William Tutte’s Hidden Past

If William Tutte is remembered at all by architects, it is for his contribution to solving the problem of Squaring the Square . (Tutte 1958) A solution using Graph Theory and Kirchhoff’s Laws for electrical flow in wires that was subsequently used in Philip Steadman’s The Automatic Generation of Minimum Standard House Plans. (Steadman 1970)

112x112squaredsquare

This last, ultimately failed enterprise is explained in some detail here.

Bill Tutte died in 2002 and his obituaries, like this from the Guardian, reveal that he also had a wider influence on world events.

Continue reading

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Missing Pullover Found

My long lost 1970s pullover, slip-over, or perhaps more properly 70s tank-top, has turned up at the back of a cupboard. Last year we had turned the house upside down looking for it.

P1020204 copy

The Missing Pullover

The design is apparently based on the colour theory of Interaction of Colour, (Albers 1963)  and illustrates the first of  The Twelve Fold Ways from Stanley’s Enumerative Combinatorics, (Stanley 1986 and 1997). The first way being n-tuples of x things with enumeration formula x to the power n.

Here there are 2 objects, an inside and an outside, and 4 colours giving 24 = 16 different combinations. Ignoring the 4 same-on-same combinations gives the 12 unique combinations numbered below.

pullover_one

Using a spreadsheet type program each of these 12 combinations is then associated with a random number function.

pullover_three

The combinations are then sorted on their associated random numbers and this is repeated as often as necessary with newly generated random numbers.

pullover_two

Repeatedly applied selections

This gives an even mixture because every combination is used before it is used again. This ensures that there are equal numbers of each combination, and therefore that equal numbers of balls of wool are required.

pullover_four

Knitted Sample

An equivalent procedure would be to put, say cardboard samples, representing each of the 12 combinations in a bag and drawing them out blindfold one-by-one until none are left, then putting all the cardboard samples back in the bag and repeating the procedure.

An even mixture would not be guaranteed if each sample was drawn out blindfold and then immediately put back in the bag before making another selection, such a method would just statistically tend towards an even distribution.

Architectural applications of this techniques to follow.

Unfortunately, as perhaps the observant will have noticed, the long lost pullover was not made in accordance with the knitted sample or the procedure above, but appears to just randomly list all the 2 colour combinations of 3 colours. I think then that it had better go back in the cupboard.

Bibliography

Albers, J., 1963. Interaction of Colour, Yale University Press.

Knuth, D.E., 2005. The Art of Computer Programming, Volume 4 Fascicle 2, Generating All Tuples and Permutations .Addison-Wesley

Stanley, R.P., 1986. Enumerative Combinatorics (Volume 1),Wadsworth & Brook.

Stanley, R.P., 1997. Enumerative Combinatorics (Volume 2), Cambridge University Press.

 

Posted in Aesthetics, Architecture, Design, Enumeration, Knitting | Tagged , , , , , , | 1 Comment

Spatial Representation of Number

Francis Galton

“…this peculiarity is found so far as my observations have extended, in about 1 out of every 30 adult males or 15 females. It consists in the sudden and automatic appearance of a vivid and invariable “Form” in the mental field of view, whenever a numeral is thought of, and in which each numeral has its own definitive place. This Form may consist of a mere line of any shape, of a peculiarly arranged row or rows of figures, or of a shaded space.” (Galton 1880)

VisualisedNumerals1

VisualisedNumerals2The term numeral is now more usually used to refer to the digits of a say  base 10 number system, with numerals 0 1 2 3 4 5 6 7 8 9. In modern parlance the visualisations above are therefore of numbers rather than numerals. They also seem to be surprisingly biased towards representing the verbal -ty number words, twenty, thirty etc. (B, S1, H also -teen word numbers, thirteen, fourteen etc. R, EN, NL, RN and ED) see earlier post about verbal number words.

Colour and brightness play a part in some representations (C, GE, GS, EN, MT, TEW) and lines shown in the diagrams sometimes do not appear in the actual mental representation (WHP, MT). The number 12 also seems to be well represented (B, WS, C, GS,MT, GH, THW, PGE, LMH, NL, RN,CH) probably because of its importance in clock counting. There is also a left-to-right and / or bottom-to-top preference in representing increasingly large numbers, but this by not universal.

