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.
If you have never come across the following idea, it may at first seem abstract, even far-fetched. Yet it is not an invention of this site but established mathematics: the tetrahedron can be continued into any number of dimensions. These generalised tetrahedra are called simplexes. In the context of number theory and the tetractys as developed on this site, the equation simplex = tetrahedron comes almost as a revelation.
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.
- Point: dimension 0
- Segment: dimension 1
- Surface (triangle): dimension 2
- Solid (tetrahedron): dimension 3
The tetrahedron, the “four-faced”, is thus the fourth link in this chain.
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.
The whole of Pascal’s triangle is therefore a diagram of the multidimensionality of the simplexes. Anyone who knows it can see here how inseparably geometry, above all the geometry of simplexes, and number theory are interwoven.
Two further observations: Pascal’s triangle can be divided into ten-point triangles with four points to a side, with inverted six-point triangles between them. And here too the structure is shaped by the alternation of even and odd numbers. That triangular numbers also come up here has, however, nothing directly to do with this division.
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 |
Compare the table row by row with Pascal’s triangle: the numbers are the same.
The tetrahedron, the pyramid on a triangular base and simplest Platonic solid. In the figure above it appears as a square with two diagonals, because there you look straight at an edge.
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. It surely shows something significant. But merely adding up points does not yet give the ten any meaning.
A second interpretation also claims to be the Pythagorean tetractys: the tetractys of music. It consists of the basic consonances, fourth 4 : 3 (= 8 : 6), fifth 3 : 2 (= 9 : 6) and octave 2 : 1 (= 12 : 6). Both interpretations leave a decisive question open: where is the ten, the sum and quintessence 1 + 2 + 3 + 4 = 10 that Philolaus describes so explicitly? In neither does it play a recognisable, meaningful role.
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. Read the tetractys of space as a sequence of multidimensional tetrahedra, add number theory, and suddenly everything appears in a different light.
With the fourth dimension begins the world of spaces that can only be thought. One could also call it the entry into the spiritual, transcendent world.
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.
Counting what bounds each figure gives a series of its own:
| Figure | Dimension | bounded by |
|---|---|---|
| Point | 0 | nothing |
| Line | 1 | 2 points |
| Triangle | 2 | 3 lines |
| Tetrahedron | 3 | 4 faces |
| 5-cell | 4 | 5 spaces (tetrahedra) |
The 5-cell, rotated in the fourth dimension and projected into space. The five tetrahedra seem to grow into one another.
Looking closely, you can at times make out a pyramid on a square base in the rotating shape, with the two diagonals lying in its ground plan. That is not entirely wrong. As it turns, the five tetrahedra seem to grow out from inside.
A three-dimensional being will never gain a true impression of four dimensions. The projection is a shadow, but it gives an inkling. Few will find the animation an eye-opener.
The Pythagoreans knew this boundary of space, but they thought beyond it. How else could they have said about the ten what Philolaus hands down? The 5-cell is especially good ground for philosophising in the spirit of the tetractys, because the pentagram comes into play here.
Behind this lies a principle that recurs again and again in the many-layered, fractal nature of the tetractys: the fourness culminating in ten is a closed order. The odd five, the first prime in the twin grid of the rhythm of six, breaks this order. Together with the even numbers divisible by five, ten and its multiples, it creates a new, more complex tetractys.
Further reading: Simplex on Wolfram MathWorld, Pentatope on Wolfram MathWorld and the Wikipedia article Simplex.
The crosser of boundaries
Here the discussion becomes explicitly philosophical. The fourth dimension seems not to exist in reality, and yet its tetrahedron contains a pentagram. Space can be filled without gaps by triangular and square forms: by pyramids and tetrahedra, by cubes and cuboids. This is the isotropic vector matrix of Buckminster Fuller, built only from triangles, hexagons and squares (see Circle, triangle and square and Gap-free filling of space). Between triangle, square and hexagon there is a gap: the pentagon. Space cannot be filled with pentagonal forms without the bodies passing through one another. 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. Planets would crash into each other, every mechanical process would be fatal. String theorists, who assume ten or eleven dimensions, argue along similar lines. 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. Time alone decides whether two bodies meet in the same place. So the pentagram appears as the principle of time, matter and life, which matches its meaning in the Hermetic teachings. The right angle marks the natural boundary of space, and the pentagon crosses it. 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. Apply the procedure of the Sierpiński triangle to the square and the pentagon: with the square the parts still fit together without gaps. From the pentagon on they overlap, and their intersection forms a pentagram, the crosser of boundaries.
The Sierpiński procedure on the square: the sub-squares meet exactly. Use the slider to switch to the pentagon, where the parts overlap and a pentagram appears in the middle.
Considering both cases carefully leads to the conclusion that the square stands in direct relation to the unit circle. The reason is its four interior angles of 90° each, which together make exactly one full circle (see Angle sums and full circles). It follows that the Pythagorean tetractys stands for unity and completeness.
Rhythm, time and life
The principle of crossing boundaries shows in number theory too. If one follows the Pythagorean idea that number is the first cause of all appearance, the rhythm of six of the twin primes, which holds without end, can be seen as the timelessly unchanging pair of space and energy, a mirrored three. The overlapping grids of the prime “disorder”, by contrast, each starting ten further on, at 25, 35, 45 …, a mirrored five, depict the dynamics of time and matter, becoming and passing away and so the principle of life (see The disorder of the primes).
| Grid | starts at |
|---|---|
| steps of 10 | 25 = 5 × 5 |
| steps of 14 | 35 = 5 × 7 |
| steps of 18 | 45 = 5 × 9 |
| steps of 22 | 55 = 5 × 11 |
| steps of 26 | 65 = 5 × 13 |
| … | … |
In the simplexes these grids appear as star figures made only of pentagrams, always with the quotient 2.5: 25/10 of five pentagrams, 35/14 of seven, 45/18 of nine. The numerators lie ten apart, the denominators four. The pentagon with its pentagram is the start figure of this principle, and its square 25 opens the disorder.
The occultist and Rosicrucian Ferdinand Maack, who also invented a three-dimensional chess, put a related thought in his book Talisman Turc in 1926 (translated here):
To live is to be periodic, to vibrate rhythmically with the universe […]. Sharp periods are harmful to the self-preservation of the individual. Therefore nature – although it owes its own preservation to the law of periodicity – seeks to weaken and blur accented periods. For this nature needs no new measures or laws, it simply lays a number of periods on top of one another! The sum of these superimposed periods appears aperiodic and makes the individual more stable, better fitted for the struggle for existence.
This is exactly how the apparent disorder of the primes arises: through the overlapping of the multiples of all prime divisors with five, without end. Behind the irregular shell lies a regular core. Whoever has the patience to draw simplexes and count their star figures will find this confirmed. How the same double pentagram appears in the cross-section of DNA is described in The tetractys at a glance.