Category Archives: Geometry

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 … Continue reading →

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William Tutte’s Hidden Past

Posted on April 16, 2014 by Graham Shawcross

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 , , , , , , | 3 Comments

Subitising

Posted on January 9, 2014 by Graham Shawcross

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 , , , , , | 1 Comment

Smartgeometry 2013

Posted on May 8, 2013 by Graham Shawcross

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 →

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Visual Thinking

Posted on December 12, 2012 by Graham Shawcross

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 →

Posted in Architecture, Geometry, Logic | Tagged , , | 6 Comments

Find Your Own Space

Posted on October 12, 2012 by Graham Shawcross

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 , , | 2 Comments

Wang Tiles and Turing Machines

Posted on October 12, 2012 by Graham Shawcross

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 , , , , | 22 Comments

Wang Tiles and Aperiodic Tiling

Posted on October 12, 2012 by Graham Shawcross

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 , , | 13 Comments

Aperiodic Tiling

Posted on October 12, 2012 by Graham Shawcross

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 , , | 6 Comments

Periodic and Non-Periodic Tiling

Posted on October 12, 2012 by Graham Shawcross

The concepts of periodic and non-periodic 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 , , , | 4 Comments