GRIEF, Ecole Nationale Supérieure d’Architecture de Bretagne, Rennes, France.
COSTEL, LETG UMR 6554 CNRS, Université Rennes 2, Rennes, France.
After a few recalls about the topological definition of dimension and some of the topics that have to do with this crucial issue, this paper focuses on the search for regular polytopes by folding, on complete, incomplete and recursive tiling, and on recursive folds, in order to travel, not only from dimension to dimension, but through dimensions.
Let’s recall first the inductive topological definition of dimension. Our space is three-dimensional (3D) because we are not forbidden to go from any part of it towards another by any point (0D), nor by any line (1D), but we may be “cut” from another part of space by a surface (2D). Any space is nD when it is cut by a (n-1)D space, but not by a (n-2) or lower dimension space. This definition does not apply only to common sense space, but is valid for any continuum, for example the “space” of colours .
We consider an “empirical continuum” defined by our perceptive system (including our kinaesthetic sense), and not a mathematical continuum, nor the physical phenomenon, which may be not continuous, or which may have another structure than our perception of it: “space” is 3D for us human beings, relatively to our visual and tactile senses, essentially, plus our “sense of displacement”, or kinaesthetic sense, i. e. the muscular perception of movement which permits to link different visual or tactile perceptions with each other. But it does not mean that “space”, as a physical object, is 3D: actually, physicists consider much more dimensions than three in their latest models, though we’ll never be able to directly perceive it.
Being inductive, this definition implies that dimension is an integer number, which matches our intuition. Being topological, it applies without having to define a metrics on space. Topology considers objects which can be stretched and deformed as we wish (without tearing them though): a square is the same as a circle for topology, it is simply a closed line. But this definition is coherent with our more usual use of dimension for space with a metrics, i. e. the coordinates (any point of a 3D space needs 3 coordinates to be located, 1 or 2 are not enough, 4 are too much), or the way measures grow when the linear size (1D) grows: area (2D) as a power 2, volume (3D) as a power 3 of the linear measure.
This definition applies also to parts of space, to objects or figures of a space. As a corollary of the topological definition of dimension, the limit (boundary, border, edge) of a part of nD space is (n-1)D: a segment (1D) has got two points (0D) as its border; the border of a disc (2D) is a circle (1D), that of a square (2D) is made of four segments (1D), that we can also consider as one folded line; the border of a sphere (as a 3D volume) is a sphere (as a 2D surface); that of a cube (3D) is made of six squares (2D) that we may consider as one folded surface. We see there an important paradox of this issue of dimension: though a segment has got a border (constituted of two points), a circle, which is also a 1D object has got no border… In the same way, the surface of a sphere (2D) has got no border: a part of a space of less than 3 dimensions can be finite (bounded) without being limited! However, we cannot conceive a volume without a border, unless it is the whole space itself…
The paradox of a bounded unlimited surface is easy to conceive, as the surface of the earth is roughly a sphere, and we know well that if we travel on it ahead of ourselves for a long enough time, we’ll come back to our starting point… But we are 3D beings, and though we are in a great part tied to the surface of the earth, we can also see it from 3D space, and observe that it has got no border because it is curved, and curved in such a way that it closes upon itself. We even can ourselves make a closed surface, for example by folding an adequately cut sheet of paper (a rough approximation of a part of a plane) into a cube (the surface of it): we can fold it because we are 3D people… For a surface to be curved, or folded, we see that it must be immersed in a space of higher dimension, namely for a surface 3D space. But how a volume could be curved or folded , as we cannot conceive of a 4D space in which it would be immersed?
The inverse of the above corollary is that any unlimited nD object defines (i. e. is the border of) at least one (n+1)D object: on a plane, an infinite line defines two half planes; a circle defines a disc and the remnant of the plane with a circular hole; on a sphere, a great circle (the equivalent of a straight line on the sphere) defines two hemispheres; on a torus, a circle defines a surface which we can assimilate to a finite cylinder. Similarly, in 3D space, an infinite plane defines two “half-spaces”, the surface of a sphere defines a solid sphere and the remnant of space with a spherical hole, etc.