In summary this effect is automatic and involuntary, located in a fixed internal visual space, is idiosyncratic, emerges in childhood, is stable over time but is far from universal. Because of this the effect is usually thought to be a specific form of synesthesia. (Hubbard et al 2005)

SNARC Effect

The Spatio-Numerical Association of Response Codes (or SNARC) effect refers to the fact that when subjects, doing some task, are presented with Arabic numbers in the range 1-9, they respond more quickly to small numbers (1-4) with the left hand and more quickly to large numbers (6-9) with the right hand. (Dehaene et al 1993)

According to Dehaene and his associates, the effect works for single (experiment 1) or two digit numbers (experiment 2) and is not affected by the subjects being left or right handed (experiment 5) or crossing hands (experiment 6) but is reduced for subjects with a right-to-left reading habit (experiment 7). It is absent for letters (experiment 4) but present for number words (experiments 8 and 9).

The effect seems to be a relative one because when the presented number range is limited to the numbers 1 to 5, people respond more quickly with the right hand to the numbers 4 and 5, the  right-hand end of the putative number line. But when the range presented is 4 to 9 people respond more quickly with the left hand to the same numbers 4 and 5, that have now become the left-hand end of the number line.

From these experiments Dehaene and his collaborators suggested that:-

“…the representation of number magnitude is automatically accessed during parity judgment of Arabic digits. This representation may be linked to a mental number line […], because it bears a natural and seemingly irrepressible correspondence with the natural left – right coordinates of external space.”

As with subitising the response is automatic but learnt, see Subitising.

But unlike subitising which is identifiable in children as young as 3 months old, the SNARC effect only appears in children’s responses when they are at least 7 years or by some accounts 9 years old.

The task in Dehaene’s two digit experiment (experiment 2) was a parity (odd or even) judgement and it has been pointed out that the parity of 2 digit numbers can be assessed entirely from the last digit, so there is some doubt that the effect extends to two digit numbers.

The effect is also observed with eye movement, responses by looking to the left being faster when recognising small numbers and faster to the right with large numbers.

A similar effect occurs when people are asked to generate numerals randomly. If their head is turned to the left they tend to generate more lower numbers and more higher numbers when their head is turned to the right.

It has also been observed that when the distance between thumb and forefinger is varied; the greater the distance being held the more likely a random numeral is to be larger than expected. (Andres et al. 2004)

Priming

The SNARC effect requires no conscious effort and in fact takes place entirely unconsciously a fact that is emphasised by the fact that a priming effect has also been observed. So if a numeral is displayed that is unrelated to the task being undertaken, then the numeral displayed can affect whether the right or left hand is used preferentially.

Most authors assume that the parity (odd or even) judgement, used in most of the reported experiments, is just such a task; but the subjects are making a judgment about the displayed number and therefore paying attention to it. I think to demonstrate a true priming effect the task being carried out should not draw attention to the displayed numeral or use it in any way.

Reading Direction

The effect also seems to be related to the cultural background of subjects, particularly their habitual reading direction. The graph below shows the difference between right-hand and left-hand reaction times for a group of Canadian students with left-to-right reading habits and displays a normal SNARC effect.

canadians

After Shaki S., Fischer M., & Petrusic W. (2009)

However, monolingual Arabic speaking Palestinians who are therefore right-to-left readers exhibit an inverse SNARC effect.

palestinians

After Shaki S., Fischer M., & Petrusic W. (2009)

And the effect almost disappears with monolingual Hebrew speaking Israelis, who read text from right-to-left but read Arabic numbers from left-to-right, .

israelis

After Shaki S., Fischer M., & Petrusic W. (2009)

This shows that the SNARC effect is susceptible to habitual reading direction, but there is also evidence that it is an effect that is fairly easy to un-train. For instance if subjects are asked to think about numbers as if they are on a clock face, where the numbers increase in an approximate right-to-left direction, then the effect is not observed. (Bächtold et al 1998)

Incidentally, the graphs above all display a particular cultural preference, which might be relevant to the SNARC effect itself. The abscissa (x axis) is shown with values that increase from left to right. And the secondary ordinates (y axis) show values that increase from bottom to top, exhibiting the lower-to-higher small-to-large number metaphor.

