The simplex – multidimensional tetrahedra
Each new vertex lifts the tetrahedron into a further dimension. Pascal's triangle is the complete table of these solids, and the 5-cell carries a pentagram within it.
The sequence point, segment, triangle, tetrahedron does not stop after the third dimension. With every further point joined to all the others, the simplex of the next dimension appears. Drawn in the plane it looks like a polygon with all its diagonals. The tetrahedron, for example, appears as a square with both diagonals when you look straight at one of its edges.
Two-dimensional projections of the simplexes: the figure with four points is the tetrahedron, seen edge-on. Five points give the four-dimensional 5-cell.
Pascal’s triangle as a table of simplexes
Along its diagonals, Pascal’s triangle contains in turn the ones, the natural numbers, the triangular numbers, the tetrahedral numbers and then the numbers of higher dimensions. At the same time every row counts the parts of a simplex: the n-dimensional simplex has as many vertices, edges, triangles and tetrahedra as row n + 1 gives.
Pascal's triangle. The odd numbers are coloured and form the Sierpiński triangle. The buttons highlight the diagonals.
| Simplex | Vertices | Edges | Triangles | Tetrahedra | Cells |
|---|---|---|---|---|---|
| Point (0D) | 1 | ||||
| Segment (1D) | 2 | 1 | |||
| Triangle (2D) | 3 | 3 | 1 | ||
| Tetrahedron (3D) | 4 | 6 | 4 | 1 | |
| 5-cell (4D) | 5 | 10 | 10 | 5 | 1 |
The tetractys of space
Point, line, triangle and tetrahedron together have 1 + 2 + 3 + 4 = 10 vertices. This “tetractys of space” is often shown to illustrate the fourness. But merely adding up points does not yet give the ten any meaning.
It looks different when you consider the edges. The triangular numbers in the triangle of points count not the vertices but the edges of the simplexes: 1, 3, 6, 10. The triangle of ten points thus corresponds to the tetrahedron of the fourth dimension with its ten edges.
The 5-cell
The four-dimensional tetrahedron is called the 5-cell or pentatope. Just as the segment consists of two points, the triangle of three segments and the tetrahedron of four triangles, the 5-cell consists of five tetrahedra. Drawn in the plane it is a pentagon with a pentagram inside, the sign by which the Pythagoreans recognised one another.
The 5-cell, rotated in the fourth dimension and projected into space. The five tetrahedra seem to grow into one another.
A three-dimensional being will never gain a true impression of four dimensions. The projection is a shadow, but it gives an inkling.
The crosser of boundaries
Here the discussion becomes explicitly philosophical. Space can be filled without gaps by triangular and square forms (see Circle, triangle and square), but not by pentagons. The pentagram in the 5-cell therefore crosses the boundary of three-dimensional space. It is the crosser of boundaries, Hermes, messenger of the gods.
The thought can be taken further: a four-dimensional world in which bodies pass through one another would destroy us. Yet perhaps we meet this interpenetration daily on a small scale, wherever living things displace, devour and replace one another. Physics calls time the fourth dimension. Whether two bodies claim the same place depends only on whether they are there at the same time. So the pentagram appears as the principle of time, matter and life. The fourth dimension stands both for entry into the transcendent and for the embodiment of spiritual principles in matter. Three-dimensional space provides the frame, and in it the divine reveals itself in matter, as matter reveals the divine.
The same motif appears in the fractal polygons: from the pentagon on the sub-figures overlap, and their intersection forms a pentagram. The square, by contrast, whose four right angles make exactly one full circle, stands for unity and completeness.