to belong to geometry, was nevertheless so designed that it did not call for the determination of a magnitude, nor could it be solved by quantitative calculation, consequently I did not hesitate to assign it to the geometry of position, especially since the solution required only the consideration of position, calculation being of no use. In this paper I shall give an account of the method that I discovered for solving this type of problem, which may serve as an example of the geometry of position.

2. The problem, which I understand is quite well known, is stated as follows: In the town of Königsberg in Prussia there is an island A called Kneiphof, with two branches of the river Pregel flowing round it. There are seven bridges a, b, c, d, e, f and g crossing the two branches of the river. The question is whether a person can plan a walk in such a way that he will cross  each of these bridges once but not more than once. I was told that while some denied the possibility of doing this and others were in doubt, no one maintained that it was actually possible. On the basis of the above I formulated the following very general problem for myself: Given any configuration of the river and branches into which it may divide, as well as any number of bridges, to determine whether or not it is possible to cross each bridge exactly once.

3. The particular problem of the seven bridges of Königsberg could be solved by carefully tabulating all possible paths, thereby ascertaining by inspection which of them, if any, met the requirement. This method of solution, however, is too tedious and too difficult because of the large number of possible combinations, and in other problems where many bridges are involved it could not be used at all. Hence I discarded it and searched for another more restricted in its scope; namely, a method which would show only whether a journey satisfying the prescribed condition could in the first instance be discovered;  such an approach, I believed, would be simpler.

KonigsbergBridges

4. My entire method rests on the the appropriate and convenient way in which I denote the crossing of bridges, in that I use capital letters A, B, C, D, to designate the various areas of land that are separated from one another by the river. Thus when a person passes from area A to area B, using either of the two possible bridges a or b, I denote this by the letters AB, the first of which denotes the area whence he came and the second the area where he arrives after crossing the bridge. If the traveller then crosses from B over bridge f into D, this crossing is denoted by the letters BD; the two crossings AB and BD performed in succession I denotes simply by the three letters ABD, since the middle letter B designates the area into which the first crossing leads as well as the area out of which the second leads.

5. Similarly, if the traveller proceeds from D across bridge g into C, I designate the three successive crossings by the four letters ABDC. The crossing of four bridges will be represented by five letters and if the traveller crosses an arbitrary number of bridges his journey will be described by a number of letters that is one greater than the number of bridges. For example, eight letters are needed to denote the crossing of seven bridges.

6. With this method I pay no attention to which bridges are used; if the crossing from one area to another can be made by one of several bridges it makes no difference which is used, so long as it leads to the desired area. Thus if a route could be laid out over the seven Königsberg bridges so that each bridge was only crossed once and only once, it would be possible to describe this route using eight letters, and in this series of letters AB (or BA) would have to occur twice since there are two bridges (a and b) connecting the areas A and B. Similarly the combination AC would occur twice and the combinations AD, BD and CD would each occur once.

7. Our question is now reduced to whether from the four letters A, B, C and D a series of eight letters can be formed in which all the combinations just mentioned occur the required number of times. Before making the effort, however, I think it well to consider whether its existence is even theoretically possible or not. For if it could be shown that such an arrangement is in fact impossible, then the effort expended on trying to find it would be wasted. Therefore I have sought for a rule that would determine without difficulty, as regards this and similar questions, whether the required arrangement of letters is feasible.

EulerOneRegion8. For the purposes of finding such a rule I take a single region A into which an arbitrary number of bridges a, b, c, d, etc. lead. Of these bridges I first consider only a. If the traveller crosses this bridge, he must either first have been in A before crossing or have reached A after crossing, so that according to the above method of denotation the letter A will appear exactly once. If there are three bridges leading to A and the traveller crosses all three, the letter A will occur twice in the expression representing his journey, whether it begins at A or not. And if there are five bridges leading to A, the expression for a route that crosses them all will contain the letter A three times. If the number of bridges is odd, increase it by one, and take half the sum; the quotient (the answer) represents the number of times the letter A appears.