Finger Counting

Apparently in all human cultures children use finger counting before being taught arithmetic. (Butterworth 1999)

FingerCounting

In a study of Scottish students, 83% right-handed, 10% left-handed and 7% ambidextrous, 66% started finger counting on their left hand (left starters) and 34% on their right hand (right starters), a significant difference. Of the left starters 92% started counting on the thumb, 3% on the index finger and 4% on the pinkie or little finger. With the right starters 80% started counting on the thumb. So in total 61% (92% of 66%) are left starters who start to count with the left hand thumb and exhibit the counting pattern illustrated above, that is a pattern that corresponds to a left-to-right number line representation. (Fischer 2008) Fischer himself suggests that there are studies that disagree with these findings. (Satoa and Lalaina 2008)

I think there may be some problems with the survey method used to collect this data; in particular the fact that the questionnaire included the diagram above (without the added numerals) showing the hands palm-up, perhaps unconsciously suggesting to respondents a left-right number line.

Statistically right starters have a small and significantly weaker SNARC effect than left starters who have a normal SNARC effect. This at least indicates that that finger counting habits  exert an influence on numerical cognition. See also Five Finger Exercises

 Sex Difference

It is reported that the SNARC effect is weaker in females than in males as shown in the graph below. (Bull, Cleland and Mitchell 2013) It is hypothesised that this may be due to the fact that the inferior parietal lobe, identified by fMRI studies as being involved in numerical cognition, is 25% bigger in men than in women, or in jest that men have a greater arm span than women.

SexDifferences

from Bull, Cleland and Mitchell 2013

Button Labels

Interesting things start to happen if labels “left” and “right” are put on the left-hand and right-hand buttons (congruently) or in Stroop like fashion put on the wrong buttons (incongruently). The arrangements are shown in the physical account on the right hand side of the diagram below, where the expected preference for the centrally displayed target number is indicated by the bold buttons. (van Dijck & Fias 2011)

/Users/grahamshawcross/Documents/blog_drafts/SNARC-effect/Incong

After van Dijck & Fias (2011)

With a size comparison task and instructions if  the target number is less than 5 press the  button labelled “left” and if the target number is greater than 5 press the button labelled “right” or vice-versa, the following results are obtained.

/Users/grahamshawcross/Documents/blog_drafts/SNARC-effect/Incong

After van Dijck & Fias (2011)

The authors believe that subjects are responding to the conceptual account on the right in the table above, where the incongruent preferences are reversed. They suggest that the SNARC effect derives from congruency between conceptual categories and not from congruency between a position on a mental line and left to right responses. They also suggest that the conceptual account listed above, maps to a more general set of left-to-right categories such as:-

/Users/grahamshawcross/Documents/blog_drafts/SNARC-effect/Incong

After Proctor & Cho 2006

This has lead some observers to believe that the SNARC effect is as much a verbal phenomena as a numerical one.

Discussion

All authors seem to agree that numbers are at least to some extent represented spatially and  that the SNARC effect is in some way a real effect.

Sequence-space synaesthetes, like Galton’s subjects, do not appear to have unusually strong SNARC responses, probably indicating that they are using a separate mental mechanism.

The SNARC effect is essentially statistical, that is not experienced by everyone or experienced in varying degrees by others. For instance cultural differences, the weaker effect in the 34% of right starting finger counters and the weaker effect in women.

There is an emerging tendency to see the SNARC effect as a verbal rather than a specifically numerical effect.

The embodied cognition paradigm suggests that all of our knowledge is represented together with the sensory and motor activity that was present during its acquisition. So it should be expected that an abstract ability such as numerical cognition inherits the functional properties of more basic perceptual and motor process such as subitisation,  finger counting, number lines and other relational metaphors. (Lakoff & Núñez 2000)

Speculation

It would be good to see if the SNARC effect could be used to prime some desired action such as a preferential turn to the left if a small number is displayed in the field of vision or to the right if a large number is displayed. But first it would be necessary to establish that a true priming effect can be established. That is to have the display of small or large numbers influence peoples’ actions without attention having been drawn to the displayed number.

For instance the famous effect of having a poster with eyes rather than flowers over a British university staff common room honesty box increasing the amount of money being put into the box. (Kaheman 2011) (Bateman, Nettles and Roberts 2006)

eyes_chart_1

If such a priming effect could be demonstrated for numbers, then it would be fairly simple to build a maze like computer game which users could walk through and only have to decide to turn left or right at each T junction where a randomly selected number was displayed. A game that could obviously be instrumented to automatically collect data on the effect.