We human beings are 3D individuals living in a 3D space. This is so obvious that we don’t daily question it. More than one hundred years ago, some authors, the most famous of them being Edwin Abbott Abbott with his novel Flatland: A Romance of Many Dimensions By A Square , undertook to make their readers more open-minded to their 3D condition by imagining, mainly, 2D beings living in a 2D space: to be aware of their possibilities and limitations makes us more conscious of our own possibilities and limitations, and of our own means of representation of 1D, 2D and 3D objects, and even 4D objects, which are not more inconceivable, after all, than 3D objects are for 2D beings (which we, as “superior” beings, can observe with some condescension). This comparison is indeed efficient and throws light on evidences and paradoxes of our 3D fate. Thomas F. Banchoff , among others, has prolonged this endeavour: this paper owns a lot to his work.
A way to go from nD to (n+1)D is folding. In the search for regular polytopes, we’ll focus on that way to go from one dimension to another, but we’ll also try to not only jump from one dimension to another, but to go through dimensions, to explore that unknown world that lies between 1D and 2D, between 2D and 3D…
Regular polytopes (polygons, polyhedra, etc.) are a favourite item of mathematicians to characterize spaces of any dimension, as soon as a metrics is defined on those spaces. It is not generally question of a regular “polytope” in 1D; it is however obvious that we can extend the definition of a regular polytope on a line, by observing that a segment answers to this definition, i. e. an object with the maximum of symmetry possible in its space. That the segment is actually the only object possible in its space makes its status of regular polytope too trivial to be generally mentioned. Going from dimension to dimension by the way of regular polytopes seems very simple. The most often referred to is to start from a point, then to get a segment by moving the point in one direction, and to go from there to a square by moving the segment in the second direction , then to a cube by moving the square in the third direction; if you consider our 3D space it is the end of the travel, but it is easy to imagine that in 4D space there is a 4th direction available, which allows to go from the cube to a hypercube by “moving” the cube in that 4th direction; and so on…
Actually there is another even simpler way to accomplish this travel. At each step, instead of moving the previous polytope in some direction, you can add a vertex (a point) to the previous polytope; then the succession of polytopes becomes: point, segment, equilateral triangle, tetrahedron, etc… In the 4th dimension we can imagine adding a vertex to the tetrahedron in the same way that we add one to the triangle in the 3rd to obtain what may be called a pentatope (since it has got 5 vertices, and five hyperfaces, or “cells”). Those simplest polytopes, which are constituted by (n+1) vertices defining triangles in any nD space are called simplexes.
So we have, in any dimension, at least, a “cube” (which we call a square in 2D), and a “simplex”. To find other ones, one way is to consider the “duals”; the dual of a polytope is the polytope obtained by placing the vertices of the new polytope in the centre of the faces of the previous one: for example, in 3D, the dual of the tetrahedron is itself a tetrahedron, but the dual of the cube provides a new polyhedron: the octahedron. Actually, it can be demonstrated that any simplex is “self-dual”, and that there exists a dual polytope of the “cube”, distinct from it, in any dimension higher than 2. So there are at least 3 regular polytopes in any dimension…
Actually, in any dimension higher than 4 those 3 regular polytopes are indeed the only ones. But in 2D, they are infinitely many regular polytopes (polygons), and they are all self-dual; in 3D there are 5 polytopes (polyhedra), one is self-dual (the tetrahedron) and the two other pairs are dual of each other (resp. the cube and the octahedron; and the dodecahedron and the icosahedron); and in 4D there are 6 (the simplex, which is self-dual, two pairs of dual polytopes, and another one which is self-dual)… This shows the complexity of dimension, behind its seemingly simple and boring enumeration: 1, 2, 3, 4,…
Among the many ways to define and produce regular polytopes, one, which goes from dimension to dimension, is that of folding. Any child knows how to make a cube starting from a sheet of paper dexterously cut into connected squares. Actually, we don’t miraculously create a volume (3D) starting from a sheet of paper (roughly 2D)… We only produce a surface that encloses such a volume, the “skin” of a cube. But as we have seen before, a closed surface defines at least one volume. Trying to fold regular polytopes around a regular polytope (all of the same kind) in any dimension is a way to find all the regular polytopes in the space of higher dimension. The restraints we have are to respect some symmetry, and to let enough looseness to be able to fold.
