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Category Archives: Geometry
Königsberg Bridges
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 … Continue reading
Posted in Architecture, Enumeration, Geometry, Logic
Tagged Architecture, Design, Enumeration, Geometry, Logic
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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 … Continue reading
Posted in Architecture, Design Methods, Geometry, Logic, Tiling
Tagged Aperiodic Tiling, Architecture, Design Methods, Geometry, Logic, Tiling, Turing Machines
3 Comments
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 … Continue reading
Posted in Aesthetics, Architecture, Brain Physiology, Design, Design Methods, Enumeration, Geometry, Randomness
Tagged Architecture, Brain Physiology, Design, Enumeration, Geometry, Randomness
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Smartgeometry 2013
The tenth annual smartgeometry event has just finished in London, and was held at the Bartlett School of Architecture. Previous years’ events have been held at Troy, Copenhagen, Barcelona, New York and San Francisco with institutions bidding for the privilege of hosting … Continue reading
Posted in Architecture, Geometry, Graphics Technology
Tagged Architecture, Geometry, Graphics Technology
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Visual Thinking
I have recently become interested in visual thinking, probably because I like making arguments visually but also as a result of having encountered problems in producing simple easy to understand diagrams. Diagrammatic Illustrations The minimal (graphics only) version above contains … Continue reading
Find Your Own Space
A randomly distributed sets of points can be considered as the lowest state of architectural order; the placing of the simplest object, a point, with the least degree of structure, being randomly placed. Well distributed point sets are important in … Continue reading
Posted in Aesthetics, Aperiodic Tiling, Architecture, Geometry, Tiling
Tagged Aesthetics, Aperiodic Tiling, Architecture
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Wang Tiles and Turing Machines
Wang pointed out that it is possible to find sets of Wang tiles that mimic the behaviour of any Turing Machine (Wang 1975). A Turing machine can compute all recursive functions, that is functions whose values can be calculated in … Continue reading
Posted in Aperiodic Tiling, Architecture, Geometry, Tiling, Turing
Tagged Aperiodic Tiling, Architecture, Geometry, Tiling, Turing Machines
22 Comments
Wang Tiles and Aperiodic Tiling
Wang originally conjectured that no aperiodic tilings could exist. Wang was interested in the decidability of the Tiling Problem; it is said to be decidable if there exists an algorithm which will yield a solution for any given set of prototiles … Continue reading
Posted in Aperiodic Tiling, Architecture, Geometry, Tiling
Tagged Aperiodic Tiling, Geometry, Tiling
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Aperiodic Tiling
This closely follows “Tilings and Patterns” (Grünbaum and Shephard, 1987), but uses coloured diagrams rather than their monochrome ones. In some ways this is simply a catalogue of aperiodic tilings, their various forms and some indication of their uses, but my interest … Continue reading
Posted in Aperiodic Tiling, Architecture, Geometry
Tagged Aperiodic Tiling, Geometry, Tiling
6 Comments
Periodic and NonPeriodic Tiling
The concepts of periodic and nonperiodic tiling are defined so as to clearly distinguish them from aperiodic tiling; the subject of a future post “Aperiodic Tiling”. Informally a tiling (of the 2D Euclidean plane) is a collection of subsets of … Continue reading
Posted in Aperiodic Tiling, Architecture, Geometry, Tiling
Tagged Aperiodic Tiling, Architecture, Geometry, Tiling
4 Comments