Finally it would be interesting to see if the SNARC effect could be exported into the real world, for instance 7 George Square, which from this position has symmetrical left and right stairways leading into the building.

PyscholgyDepartment

Bibliography

Andres, M. , Davare, M. , Pesenti, M. , Olivier, E. & Seron, X. (2004) Number magnitude and grip aperture interaction. Neuroreport 15, 2773–2777 .

Bächtold D, Baumüller M, Brugger P. (1998)  Stimulus-response compatibility in representational space Neuropsychologia 36(8):731-5

Bateson M., Nettle, D., Roberts, G. (2006) Cues of Being Watched Enhance Cooperation in a Real World Setting Biology Letters 2 412‐14

Bull, R., Cleland, A.A., & Mitchell, T. (2013) Sex Differences in the Spatial Representation of Number Journal of Experimental Psychology: General 142(1) 181-192

Butterworth B. (1999) The Mathematical Brain Macmillan, London

Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and number magnitude. Journal of Experimental Psychology: General 122: 371-396.

van Dijck J. P. & Fias W. (2011) A working memory account for spatial-numerical associations Cognition 119(1) 114-119

Fischer M. H. (2008) Finger counting habits modulate spatial-numerical associations cortex 44 386-392

Fischer M. H. (2011) The spatial mapping of numbers – its origin and flexibility in Language and Action in Cognitive Neurosciences, eds Coello Y., Bartolo A., editors. London, Psychology Press

Fischer M. H., Castel A. D., Dodd M. D., & Pratt J. (2003). Perceiving numbers causes spatial shifts of attention. Nature Neuroscience, 6, 555–556.

Galton F. (1880) Visualised Numerals Nature Vol. 21 252-6, 494-5

Galton, F. (1881) The Visions of Sane Persons Popular Science Monthly Volume 19 August 1881

Hubbard E. M, Manuela Piazza, M., Pinel, P. & Dehaene, S. (2005) Interactions between number and space in parietal cortex Nature Reviews Neuroscience 6, 435-448 (June 2005)

Kahneman D. (2011) Thinking Fast, Thinking Slow. Allen Lane. p 57

Lakoff G. & Núñez R. E. (2000) Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being Basic Books

Proctor R. W. & Cho Y. S. (2006) Polarity correspondence: A general principle for performance at speeded binary classification tasks. Psychological Bulletin 132(3) 416-442

Satoa M. and Lalaina M. (2008) On the relationship between handedness and hand-digit mapping in finger counting cortex 44 393-399

Shaki S., Fischer M., & Petrusic W. (2009). Reading habits for both words and numbers contribute to the SNARC effect Psychonomic Bulletin & Review, 16 (2), 328-331

Wood G., Nuerk H. C., Willmes K., Fischer M. H. (2008). On the cognitive link between space and number: a meta-analysis of the SNARC effect. Psychol. Sci. Q. 50, 489–525.

Zebian S. (2005) Linkages between number concepts, spatial thinking and directionality of writing: The SNARC effect and the reverse SNARC effect in English and Arabic monliterates, biliterates and illiterate Arabic speakers. Journal of Cognition and Culture 8(1–2) 165-190

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Five Finger Exercises

Written and spoken numbers are represented differently. In English numbers are usually written with Arabic numerals or as a transliteration of the spoken version, for example 342 or three hundred and forty two.

Rod counting provides a written representation of number and a mechanical calculating process that like the abacus takes advantage of the subitising effect, the ability to just glance at a small group of objects and without effort be immediately aware of how many objects are in the group. Continue reading

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Chocolate Fireguards

A very short post on Chocolate Fireguards, which as the name suggests are objects which subvert their own function.

The first example is a real fireguard, though not one actually made of chocolate. It is an example of an object part of which unitentionally subverts its own function.

Graham Shawcross: Scoughall Fireguard 2010

Graham Shawcross: Scoughall Fireguard 2010

Continue reading

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Subitising

The school report of Emily, our 4 year old grandchild, said that she could subitise up to the number 6, and I had no idea what this meant.