Let’s start in 2D, and let’s try first to place regular polygons around another one, by patching them edge to edge. You can place 3 triangles around a triangle, and by folding them up, you obtain the tetrahedron (Fig. 1.a). You can place 4 squares around a square and by folding, you obtain an open cube that you close with a 6th square (Fig. 1.b). You can place 5 pentagons around a pentagon, and by folding those, you obtain half of a dodecahedron, that you can close by another symmetric half, rotated (Fig. 1.c). Now if you place hexagons around an hexagon, there is no place left to fold them (as they tile 2D space, as we’ll see further). If you try with heptagons, etc, there is even superposition, so you cannot fold them.
Now, let’s return to the square and the triangle and let’s try to add other squares (resp. triangles), touching now by the vertices. With the square, as soon as you add those 4 new squares, the plane is filled and you cannot fold them out. But with the triangle, there are two new possibilities: you can add three new triangles, and by folding, you obtain an open octahedron, which you close with a last triangle (Fig. 1.d). Or you can add 6 new triangles (2 at each vertex) and you obtain the half of an icosahedron, which you complete with another half, by rotating it, as for the dodecahedron (Fig. 1.e). Now if you try to add 3 new triangles, you fill up the plane, and you cannot fold them out.
Figure 1 shows the start of the folding patterns (above), and the Schlegel diagrams (below) of the regular polygons: Schlegel diagrams are actually usual perspectives, but where the camera is placed very near from the object, and with a very great focal angle, so that the front face embraces all the other ones:
Figure 1 a b c d e
Now let’s return back to 1D. Instead of playing with paper, scissors and tape, a child can play with sticks or matches (all these games are dangerous anyway). We can place two segments around one, and that is all we can do, by the way, as segments have two vertices, but have no “edges” in 1D space. We can fold them easily around their extremities (we may want to tie them, but loosely), and they kindly rejoin each other to make a triangle. Now, to make other polygons, we can fold them less, and let some space for a 4th segment, to get a quadrilateral. But here we encounter a problem that did not exist in going from 2D to 3D, which is that we do not compulsorily obtain a square, or that this square collapses easily upon itself… Actually, all polygons made of more than 3 segments collapse, where any convex polyhedron is uncollapsable: here is one the mysteries of dimension… Anyway, let’s try to arrange our quadrilateral into a fair square, and let’s resume our game. We see that we can add as many segments as we want and thus obtain all the (regular, if we are careful) polygons.
Going from 3D to 4D, we see that we can place 4 tetrahedra around one, and we have only to fold them in 4D space to obtain the 4-simplex (Fig 2.a). And we can place 6 cubes around one, and fold them to obtain an open hypercube, which we’ll close with an 8th cube to obtain the hypercube (Fig. 2.b). We can also place 8 octahedra around one, this will produce a polytope made of 24 octahedra, which is called 24-cell (Fig. 2.d,e,f). And we can put 20 dodeacahedra around one, to get a more complicated polytope, the 120-cell (Fig. 2.c).
Figure 2 a b c d e f
Now, can we place polyhedra, not only against faces of another one, but between these, against edges, as we have done against vertices of polygons. It is only possible if we start from the tetrahedron. We can place 4 more tetrahedra this way to get the 16-cell (Fig. 3.a,b,c). And there is even some place left (not much, but enough…), to put one more tetrahedron, and this leads to the most complex 4D polytope, the 600-cell (Fig. 3.d,e,f).
Figure 3 a b c d e f
Trying to find ways to fold out polygons into polyhedra, we have encountered situations where the plane was filled, namely with equilateral triangles (with 12 of them around one), with squares (with 8 of them around one), with hexagons (with 6 of them around one), and that’s all (higher polygons overlaid). Those situations correspond to the possible regular tilings of the plane.
On a line, any segment defines a “tiling”. On a circle, though, the segment (which is an arc) must be a divider of the circumference. On the plane, the equilateral triangle (Fig. 4.a), the square (Fig. 4.b), and the hexagon (Fig. 4.c) define tilings:
Figure 4 a b c
On the sphere, tiling is even more restricted than on the circle: only the regular polyhedra define a tiling, or tessellation, of the sphere:
Let’s return to planar tiling, and let’s examine those patterns, according to some regularities. For example, let’s compare the distribution of the vertices and of the centres of the polygons: we see that for the squares they are the same, but that for the triangles and the hexagons they swap: we could say, referring to the vocabulary used for polyhedra, that the square (orthogonal) tiling is self-dual, but that the triangular and the hexagonal ones are duals of each other.
