Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 41 (2000), No. 2, 589-595.

Self-similar Simplices

Eike Hertel

Mathematisches Institut, Friedrich-Schiller-Universität Jena Ernst-Abbe-Platz 1–4, D-07743 Jena, Germany e-mail: hertel@minet.uni-jena.de

Abstract. A Euclidean d-simplex S is called k-self-similar if S can be dissected into k ≥ 2 simplices, each similar to S. Each triangle (d = 2) is k-self-similar for k = 4 and k ≥ 6 whereas for d > 2 most d-simplices are not self-similar. A first class of 3-simplices which are m^{3}-self-similar for all positive integers m is characterized.

1. Introduction

The concept of self-similarity comes from fractal geometry, cf. [2]. Let φ_{i} be similarities of the Euclidean d-space ℝ^{d}, i.e.

⋀_{x,y∈ℝ^{d}} (|φ_{i}(x) − φ_{i}(y)| = λ_{i}|x − y|)

where 0 < λ_{i} < 1 (i = 1, …, k). Then a subset M of ℝ^{d} is called k-self-similar (k ≥ 2) if M is invariant under φ_{1}, …, φ_{k}, i.e. if

M = ⋃_{i=1}^{k} φ_{i}(M).

We look for k-self-similar d-simplices. A d-simplex S is the convex hull of d + 1 affinely independent points p_{0}, …, p_{d} ∈ ℝ^{d}:

S = conv{p_{0}, …, p_{d}} := {x ∈ ℝ^{d} : x = ∑_{i=0}^{d} λ_{i}p_{i} ∧ ∑_{i=0}^{d} λ_{i} = 1 ∧ λ_{i} ≥ 0 (i = 0, …, d)}

(we don’t distinguish points and vectors in notation). Thus, the set vert(S) := {p_{0}, …, p_{d}} is the set of vertices of S. Another way of specifying a d-simplex will be useful, namely

S = ⟨p_{0}; a_{1}, …, a_{d}⟩ = {x : x = p_{0} + ∑_{1}^{d} λ_{i}a_{i} ∧ 1 ≥ λ_{1} ≥ ⋯ ≥ λ_{d} ≥ 0},

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag

where a_{i} := p_{i} − p_{i−1} (i = 1, …, d) denote the edges (edge-vectors) of a maximal simple edge path beginning in the vertex p_{0} (cf. Figure 1).

Figure 1

Furthermore, we say a set S is dissected or S admits a dissection into sets S_{1}, …, S_{k}

S = ∑_{i=1}^{k} S_{i} :⟺ S = ⋃_{i=1}^{k} S_{i} ∧ int(S_{i} ∩ S_{l}) = ∅ (i ≠ l).  (1)

2. General k-self-similarity

Now, we define the (general) self-similarity of simplices in a slightly more special manner than above:

Definition 1. A d-simplex S is called k-self-similar if S admits a dissection into k ≥ 2 simplices, each similar to S.

For d = 2 one has a complete classification of self-similar simplices (triangles), cf. [3, 6]:

Proposition 1. a) Each triangle is k-self-similar with k = 4 and k ≥ 6.

b) A triangle S is 2-self-similar if and only if S is a right triangle.

c) A triangle S is 3-self-similar if and only if S is a right triangle.

d) A triangle S is 5-self-similar if and only if S is a right triangle or S has angles of size \frac{2π}{3} and \frac{π}{6}.

Figure 2

Figure 3

Figures 2 and 3 show the sufficency of the conditions in Proposition 1 (the principles of construction). Also, in the two-dimensional case each simplex is (general-) self-similar. It’s remarkable that the situation for d > 2 is quite different:

Lemma 1. Most d-simplices are not k-self-similar for d > 2.

Indeed, each k-self-similar d-simplex S admits a dissection (tiling) of the whole space ℝ^{d}. By a theorem of Debrunner [1], such a simplex S must be equidissectable to a d-cube. Hence, S has vanishing Dehn-functionals (Lemma 1 holds for any d-polyhedra with d ≥ 3).