Subitising is a technical term that comes from the Latin root subito meaning suddenly or immediately. It describes the ability to just glance at a small group of objects and without effort be immediately aware of how many objects are in the group. For instance you can look at a dice and realise that three is showing without having to count the number of dots.

Dice-Three

The strong Gestalt of the arrangement of the pips (spots) on dice almost certainly aids subitisation due to the high degree of symmetry and the gridded nature of their location.

/Users/grahamshawcross/Documents/blog_drafts/subitising/Subitisi

The task gets a little more difficult with randomly located spots. These were located with a Dart Throwing Poisson Disk distribution to avoid overlapping pips (see Find Your Own Space for details http://grahamshawcross.com/2012/10/12/find-your-own-space/)

/Users/grahamshawcross/Documents/blog_drafts/subitising/Subitisi

And perhaps even more difficult if the pips are arbitrarily colour coded.

/Users/grahamshawcross/Documents/blog_drafts/subitising/Subitisi

Or allowed to overlap, with their location just randomly selected with the proviso that each pip must be completely within the bounding square; pips that would overlap the boundary just being rejected. This ignores the possible pathological cases were randomly selected locations are more or less coincident.

/Users/grahamshawcross/Documents/blog_drafts/subitising/Subitisi

It should not be surprising that this arrangement looks rough and clustered (see The Aesthetics of Aperiodic Tiling http://grahamshawcross.com/2012/11/02/aesthetics-of-aperiodic-tilings/ )

Subitising ability is usually measured by recording how quickly the number of objects in a group is recognised. My assumption with the examples above is that with each example it progressively takes longer to identify the numbers represented, but that it is still possible to do so.

The objects to be counted can obviously be anything at all, all be different, and not have the strong Gestalt form of primary coloured discs, as used above. These strong forms probably aid feature recognition as thought of in Attention Theory. On the other hand the effect of a regular canonical arrangement of  pips is summarised in the graph below. (Piazza et al. 2002)

/Users/grahamshawcross/Documents/blog_drafts/subitising/Subitisi

As perhaps might be obvious, with 1-4 spots there is little difference between random and canonically arrangements but with 6-9 spots there is a marked difference, showing that the canonical arrangement aids subitisation.

Subitising Range

Both reaction time and accuracy are subject to a subitisation effect. The graph below summarises the effect on accuracy were the number of spots presented ranged from 1 to 200, and in this case where subjects were prompted to carry out the task as rapidly as possible. When subjects were asked to concentrate on accuracy the results were very similar. (Kaufmann et al. 1949)

Screen Shot 2014-01-07 at 21.29.04

Accuracy is high, and reaction times low, when the number of dots range from 1 to 5, 6 or 7, the subitising range. Accuracy falls off rapidly above 10 spots with larger numbers startlingly underestimated. Within the subitising range response times increase by approximately 40-100ms per extra item whilst outside the range they increase by 250-350ms per item. These rates are somewhat higher for children but similarly separated.

The subitising range also increases with age, 3 week old children can subitise 1-3 objects and 7 year old children 1-7. (Dehaene & Cohen 1994) Subitising is thus a learnt automatic response. That is a skill that requires no conscious effort but needs to be learnt.

Neurophysiological Basis

There seems to be evidence that subitisation is a separate but overlapping neurophysiological feature of enumeration. Part of the evidence for this is that people with Ballint’s syndrome, who are unable to perceive visual scenes properly, or position objects in space, cannot accurately enumerate objects outside the subitising range but within the range can subitise normally. The disorder is associated with an area of the brain responsible for spacial shifts of attention, something that is thought necessary for counting. Some research has questioned this, suggesting that attention is also required for subitisation. (Piazza et al. 2002)

There is evidence that new-born children have an innate subitising ability and that this is shared with other genera such as fish, indicating that subitisation has a deep primitive evolutionary basis.

Dominoes and Cards

With the addition of a blank half tile the same regular pip arrangement familiar from dice is also used in the design of dominoes.

/Users/grahamshawcross/Documents/blog_drafts/subitising/Subitisi

A slightly different arrangement is used with playing cards where a different and pictorial mnemonic strategy is used for what might be considered the numbers above 10, the Jack, Queen and King. The 7, 8, 9 and 10 are also more difficult to subtise than the lower numbers and the numerical cues in the corner of the cards are probably made more use of with these cards.