9. Let us now return to the Königsberg problem. Since there are five bridges leading to (and from) island A, the letter A must occur three times in the expression describing the route. The letter B must occur twice, since  three bridges lead to B; similarly D and C must both occur twice. That is to say, the series of letters that represents the crossing of the seven bridges must contain A three times and B, C, and D  each twice. But this is quite impossible with a series of eight letters, for the sum of the required letters is nine ( 3 + 2 + 2 + 2 = 9). Thus it is apparent that a crossing of the seven bridges in the manner required cannot be effected.

10. Using this method we are always able, whenever the number of bridges leading to a particular region is odd, to determine whether it is possible in a journey to cross each bridge exactly once. Such a route exists if the number of bridges plus one is equal to the sum of the numbers which indicate how often each individual letter must occur. On the other hand if this sum is greater than the number of bridges plus one, as it is in our example, then the desired route cannot be constructed. The rule that I give for determining from the number of bridges that lead to A how often the letter A will occur in the route description is independent of  whether these bridges all come from a single region B or from several regions, because I was considering only the region A, and attempting to determine how often the letter A must occur.

11. When the number of bridges leading to A is even, we must take into account whether the route begins in A or not. For example, if there are two bridges that lead to A and the route starts from A, then the letter A will occur twice; once to indicate departure from A by one of the bridges and a second time to indicate the return to A by the other bridge. However, if the traveller starts his journey in another region, the letter A will only occur once, since by my method of description the single occurrence of A indicates an entrance into as well as a departure from A.

12. Suppose, as in our case, there are four bridges leading into the region A, and the route is to begin at A. The letter A will then occur three times in the expression for the whole route, while if the journey had started in another region, A would only occur twice. With six bridges leading to A, the letter A will occur four times if A is the starting point, otherwise only three times. In general, if the number of bridges is even, the number of occurrences of the letter A, when the starting region is not A, will be half the number of bridges; when the route starts from A, one more than half.

13. Every route must, of course, start in some one region. Thus from the number of bridges that lead to each region I determine the number of times that the corresponding letter will occur in the expression describing the whole route as follows: When the number of bridges is odd, I increase it by one and divide by two; when the number is even I simply divide it by two. Then if the sum of the resulting numbers is equal to the number of bridges plus one, the journey can be accomplished, though it must start in a region approached by an odd number of bridges. But if the sum is one less than the number of bridges plus one, the journey is feasible if its starting point is approached from a region with an even number of bridges, for in that case the sum is again increased by one.

14. My procedure for determining whether in any system of rivers and bridges it is possible to cross each bridge exactly once is as follows: First I designate the individual regions separated from one another by water A, B, C, etc. Second, I take the total number of bridges, increases it by one, and write the resulting number at the top of the page. Third, under this number I write the letters A, B, C, etc. in a column, and opposite each letter note the number of bridges that lead to that particular region. Fourth, I place an asterisk again each letter that has an even number next to it. Fifth, in a third column I write opposite each even number half of that number and opposite each odd number half the sum formed by that number plus one. Sixth, I add up the last column of numbers. If the sum is one less or equal to the number at the top, I conclude that the  required journey can be made. But it must be noted that when the sum is one less than the number at the top, the route must start from a region marked with an asterisk, and when the two numbers are equal, it must start from a region that does not have an asterisk.

For the Königsberg problem, I would set up the tabulation as follows:

/Users/grahamshawcross/Documents/blog_drafts/konigsberg-bridges/

The last column now adds up to more than 8, and hence the required journey cannot be made.

