Three dimensions - plus

[Barrie Christian]

We easily recognize symmetry in two dimensions. Reflectional symmetry is the easiest, but with practice the other types of symmetry can be spotted.
It is understandable that, when it comes to recognizing patterns, we simplify the world into two dimensions - those patterns on a rug, on a floor, or a wall.
The world, and the buildings around us, however, exist in three dimensions.
How much harder, then, is it to pick out symmetry in those objects.
Over two thousand years ago the Greeks defined 5 basic three dimensional objects - the regular Platonic solids - the cube, tetrahedron, octahedron, dodecahedron, and icosahedron.
Studying these objects reveals intriguing mathematical relationships between the numbers of faces, edges and nodes. The Greeks, and others since, attributed other meanings to these objects, such as beauty, truth, purity, as well as the basic elements.
Curiously, although the circle was an established two-dimensional figure to the Greeks, the sphere was not included among the Platonic solids.
The complexities of considering the symmetries of these three dimensional objects are dramatically greater than their 2D versions.
[attachment=1]Platonic solids.jpg[/attachment][attachment=0]cube_symmetry.jpg[/attachment]

[Barrie Christian]

Most of us are familiar with co-ordinates, either from spreadsheet charts or maps.
They take the form x,y to locate a place in 2D space.
In 3D space, the co-ordinates are x,y,z.
[attachment=0]XYZ-cubes.jpg[/attachment]
Many among you will spot that this formulation need not stop at 3 dimensions.
1,1,1,1 defines a point in 4 dimensional space. A 4D cube is called a hypercube or tesseract. Continuing in the same manner, more and more dimensions can be defined. Of course it is impossible to visualize an object which exists in n-dimensional space.
Mathematicians exploring symmetry succeeded in defining an object in 196,883 dimensional space, with aburdly many symmetries.
(The full details of this objects symmetries, and all the others in fewer dimensions, are contained in the book 'Atlas of Finite Groups').

[Barrie Christian]

If you were wondering what relevance to architecture are extra dimensions, this will help.
I owe the connection here to Marcus du Sutoy, "Moonshine".
There is a way to represent a four dimensional cube (called a tesseract or hypercube).
He points out that La Grande Arche in Paris is such a representation of a 4-dimensional cube.
[attachment=2]Hypercube.png[/attachment]
[attachment=1]La Grande Arche.jpg[/attachment]
The 3rd image is a fascinating representation of 4-dimensions.
[attachment=0]Tesseract.gif[/attachment]