This duality may be expressed in another way. If we apply the principle of the distance map  to the three distributions of centres, we see that the Voronoï diagrams we get are those so-called dual tilings.
We can also consider tiling as a cellular automaton, and launch a growth process. Usually CA consider cells as pixels on an orthogonal grid. But we can consider any tiling polygon as well. Starting with a polygon, we apply the rule that we’ll add all the same polygons we can that touch the previous ones at each step. We obtain different patterns according to the neighbourhood chosen to apply the rule: as in a CA we can consider only the edges (Fig. 7.a,c), or the edges and the vertices (Fig. 7.b,d). For the hexagon, there is no difference (Fig. 7.e):
Figure 7 a b c d e
We see that this growing process leads to a pattern similar to the starting polygon, only in one case: the square, and with the second rule.
Those polygons are the only ones that actually tile the plane. But why not try the same growth process with a pentagon? It leads to an incomplete tiling:
This experiment with the pentagon gives us the idea to try other incomplete tilings, even with the polygons that tile the plane. It is produced by adding a new rule to the previous one: we’ll not add a polygon if it touches more than one of the previous generation. These incomplete tilings with the triangle (Fig. 9), the square (Fig. 10), the hexagon (Fig. 11), and the pentagon (Fig. 12) provide interesting patterns:
Figure 9 Figure 10
Figure 11 Figure 12
Now, returning to our first patterns, we notice that the square produces a square, but that the hexagon does not produce an hexagon, and the equilateral triangle neither. That has to do with self-similarity: we know that the square is self-similar (i. e., that it can be filled up with reduced copies of itself), but that the hexagon is not. But we thought that the equilateral triangle was self-similar too, didn’t we? Well, it is, but upon the condition that we accept to change the orientation of one the reduced copies…
Self-similarity is another way of defining dimension. Let’s observe the patterns we got before in this way. In 1D, when we have added two segments around one, we have obtained a segment filled up with 3 reduced copies of itself, 3 times smaller. In 2D, the square is filled up with 9 copies of itself, reduced by 3; the alternative pattern of the equilateral triangle (where we permit a change of orientation) shows a triangle filled up with 4 triangles 2 times smaller. In 3D, the only self-similar polyhedron, the cube, is filled with 27 cubes 3 times smaller. The dimension (self-similarity dimension, which in those cases matches the topological one) is obtained by dividing the logarithm of the number of copies by the logarithm of the number by which those copies are reduced: for the segment, we get log 2 / log 2 = 1; for the square: log 9 / log 3 = log 32 / log 3 = 2 log 3 / log 3 = 2; for the triangle: log 4 / log 2 = log 22 / log 2 = 2 log 2 / log 2 = 2; for the cube: log 27 / log 3 = log 33 / log 3 = 3 log 3 / log 3 = 3: which is coherent with their topological dimension.
So we can consider another way of tiling, which is recursive tiling. We know that the triangle and the square fill the plane so their recursive tiling is not very satisfying. But we can do it with a triangle from which we have removed the middle reduced one, and we obtain the well known Sierpinski gasket:
But we have seen that the pentagonal tiling did not fill the plane. So we can apply the recursive tiling to it:
Now, in 3D, only the cube tiles (or “tessellates”) space, and it is also the only one polyhedron that is self-similar. The tetrahedron could be a good candidate for self-similarity. After all, it is the equivalent of the equilateral triangle which is self-similar. Unhappily, when you stack tetrahedra in the way you stack triangles in the plane, the hole in the middle is not in the form of a tetrahedron, or any number of tetrahedra, but is in the form of an octahedron!