15. Let us take an example of two islands with four rivers forming the surrounding water.

Imaginary

Fifteen bridges marked a, b, c, d, etc. cross the water round the two islands and the adjoining rivers. The question is whether a journey can be arranged that will pass over all the bridges , but not over any more than once. I begin by marking the regions that are separated from one another by water with the letters A, B, C, D, E and F, there are six of them. Second, I take the number of bridges (15) add one and write the number (16) uppermost. Third, I write the letters A, B, C, etc in a column and opposite each letter write the number of bridges connecting with that region, e.g. eight bridges for A, four for B, etc. Fourth, the letters that have even numbers against them are marked with an asterisk. Fifth, in a third column I write the half of each corresponding number, or if the number is odd, I add one to it, and put down half the sum. Sixth, I add the numbers in the third column and gets 16 as the sum.

/Users/grahamshawcross/Documents/blog_drafts/konigsberg-bridges/

The sum of the third column is the same as the number 16 that appears above, and it follows that the journey can be effected if it begins in regions D or E, whose symbols have no asterisks. The following expression represents such a route:

EaFbBcFdAeFfCgAhCiDkAmEnApBoElD

Here I have indicated, by small letters between the capitals, which bridges are crossed.

16. By this method we can easily determine, even in cases of considerable complexity, whether a single crossing of each bridge in sequence is possible. But I should now like to give another and much simpler method which follows quite easily from the preceding, after a few preliminary remarks. In the first place I note that the sum of the numbers written down in the second column is necessarily double the actual number of bridges. The reason is that in the tabulation of the bridges leading to the various regions each bridge is counted twice, once for each of the two  regions that it connects.

17. From this observation it follows that the sum of the numbers in the second column must be an even number, since half of it represents the actual number of bridges. Hence, if any of the numbers opposite the letters A, B, C, etc. are odd an even number of them must be odd. In the Königsberg problem for instance, all four of the numbers opposite the letters A, B, C, D, were odd, while in the example just given only two of the numbers were odd, namely those opposite D and E.

18. Since the sum of the numbers opposite A, B, C, etc. is double the number of bridges, it is clear that if this sum is increased by two in the latter example and then divided by two, the result will be the number written at the top. When all the numbers in the second column are even, and half of each is written down in the third column, the total of this column will be one less than the number at the top. In that case it will always be possible to cross all the bridges. For in whatever region the journey begins, there will be an even number of bridges leading to it, which is the requirement.

19. Further, when only two of the numbers opposite the letters are odd, and the others even, the required route is possible provided it begins in a region approached by an odd number of bridges. We take half of each even number, and likewise half of each odd number after adding one, as our procedure requires: the sum of these halves will then be one greater than the number of bridges, and hence equal to the number written at the top. But, when more than two of the numbers in the second column are odd, it is evident that the sum of the numbers in the third column will be greater than the top number and hence the desired journey is impossible.

20. Thus for any configuration that may arise the easiest way of determining whether a single crossing of all the bridges is possible is to apply the following rules:

If there are more than two regions which are approached by an odd number of bridges, no route satisfying the required conditions can be found.

If, however, there are only two regions with an odd number of approach bridges the required journey can be completed provided it originates in one of these regions.

If, finally, there is no region with an odd number of approach bridges, the required journey can be effected, no matter where it begins.

These rules solve completely the problem initially proposed.

21. After having determined that a route actually exists we are left with the question of how to find it. To this end the following rule will serve: Wherever possible we mentally eliminate any two bridges that connect the same two regions; this usually reduces the number of bridges considerably. Then, and this should not be too difficult, we proceed to trace the required route across the remaining bridges. The pattern of this route, once found, will not be substantially affected by the restoration of the bridges which were first eliminated from consideration, as a little thought will show. Therefore I do not think  I need say no more about finding the routes themselves.

Commentary

After overcoming his initial reluctance to solve such a banal problem, in Section 1, Euler relates the problem to an emerging area of mathematical enquiry, the geometry of position. In Section 2, he states the problem clearly and labels the island Kneiphof A and the bridges  a, b, c, d, e, f and g. In Section 3, although Euler recognises that the problem could be solved by enumerating all routes, he rejects this method as being too tedious, too difficult and not capable of being extended.