3. Perfect k-self-similarity

Now we restrict our consideration to a first special case of k-self-similar simplices.

Definition 2. A simplex S is said to admit a perfect k-self-similar dissection, or, in short, S is called k-perfect, if S admits a dissection (1) into k ≥ 2 simplices S_{i} that are mutually incongruent (but similar to S).

For d = 2 one has the following results:

Proposition 2. a) Each non-equilateral triangle is 2m-perfect for all m ≥ 4.

b) The equilateral triangle is non-k-perfect for any k ≥ 2.

c) A triangle S is k-perfect for all k ≥ 2 if and only if S is a non-isoceles right triangle.

Figure 4

The principle of construction in case a) is shown in Figure 4, cf. [7]. Statement b) is a consequence of the fact that there is no dissection of ℝ^{2} into mutually incongruent equilateral triangles, one of them being minimal [8], cf. also [10]. Concerning statement c) see Figure 3 (k = 3). What happens in the situation of d > 2? We have only the following

Conjecture 1. For d ≥ 3 there isn’t any perfect d-simplex.

4. Reptiles

Finally, we consider the following special case of self-similarity:

Definition 3. A d-simplex S is called a replicating tile, or, in short, S is called a k-reptile if S admits a dissection (1) into k ≥ 2 simplices S_{i} that are mutually congruent (and similar to S).

For d = 2 the k-reptiles (triangles) are well known, cf. [9]:

Proposition 3. A triangle S is a k-reptile if and only if

a) k = m^{2} (m ≥ 2; any triangle), or

b) k = 3m^{2} (m ≥ 1; S is a right triangle with acute angles \frac{π}{3} and \frac{π}{6}), or

c) k = m^{2} + l^{2} (m, l ≥ 1; S is a right triangle with cathetuses in the length ratio m : l.

Examples for the three cases are shown in Figure 5.

Figure 5

Thus, each triangle S is a k-reptile (with k = m^{2}). The corresponding dissection should be called standard: Divide each side (edge) of S by m − 1 points into m parts of equal length. Then dissect S by straight lines through these points parallel to the sides of S.

The situation for dimensions d ≥ 3 is rather more difficult (see Lemma 1). But we can apply the above standard dissection to a 3-simplex: Divide each edge p_{i}p_{k} of the tetrahedron

S = conv{p_{0}, p_{1}, p_{2}, p_{3}}

into congruent parts and dissect S by planes through these points parallel to the facets of S. If we assume that S can be dissected in this way (on the analogy of proposition 3 a) into m^{3} congruent tetrahedra, each similar to S, then S also admits such a dissection into 8 = 2^{3} tetrahedra. Let m_{i} be the midpoints of the edges of S (i = 1, …, 6), cf. Figure 6.

Figure 6

Thus, S is dissected into the four tetrahedra

S_{0} := conv{p_{0}, m_{1}, m_{4}, m_{6}}, S_{1} := conv{m_{1}, p_{1}, m_{2}, m_{5}},
S_{2} := conv{m_{2}, p_{2}, m_{3}, m_{6}}, S_{3} := conv{m_{3}, p_{3}, m_{4}, m_{5}}

and the “middle octahedron” O := conv{m_{1}, …, m_{6}}. Obviously, the middle octahedron of any tetrahedron is centrally symmetric. We need the following

Lemma 2. If a centrally symmetric octahedron O is divided into four tetrahedra

O = S_{1} + S_{2} + S_{3} + S_{4}

then two of them form a quadrangular pyramid, and hence the others do as well.

Proof. Each edge of S_{i} is either an edge of O or its relative interior is in the interior of O. Hence, each triangular facet of O is an “outer” facet of exactly one of the simplices S_{i}. We consider any vertex p of O = conv{a, b, c, d, p, q}. At p there meet outer facets of a) four, b) three, or c) two tetrahedra.