/Users/grahamshawcross/Documents/blog_drafts/subitising/Subitisi

The Abacus

The ability to subitise up to 5, plays an important part in the design (or evolution) of the abacus. In the Chinese and Japanese abacus the singles are counted up to 5 and recorded as sets of five as illustrated in the diagram below which also shows the carry mechanism.

/Users/grahamshawcross/Documents/blog_drafts/subitising/Subitisi

The carrying mechanism is illustrated by adding 2 to the displayed number 715408.

Step 1 is to move 2 beads up the singles section (row) of the units wire (column). This leaves 5 beads recording the number 5.

The number 5 can be represented by 5 beads at the top of a singles section or 1 bead at the bottom of the corresponding fives section.

But the bead in the fives section of the units column has already been used to represent the number 8, as 3 singles plus 1 five.

Step 2 is to move all 5 beads down to the bottom of the singles section of the units column.

Step 3 compensates for this by adding 5 to the units fives row. This is done by moving the fives bead to the top of the column preparatory to recording the carry.

Step 4 records the carry by moving 1 bead up the singles section of the tens column.

This is a very visual way of doing maths that relies heavily on being able to to effortlessly recognise up to five objects.

Cognitive Evaluation

Those of us getting on in years are more likely to encounter subitisation as part of an assessment of cognitive ability as recommended in SIGN 86. Management of Dementia. A National Clinical Guideline. (SIGN 2006) which recommends the use of the Addenbrook’e Cognitive Examination (ACE-R) part of which is illustrated bellow. (Mioshi, Eneida, et al. 2006)

AddenbrookesCognitiveTest

Given the previous discussion these high numbers, displayed without any canonical patterning, represent quite difficult tests of cognitive functioning. The distributions also look suspiciously un-random with very little clustering as should be expected. I have noticed that in psychological experiments very little attention is given to the design of random arrangements.

Discussion

It is interesting that the primitive ability to subitise seems to form the basis of the design, or design evolution, of cultural objects such as dice, dominoes, playing cards and abacuses. Reinforced in the case of the games by strong Gestalt type patterning.

In the design of playing cards a completely different pictorial mnemonic strategy is used for the face cards and an auxiliary aid added for all cards but of most use for numbers usually considered to be outside the subitising range. The design thus subtly takes advantage of subitising whilst also using other strategies and auxiliary cues where subitising would be unlikely to work.

The design and efficiency of the Chinese and Japanese abacus, relies directly on the  ability of an individual to visualise one to five objects without conscious effort.

Neurophysologically subitising seems to relate to enumeration in a way that is similar to the relationship between face recognition and visual perception. That is subitisation and face recognition are both separately identifiable, specific skills that fit seamlessly into a more generalised skill.

A recognition of subitising in architectural design could prove useful in making groups of architectural features and units more easily read and understood by utilising an automatic, unconsious  visual response in the observer, with playing card design providing a useful exemplar.

In aesthetics subitisation helps to identify Gestalt groupings particularly those relating to proximity, form or colour.

See http://grahamshawcross.com/2012/01/25/aesthetics/ for what amounts to a discussion of such Gestalt based aesthetics related to housing design.

Bibliography

Dehaene, S., & Cohen, L. (1994). “Dissociable mechanisms of subitizing and counting: neuropsychological evidence from simultanagnosic patients”. Journal of Experimental Psychology: Human Perception and Performance 20 (5): 958–975.

Kaufmann, E. L., Lord, M. W., Reese, T. W. & Volkmann, J. (1949) The Discrimination of Visual Number The American Journal of Psychology Vol. 62, No. 4. pp 498-525

Mioshi, Eneida, et al. 2006 “The Addenbrooke’s Cognitive Examination Revised (ACE‐R): a brief cognitive test battery for dementia screening.” International journal of geriatric psychiatry 21.11 (2006): 1078-1085.

Piazza, M., Mechelli, A., Butterworth, B., Price, C. 2002. “Are Subitizing and Counting Implemented as Separate or Functionally Overlapping Processes” NeuroImage 15 435-446

SIGN 86, 2006, Management of Patients with Dementia: A National Clinical Guideline Scottish Intercollegiate Guidelines Network. NHS Quality Improvement Scotland

 
 
Posted in Aesthetics, Architecture, Brain Physiology, Design, Design Methods, Enumeration, Geometry, Randomness | Tagged , , , , , | Leave a comment

Severely Constrained Design

When a design problem is severely constrained it becomes possible to generate all solutions to the problem. That is, it is possible to close out the problem.