In Sections 4, 5 and 6, Euler establishes his method of notation, labelling the separate  land areas A, B, C,  and D, and paths or journeys, from one area to another, as strings of these letters ABDC etc. paying no attention to which bridge might be used to get from one area to another. He then notes that with his notation, to represent the crossing of the seven bridges needs a string of eight letters, and that the combination AB or (BA) would need to appear twice as would the combination AC (each with two bridges), whilst the combinations AD, BD and CD would each appear once (each with only one bridge).

In Section 7, Euler notes that the problem is now reduced to whether from the four letters, A, B, C, and D a series of eight letters can be formed in which all the combinations just mentioned can occur. But before wasting effort on looking for possible arrangement he wishes to see if he can find a simple rule that says whether any arrangement is possible. In Section 8, he imagines a single area A into which an arbitrary number of bridges a, b, c, d, etc. lead. He first considers only bridge a, and notes that if the traveller crosses this bridge he must either have come from A or arrived in A and therefore the letter A will appear exactly once. If there are three bridges leading to A the letter A will appear twice and if there are five bridges it will appear three times.

So the problem as stated requires that there are only eight letters in the string representing the required route, but also requires that A appears three times and B, C, and D each appear twice, a total of nine times ( 3 + (3 x 2) = 9). The requirements are contradictory and therefore no route is possible.

Having solved the presented problem, Euler then proceeds to generalise his method and provides a step by step algorithm for solving problems with any number of islands and any number of bridges. But Euler is still not satisfied and proceeds to produce a yet simpler set of three rules that determine in any situation, if all bridges can be crossed once without any being crossed twice.

Planar Graphs

Whilst Euler’s paper is widely recognised as the origin of Graph Theory, it is interesting to note that it does not actually contain anything that looks like a graph, its major contribution being the notation adopted. By long custom, in Euclidean Geometry capital letters, A, B, C etc. denote points, and lowercase letters, a, b, c etc. edges. Euler uses capital letters to represent areas and lowercase letters to represent bridges (the connections between areas). This facilitates the idea that areas can be represented by points and connections by edges. So that the Königsberg Bridges problem can be represented graphically as follows:-

/Users/grahamshawcross/Documents/blog_drafts/konigsberg-bridges/

With graphs like this, one can easily count the number of nodes that have an odd number of edges entering them, marked red above. If there are more than 2 nodes with an odd number of edges then no route is possible, but if there are just 2 nodes with an odd number of edges then it is possible to find a route, provided that you start from one of the odd nodes.

/Users/grahamshawcross/Documents/blog_drafts/konigsberg-bridges/

As noted by Euler, where there are two (or any even number of) bridges between two regions these can be ignored because the traveller can use one bridge to go from say A to C and the other one to come back from C to A, completing a cycle, eliminating both bridges and leaving him where he started from, and able to proceed to another bridge.

In recognition of Euler’s contribution to Graph Theory, routes that visit each node once but none more than once are called Euler Cycles. Euler’s solution also applies to another banal problem, that I learnt at school, drawing figures without removing your pencil from the page.

/Users/grahamshawcross/Documents/blog_drafts/konigsberg-bridges/

A coloured map can be represented as a planar graph if the coloured regions of the map are replaced by same coloured vertices in the graph, and vertices are joined by an edge in the graph if their corresponding regions in the map share an edge.

/Users/grahamshawcross/Documents/blog_drafts/bill_tutte/FourColo

In fact the planar graph is a dual of the map, and the map can be generated from the graph. Both map and graph obey, another contribution of Euler to Graph Theory, his formula for networks; regions + vertices – edges = 1. In the examples above 7 regions + 11 vertices – 17 edges = 1 for the map and 10 regions + 8 vertices – 17 edges = 1 for the graph.

