About

Welcome to the Fourth Dimension!

We can’t quite get you there but we can imagine what we could see if we could.

Think of sunlight shining on a cube made out of sticks.

What would we see on a wall behind the cube? It would be a ‘shadow’ of the cube. In the case of the cube, our brains have little difficulty in piecing together what the cube probably looks like from its shadow.

Actually, the ‘standard’ way of sketching a cube onto a (2-dimensional) piece of paper is not really a faithful representation – think about it!

Now suppose we jumped into a fourth dimension.

Most of us can’t imagine doing that but we can do something analogous to what we did in 3 dimensions.

Shining sunlight on the 4-D object will produce a ‘shadow’ which is 3 dimensional, just like shining sunlight on a 3-D object produces something that is 2-dimensional.

The UCL Knowledge Lab Polytope is the projection from the fourth dimension into 3D space of a 4D polytope – the equivalent in the fourth dimension of a polyhedron in three dimensions. The design of the structure is not a design at all, but a mathematical necessity that turns out to fit into the Zometool system automatically.

The actual polytope is a 4-dimensional analogue of the greater rhombicosidodecahedron. Rather than squares, hexagons and decagons joined edge to edge, it is built from truncated octahedra, decagonal prisms and greater rhombicosidodecahedral cells joined face to face.

The polytope is composed of 2640 cells: 120 truncated icosidodecahedra, 600 truncated octahedra, 720 decagonal prisms, and 1200 hexagonal prisms. It has 14400 vertices, 28800 edges, and 17040 faces (10800 squares, 4800 hexagons, and 1440 decagons). It is the largest nonprismatic convex uniform 4-polytope.

The Mathematics

The object is an orthogonal 3D projection of the ‘largest’ uniform 4D polytope. The description ‘largest’ means having the most vertices and edges. It is known by various names, but omnitruncated 120/600-cell seems to be the best known.

In fairly simple terms, the Coxeter group H₄ has a fundamental region bounded by four mirrors in 4-space, forming a spherical simplex. Picture a tetrahedron, and you will be close enough. To form a polytope:

• Place a single vertex within the fundamental region, including its bounding mirrors.
• Reflect the vertex in all mirrors in which it is not lying.
• Connect the vertex to its neighbouring reflections with edges.
• Now consider all reflections of mirrors, vertices, and edges.
• If the group is finite, the polytope will be finite.

For the polytope to be “uniform”, all edges must be the same length. You can convince yourself that there are 15 ways to form a uniform polytope given a set of four mirrors… the vertex may be on any of four corners of the simplex, on any of six edges, on any of four faces, or completely inside the simplex.

The sculpture is the one generated by putting the vertex completely inside, so it is reflected in all four mirrors. This results in 14400 vertices – the order of the H₄ group. However, the projection from 4D to 3D results in a division by two: Every Zome ball, except those at the surface, is actually representing two separate vertices in the 4D object. Think of a polar projection of both hemispheres of the Earth: Every point in the projection represents a pair of points, one from the northern hemisphere and one from the southern.

Because of the projection into our ‘hyperplane’, the central cell is the only one that is not distorted:

• It could be said that we are looking at that one ‘straight on’.
• All the other cells are at various hyper-angles to our hyperplane, so they appear successively more ‘squashed’ as we approach the ‘outside’ of the model.
• On the very outside are 30 greater rhombicosidodecahedra that are completely flat; in other words, they are perpendicular to our hyperplane.
• As is a symmetrical projection (we couldn’t build it with Zome, otherwise!) many of the edges “double up.”

The model is the largest Zome constructible member of the H₄ group.