In the late 1960s and early 1970s the design of British Local Authority 2 and 3 story house types became so severely constrained that it became feasible to think about generating all possible house type designs.

Three different groups became involved with this endeavour: the National Building Agency (NBA),  the Scottish Special Housing Association (SSHA) in collaboration with the Edinburgh University Architecture Research Unit (ARU) and the University of Cambridge Land Use and Built Form Studies Group.

The Ministry of Housing and Local Government Circular 36/69  brought together and modified a number of important earlier pieces of work. (MoHLG 1969a)

It made the minimum whole house Parker Morris space standards into de-facto maxima. (MoHLG 1961)

It also introduced lists of furniture, from Design Bulletin 6: Space in the Home (MoHLG 1963), which had to be accommodated in each type of room.

Design Bulletin 16: Dimensional Coordination in Housing (MoHLG 1969b) also made 300mm external and 100mm internal planning grids and 2600mm floor-to-floor heights mandatory; the latter enabling the plan size of vertical elements like stairs to be standardised.

It soon became apparent to people designing house types that the task was becoming so severely constrained that it was very difficult or impossible to design certain house types without breaking one or more of these rules.

Generic House Types

In the late 1960s the National Building Agency (NBA) was concerned about low house building productivity. The NBA attempted to close the problem out by generating an array of all possible house types with 300mm increment shell sizes that met Circular 36/69 requirements and then proposed a reduced set that would give additional productivity gains to contractors. (National Building Agency, 1965).

nba_house_types_pair_small
Unfortunately later research by the Building Research Establishment (BRE) showed that the shell was not the main determinant of house building productivity. The initiative was not widely taken up and had very little effect, although the tools developed in the effort were sometimes used by others.

SSHA / ARU House Design Program

The Scottish Special Housing Association (SSHA) and the Edinburgh University Architectural Research Unit (ARU) took a different approach and developed a Computer Aided Design program, called House Design. (Bijl et al 1971)

This 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 any further interaction, by reference to a complete set of standard component and assembly details. That is all possible assembly details had been identified, detailed and quantified in advance of their being required.

Design Bulletin 16 recommended (probably required) that internal components be located with one of their finished faces on a 100mm grid line. Somewhat surprisingly it was thought that this would aid the location of components on site.

However it greatly increased the number of possible assembly details that were required. The enumerative analysis that follows was used to justify locating interior components symmetrically within their grid space 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.

/Users/grahamshawcross/Documents/application/nosology/dwgs/KitOf

The following diagram shows the ways the components of the circled arrangement can meet when the components are located with their faces on grid.

/Users/grahamshawcross/Documents/application/nosology/dwgs/KitOf

In House Design the components were either a load bearing partition (structurally 74mm wide) , a non-load bearing partition (structurally 50mm wide) or an internal door. Giving a requirement for a library of the following assembly details.

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

Locating internal components centrally in their 100mm grid space meant that location plans could be automatically dimensioned, with dimensions to the structural face of the components; that is before plasterboard etc were applied and exactly as site operatives handled them. This had the added advantage that grids could be ignored on site; their work having been done in organising the system, they were no longer needed.

HouseDesignDimensionedGF

SSHA Automatically Dimensioned House Type Plan

Automatic Generation of Minimum Standard House Plans

The paper The Automatic Generation of Minimum Standard House Plans, (Steadman 1970) proposed a more fundamental method of generating all possible minimum standard house plans. As Steadman put it:

“it would be possible – given requirements for minimum sizes for rooms and constraints on the permissible shapes they might take, as well as ‘adjacency requirements’ – to produce quite systematically all possible plans in which those requirements were satisfied.”

The intention of this study was therefore to systematically generate all possible standard house plans.

It follows a method used to find squares composed of smaller unique sized squares, using Graph Theory and Kirchhoff’s Laws for electrical flow in wires. (Tutte, 1958).

112x112squaredsquare

The diagrams below show how a plan can be represented as a graph. The graph below shows the vertical distribution of the room dimensions in the plan. A similar graph, usually the dual of this, can be produced for the  horizontal distribution.