Anyway that does not prevent us to try recursive 3D tiling (or tessellation) with those 4 tetrahedra:
The stacks of polyhedra we did to find the 4D polytopes may also lead to recursive tessellations. Let’s take the one that lead to the 24-cell. It is enclosed in a bigger octahedron. So we can repeat the multiplication of octahedra on this first stack:
Incomplete recursive tilings and tessellations allow us to reach paradoxical objects because their self-similar dimension is not an integer. For example, if we apply the definition mentioned above to the Sierpinski gasket (Fig. 14), which is made of 3 copies of itself, reduced by 2, we get: log 3 / log 2 = 1.58…; for the recursive pentagonal tiling (Fig. 15), we find 1.86… With the recursive tetrahedral tiling (Figure 16), we encounter a new paradox: d= log 4 / log 2 = 2. This figure is equivalent to a surface! Indeed, if you rotate the tetrahedron above and move it down, you see that you actually end up with a plane. If you add a fifth tetrahedron below, you obtain a fractal dimension of log 5 / log 2 = 2.32…:
Those recursive incomplete tilings lessen the dimension of the object from which they start: from 2D to somewhere between 1 and 2; from 3D to somewhere between 2 and 3. We’ll see with recursive folds processes that increase the dimension.
Our complacent child is now bored with cutting paper and taping it, and he crumples the sheet of paper to throw it in the wastebasket (we have removed the matches, though…). Now that is interesting because in this way he has produced some sort of ball, which is actually nearer from a real 3D form than the skins of polyhedra he obtained with much effort before. Well, if we are fastidious, we’ll not accept that this “ball” is a real 3D volume, but we must accept that it is a good approximations of it. Examining a crumpled sheet of paper, we see that it is not only folded, but that the folds are re-folded, and re-folded, and that it is actually a recursive fold.
The most known recursive fold of a line is the von Koch curve, of self-similar dimension log 4 / log 3 = 1.26…:
Recursive folds are ways to obtain figures which are of a dimension between 1 and 2, but can we actually travel all the way from 1D to 2D? Yes, because there are extreme cases where the self-similar dimension is equal to 2; We can change the angle in the von Koch construction (Fig. 20.a: 15° d=1.012… b: 30° d=1.05… c: 45° d=1.13… d: 60° d=1.26… e: 75° d=1.50… f: 90° d=2), to get a curve that fills a part of the plane:
Figure 20 a b c d e f
There are even such recursive folds that fill a 3D volume. The Hilbert curve fills a cube:
The von Koch construction reminds us of the way we have folded a line to make a triangle. We can try another recursive fold based on the square:
You can fold a line any way. But folding a plane is more difficult, because you must do folds in alternate senses. So we’ll admit to call folds operations that are not actually folds, but which consist of pulling vertices of a plane outside the plane. We can for example try to adapt the previous folds we made from a line to a triangular or square surface:
Note that in the case of the triangular folded surface, the “peaks” touch on the outside, so the complexity is better seen from below. The fractal dimension of this “surface” is log 6 / log 2 = 2.58…, that of the square one is log 13 / log 3 = 2.33…
Folding is a very good way to creativity, as proves the Japanese art of origami or the experiment in “folding as a morphogenetic process in architectural design” proposed by Sophia Vyzoviti . Tilings have also been much exploited, for example in islamic art. The aim of this paper was just to propose some leads to see forms, even such familiar forms as regular polygons and polyhedra, in a different way, to see them as part of a transforming process . Many of these leads could be pursued much farther. Those experiments make us also more aware of what is dimension, of what link dimensions with each other, and of what could lie between them.
 Abbott, Edwin Abbott, Flatland: A Romance of Many Dimensions By A Square, first published by Seeley & Co, London, 1884
 Banchoff, Thomas F., Beyond the Third Dimension: Geometry, Computer Graphics and Higher Dimensions, Scientific American Library, 1990
 Corcuff, Marie-Pascale, “Generative processes and the question of space”, GA 2005 proceedings
 Soddu, Celestino, “Gencities and visionary worlds”, GA 2005 proceedings
 Vyzoviti, Sophia, Folding Architecture: Spatial, Structural and Organizational Diagrams, BIS Publishers, 2003
 Strictly speaking, we should say “infinite or closed line (resp. surface), and each time we say “any point (resp. line, etc.)” in this inductive definition, we should add “any finite set of points (resp. lines, etc.)”
 This way is slightly different from the one usually used, where you fold regular polytopes around a vertex (in 2D), and around an edge (in 3D), see .
 Figures 2 and 3 show only the starting folding pattern of the construction of polytopes (as Figure 1, above)
 and one case where d = 1: when the angle is equal to 0°