In case a) the four simplices must be

S_{1} = conv{a, b, p, q}, S_{2} = conv{b, c, p, q},
S_{3} = conv{c, d, p, q} and S_{4} = conv{d, a, p, q}.

Hence S_{1} + S_{2} is the pyramid P = conv{a, p, c, q, b} with the parallelogram apcq as base.

Figure 7

In b), for example, S_{1} has facets abp and bcp while cdp is a facet of S_{2} and dap is a facet of S_{3}. Then S_{2} + S_{3} must form the pyramid apcqd with the base apcq. In c), let abp and bcp be facets of S_{1} and cdp and dap facets of S_{2}. Then S_{1} + S_{2} is the pyramid abcdp with the base abcd which completes the proof.

Now we go back to the standard dissection of the tetrahedron S into four tetrahedra S_{i} and the middle octahedron O. If S is an 8-reptile then O must admit a dissection into four mutually congruent tetrahedra

O = T_{1} + T_{2} + T_{3} + T_{4},

each of them similar to S. Without loss of generality we will assume that P =conv{m_{1}, ..., m_{5}} is the pyramid in accordance with Lemma 2, cf. Figure 6. This pyramid is dissected into two tetrahedra T_{1} and T_{2}, each congruent to the tetrahedron

S_{1} = conv{m_{1}, p_{1}, m_{2}, m_{5}}.

Hence, P +S_{1} = T_{1} +T_{2} +S_{1} is a triangular prism that can be dissected into three mutually congruent tetrahedra. Then S_{1}, and also S, must be a Hill-tetrahedron, cf. [5, 4]. A d-simplex

S = ⟨p_{0}; a_{1}, a_{2}, …, a_{d}⟩

is called a Hill-simplex (of the first type) if there exist real numbers c > 0 and α (0 < α < \frac{2π}{3}) with

a_{i} · a_{k} = \begin{cases}c^{2} & for i = k, \\ c^{2} cos α & for i ≠ k. \end{cases}

Therefore, in contrast to the twodimensional case, we have only a very special class of 3simplices which are m^{3}-reptiles by the standard construction:

Theorem. Any 3-simplex S is an m^{3}-reptile using the standard dissection if and only if S is a Hill-simplex.

Finally, we postulate the following

Conjecture 2. A tetrahedron S is a k-reptile if and only if S is a Hill-simplex.

References

[1] Debrunner, H. E.: Über Zerlegungsgleichheit von Pflasterpolyedern mit Würfeln. Arch. Math. 35 (1980), 583–587.

[2] Falconer, K. J.: Fractal Geometry. Mathematical Foundations and Applications. John Wiley & Sons, New York 1990.

[3] Freese, R. W.; Miller, A. K.; Uskin, Z.: Can every triangle be divided into n triangles similar to it? Am. Math. Monthly 77 (1990), 867–869.

[4] Hertel, E.: Hill-Tetraeder. Fo-Erg. FSU Jena Math/Inf/99/23.

[5] Hill, M. J. M.: Determination of the Volumes of certain Species of Tetrahedra. Proc. London Math. Soc. 27 (1896), 39–53.

[6] Kaiser, H.: Selbstähnliche Dreieckszerlegungen. Forschungsergebnisse Universität Jena, Preprint 1990.

[7] Kaiser, H.: Perfekte Dreieckzerlegungen. Elem. Math. 46 (1991), 106–111.

[8] Scherer, K.: The impossibility of a tesselation of the plane into equilateral triangles whose sidelengths are mutually different, one of them being minimal. Elem. Math. 38 (1983), 1–4.

[9] Snover, St. L.; Waiveris, Ch.; Williams, J. K.: Rep-tiling for triangles. Discrete Math. 91 (1991), 193–200.

[10] Tuza, Z.: Dissection into equilateral triangles. Elem. Math. 46 (1991), 153–158.

Received August 13, 1999