HouseGraph1

The distance from the bottom edge of the plan is added to each node of the graph (circled below)

HouseGraph2

The ‘current’ in the Kitchen ‘wire’ is its horizontal dimension 7. The ‘voltage’ at the two ends of the wire are 21 and 7. Their difference is 14 which is the vertical dimension of the room.

HouseGraph3

In spite of the NBA having earlier produced minimum room layouts for all room types (see bedroom example below) this interesting proposal seems never to have been implemented.

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Further work on this idea was reported in Synthesis and Optimization of Small Rectangular Floor Plans (Mitchell, Steadman and Ligett 1976), and this went a long way to explaining why this interesting method was not implemented.

Synthesis and Optimization of Small Rectangular Floor Plans

The analysis in this paper is based on the dimensionless dissection of a rectangle into n parts, the assignment of room names to the parts and then satisfying adjacency requirements between rooms.

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The number of possible dissections grows exponentially with the number of rectangles, as illustrated below for 1 to 8 rectangles.

Workbook1

But for 9 rectangles it was estimated that there were approximately 25000 possible dissections. A future post will show how this figure can be accurately calculated.

Most realistic house types  have 8 or more spaces (rectangles) per floor as exemplified by the SSHA house type plan shown above. The requirement for doors to be only in certain restrained locations also means that further transition-spaces would be needed.

There also seemed to be difficulties in reconciling adjacency and dissections graphs.

Whilst stating that the method is easily extended to multi-storey buildings the only examples given are of single storey trailer designs.

This leaves a catalogue of the bisections of a rectangle into 2 to 6 rooms.

DissectionPlansThat is a list of dissections unsuitable for generating most house types.

Conclusions

The Ministry of Housing and Local Government Circular 36/69 unintentionally brought together all the requirements for the design of 2 and 3 storey local authority house types to be automated.

Most importantly this included the adoption of Parker Morris space standards as de-facto maxima, the requirement for particular room types to be able to accommodate standard lists of fixed sized furniture and the mandatory imposition of dimensional coordination.

The National Housing Agency Generic House Types were manually produced. The furniture lists of Circular 36/69 were built up into collections of minimum sized rooms (as in bedroom example above). These in turn were assembled into the house types. However a desire to exactly match Parker Morris space standards, particularly the complex storage requirements often lead to the integrity of the shell being violated as shown below.

NBA_Storage_KeyAnd as mentioned earlier, later research by BRE showed that the shell was not the main determinant of house building productivity.

Based on its complete set of assembly details, the Scottish Special Housing Association / Edinburgh University ARU House Design Program became a fully operational system that automatically produced all location, component and assembly drawings and bills of quantities from interactive screen based input. Its demise came about through the Thatcher Government abandoning Parker Morris space standards and Local Authority house building in general.

The University of Cambridge Land Use and Built Form Studies perhaps more intellectually ambitious Automatic Generation of Minimum Standard House Plans never  became a working system. This was probably because they had no one with experience of actually having designed house types and apparently did not know of the work done by the NBA. Even working at the MoHLG R+D Group we had to obtain their minimum room layouts surreptitiously. But more importantly they had no way of dealing with the explosion of intermediate and preliminary results their method required. A problem that will be addressed in later posts.

Bibliography

Bijl, A., Renshaw, T., Barnard, D. et al., 1971. ARU research project A25/SSHA-DOE: house design ; application of computer graphics to architectural practice

Mitchell, W.J., Steadman, J.P. & Liggett, R.S., 1976. Synthesis and optimization of small rectangular floor plans. Environment and Planning B Planning and Design, 3(1), pp.37 – 70.

MoHLG, 1969a. Circular 36/69. London: Her Majesty’s Stationary Office.

MoHLG, 1963. Design Bulletin 6 Space in the Home. London: Her Majesty’s Stationary Office.

MoHLG, 1969b. Design Bulletin 16 Dimensional Coordination in Housing. London: Her Majesty’s Stationary Office.

MoHLG, 1961. Homes for Today and Tomorrow (The Parker Morris Report). London: Her Majesty’s Stationery Office.

National Building Agency, 1965. Generic Plans: Two and Three Storey Houses London: The National Building Agency

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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