The planar graph is therefore an exact equivalent of the map and the Four Colour Conjecture can be expressed as seeing if the vertices of the graph can be coloured with 4 colours so that no 2 adjacent vertices have the same colour.

More importantly planar graphs lend themselves to computer manipulation, and this in turn has lead to a, somewhat disputed, computer enabled proof of the four colour theorem. See here for more detail.

Directed Graphs

With planar graphs an edge (or arc) between two nodes A and B, is bi-directional and acts like a bridge that allows movement from B to A as well as from A to B, and can therefore only represent a symmetrical  relation between A and B, like adjacency. On the other hand, in directed graphs, edges are defined as having a direction, so that an edge from A to B acts like a turnstile or valve allowing movement from A to B but preventing movement from B to A. This allows values to be added to each edge, so that flows, electrical currents, dimensions etc. can be represented and networks modelled.

 

HouseGraph2

See here for details of Kirchhoff’s Laws for electrical flow in wires being applied to the automatic generation of house plans. Directed graphs also form the basis of Precedence Graphs, Critical Path Analysis  and modern Graphical Programming Systems like Grasshopper where inputs are on the left hand edge of each block, outputs on the right and scripted procedures can be added to the block itself.

 

Grasshopper_MainWindow (1)

Bibliography

Appel, K., Haken, W. and Koch, J.  (1977) Every Planar Map is Four Colorable. I: Discharging. Illinois J. Math. 21, 429-490

Appel, K. and Haken, W. (1977 ) Every Planar Map is Four-Colorable, II: Reducibility. Illinois J. Math. 21, 491-567.

Euler, L. (1736) Solutio problematis ad geometriam situs pertinentis. Comment. Acad. Sci. U. Petrop. 8, 128-140. Reprinted in Opera Omnia Series Prima, Vol. 7. pp. 1-10, 1766.

Hopkins B. and Wilson R. (2004) The Truth about Königsberg. College Mathematics Journal , 35, 198-207

March, L. and Steadman, P. (1971) The Geometry of Environment. RIBA Publications Ltd.

Newman J. R. (1953) The Königsberg Bridges. In Mathematics: An Introduction to its Spirit and Use, Scientific American. June 1978 22 121-124.

Steadman, P. (1970) The Automatic Generation of Minimum Standard House Plans Working Paper 23 University of Cambridge Land Use and Built Form Studies

Tutte, W. T. (1958) Squaring the Square from ‘Mathematical Games’ column, Scientific American Nov 1958.

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Weber’s Law

Posted on November 18, 2015 by Graham Shawcross

Weber’s Law expresses a general relationship between an initial stimulus, a quantity or intensity, and the increased stimulus required for a change in the stimulus to be detected.

The task is to tell apart, or discriminate, two things that differ by only a slight amount.

Weber’s original 1834 observation was that if you are judging if two objects differ in weight, then two heavy objects must differ by a greater amount than two lighter ones if a difference in weight is to be detected.  That is heavier weights are harder to discriminate than lighter ones.

The ability to easily discriminate two numbers increases with the numerical distance between them. We react more quickly and make fewer mistakes the greater the distance is between the numbers being compared. This is the distance effect.

Distance

So it is easier to compare 2 and 8 (on the left) than it is to compare 8 and 9 (on the right), with separation distances of 6 (8 – 2) and 1 (9 – 8) respectively.

If the distance between the two numbers remains the same then it is easier to compare two small numbers than it is to compare two larger numbers. This is the size effect.

SizeEffect

So it is easier to compare 3 and 4 (on the left) than it is to compare 8 and 9 (on the right), both comparisons having a separation distance of 1 (4 – 3 and 9 – 8 respectively).

The Just Noticeable Difference

The just noticeable difference, or difference threshold, is the least amount that a stimulus needs to be changed by, in order to produce a noticeable variation in the sensory experience that the stimulus is producing.

Weber’s Law states that the just noticeable difference is a constant fraction of the stimulus intensity.

                                  ΔI = Kw×I

Where ΔI is the just noticeable intensity difference, Kw is a constant factor (the Weber factor) and I is the stimulus intensity.

Example 1: 3-Day-Old Chick Experiments details here

Compound

In the first experiment 3-day-old chicks were trained on the number 5 and then exposed to the numbers 2 and 8, that is to numbers at equal numerical distances from the training value 5 (5 – 2 = 3 and 8 – 5 = 3). Similarly in the second experiment the training value was 20 and the chicks were exposed  to the numbers 8 and 32, equally distant from the trained value 20 (20 – 8 = 12 and 32 – 20 = 12).

In both cases avoiding results being contaminated by the distance effect.

Example 2 Jevons’es Data details here

In 1871 the early economist and logician William Stanley Jevons published an article in Nature “The Power of Numerical Discrimination” (Jevons 1871)

Jevons’es experimental procedure was to casually throw black beans at a 4½” diameter white target and then:-

At the very moment when the beans came to rest, their number was estimated without the least hesitation, and then recorded together with the real number obtained by deliberate counting”

/Users/grahamshawcross/Documents/blog_drafts/jevons/jevons.dwg

Jevons concluded that for him the absolute limit of discrimination was 4, but recognised that the limit probably varied for other people and might perhaps be taken as 4½ if that made any sense.

His estimate corresponds fairly accurately to modern estimates of the subitising range for randomly located dots. See Subitising.

Jevon’s data corresponds to Weber’s Law with accuracy getting worse as set size increases.

ΔN = (N – 4½) ⁄ 9

Where ΔN is the numerical error, 4½ is the limit of discrimination and the offset on the set size axis and the Weber Fraction is 1 ⁄ 9.

Example 3:Bayesian Analysis  details here

When baboons watch peanuts being counted into an opaque cylinder and then some more into a second cylinder, they move towards the second cylinder when it begins to have more peanuts in it than the first cylinder.

The Bayesian analysis of this switching behaviour proposes a parameterised model and uses data to infer a probability distribution for the value of each parameter.

WeberFactors

The analysis recovers Weber fractions from the switch trials that are similar to the Weber fraction obtained with simple fits across all the trials, although the wide variability of these values is consistent with non-exact representations of the quantities in the first and second cylinders.

Fechner’s Law or Scale

In 1860 Fechner  proposed an extension to Weber’s Law. This states that as the stimulus intensity increases, it takes greater and greater changes in intensity to change the perceived magnitude by some constant amount. 

S = k  log(I)

Where S is the perceived magnitude, k is a modality and dimension constant and I is the stimulus intensity.

Applicability

Weber’s Law is applicable to a variety of sensory modalities (brightness, loudness, mass, line length, etc.).  The value of the Weber fraction varies across these modalities but tends to be relatively constant within any particular modality.

It tends to be inaccurate for extremely small or extremely large stimuli values.

The concept of a just noticeable difference (JND) has commercial applications. For instance to ensure that negative changes (reduction in product size or increases in price) remain below the JND and are not apparent to customers. Or so that product improvements (better packaging, larger size or lower price) are apparent to customers but are not too wastefully expensive to produce. That is that they are at or just above the JND.

Bibliography

Cantlon, J. F., Piantadosi, S. T.,Ferrigno, S., Hughes, K. D.,Barnard, A. M. (2015) The Origins of Counting Algorithms. Psychological Science 1-13

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013).Bayesian data analysis (3rd ed.). Boca Raton, FL: CRC Press.

Jevons, W.S., (1871) The Power of Numerical Discrimination Nature, Thursday February 9, 1871.

Regani R., Vallortigara G., Priftis K., & Regolin L. (2015) Number-space mapping in the newborn chick resembles humans’ mental number line Science. DOI:10.1126/science.aaa1379

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