1 Introduction
1.1 Objectives and Motivation
1.2 Related Work
- It should be easy to find the simplex that shares a facet with a given simplex.
- It should be possible to label the vertices of all the simplices at the same time with indices \(0,\ldots ,d\), in such a way that each of the \(d+1\) vertices of a \(d\)-simplex has a different label.
- The triangulations should be monohedral, meaning that all simplices are congruent. Here, by congruent, we mean that the simplices are identical up to reflections and rotations.
- All simplices should be isotropic, meaning that they should be roughly the same in all directions.
1.3 Contribution
1.4 Future Work
2 Roots, Groups and Lattices: General Background
2.1 Graphs and Cartan Matrices
- \(\left\langle \cdot ,\cdot \right\rangle _{C}\) is positive definite, that is it defines an inner product.
- The root set \(R_\varSigma \) is finite.
- The Weyl group \(W_\varSigma \) is finite.
2.2 Root Systems in Euclidean Space
- \(0 \notin R\) and R contains a basis of \({\mathbb {R}}^{d}\),
- for all \(r\in R\), \(\sigma _r(R) \subset R\).
- For any two distinct simple roots \(r, s \in S\), we have \(\left\langle r,s \right\rangle \le 0\).
- The set S forms a basis of \({\mathbb {R}}^{d}\).
- The Weyl group is generated by the reflections associated to the simple roots in S.
\(4 \cos ^2 \phi \) | n(s, r) | n(r, s) | \(\varphi \) | length relation |
---|---|---|---|---|
4 | 2 | 2 | 0 | \(\Vert s\Vert = \Vert r\Vert \) |
4 | \(-\,2\) | \(-\,2\) | \(\pi \) | \(\Vert s\Vert = \Vert r\Vert \) |
3 | 3 | 1 | \(\pi /6\) | \(\Vert s\Vert = \sqrt{3} \Vert r\Vert \) |
3 | \(-\,3\) | \(-\,1\) | \(5 \pi /6\) | \(\Vert s\Vert = \sqrt{3} \Vert r\Vert \) |
2 | 2 | 1 | \(\pi /4\) | \(\Vert s\Vert = \sqrt{2} \Vert r\Vert \) |
2 | \(-\,2\) | \(-\,1\) | \(3 \pi /4\) | \(\Vert s\Vert = \sqrt{2} \Vert r\Vert \) |
1 | 1 | 1 | \(\pi /3\) | \(\Vert s\Vert = \Vert r\Vert \) |
1 | \(-\,1\) | \(-\,1\) | \(2\pi /3\) | \(\Vert s\Vert = \Vert r\Vert \) |
0 | 0 | 0 | \(\pi /2 \) | (undetermined) |
- \(\mathbf {A_{d}}\) (in \({\mathbb {R}}^{d+1}\)): \(s_1 = e_1 - e_2\), \(s_2 = e_2 - e_3\), ..., \(s_{d} = e_{d} - e_{d+1}\).
- \(\mathbf {B_{d}}\): \(s_1 = e_1 - e_2\), \(s_2 = e_2 - e_3\), ..., \(s_{d-1} = e_{d-1} - e_{d}\), \(s_{d} = e_{d}\).
- \(\mathbf {C_{d}}\): \(s_1 = e_1 - e_2\), \(s_2 = e_2 - e_3\), ..., \(s_{d-1} = e_{d-1} - e_{d}\), \(s_{d} = 2e_{d}\).
- \(\mathbf {D_{d}}\): \(s_1 = e_1 - e_2\), \(s_2 = e_2 - e_3\), ..., \(s_{d-1} = e_{d-1} - e_{d}\), \(s_{d} = e_{d-1} + e_{d}\).
- \(\mathbf {E_6}\) (in \({\mathbb {R}}^8\)): \(s_1 = \frac{1}{2}(e_1 + e_8) - \frac{1}{2}(e_2 + e_3 + e_4 + e_5 + e_6 + e_7)\), \(s_2 = e_1 + e_2\), \(s_3 = e_2 - e_1\), \(s_4 = e_3 - e_2\), \(s_5 = e_4 - e_3\), \(s_6 = e_5 - e_4\).
- \(\mathbf {E_7}\) (in \({\mathbb {R}}^8\)): \(s_1 = \frac{1}{2}(e_1 + e_8) - \frac{1}{2}(e_2 + e_3 + e_4 + e_5 + e_6 + e_7)\), \(s_2 = e_1 + e_2\), \(s_3 = e_2 - e_1\), \(s_4 = e_3 - e_2\), \(s_5 = e_4 - e_3\), \(s_6 = e_5 - e_4\), \(s_7 = e_6 - e_5\).
- \(\mathbf {E_8}\): \(s_1 = \frac{1}{2}(e_1 + e_8) - \frac{1}{2}(e_2 + e_3 + e_4 + e_5 + e_6 + e_7)\), \(s_2 = e_1 + e_2\), \(s_3 = e_2 - e_1\), \(s_4 = e_3 - e_2\), \(s_5 = e_4 - e_3\), \(s_6 = e_5 - e_4\), \(s_7 = e_6 - e_5\), \(s_8 = e_7 - e_6\).
- \(\mathbf {F_4}\): \(s_1 = e_2 - e_3\), \(s_2 = e_3 - e_4\), \(s_3 = e_4\), \(s_4 = \frac{1}{2}(e_1 - e_2 - e_3 - e_4)\).
- \(\mathbf {G_2}\) (in \({\mathbb {R}}^3\)): \(s_1 = e_1 - e_2\), \(s_2 = -2e_1 + e_2 + e_3\).
2.3 Root Lattices
2.4 Affine Reflection Groups
\({\tilde{A}}_{d}\), given in \({\mathbb {R}}^{d+1},\, d\ge 2\) |
\(\begin{array}{l} u_0 = \left( 0^{\{d+1\}}\right) \\ u_k = \left( (-\frac{d+1-k}{d+1})^{\{k\}}, (\frac{k}{d+1})^{\{d+1-k\}}\right) , \quad \forall k \in \left\{ 1,\ldots ,d \right\} \end{array}\)
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\({\tilde{B}}_{d}\), given in \({\mathbb {R}}^{d},\,d\ge 3\) |
\(u_0 = \left( 0^{\{d\}}\right) \)
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\(u_1 = \left( 1, 0^{\{d-1\}} \right) \)
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\(u_k = \left( \frac{1}{2}^{\{k\}}, 0^{\{d-k\}} \right) , \forall k \in \left\{ 2,\ldots ,d \right\} \)
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\({\tilde{C}}_{d}\), given in \({\mathbb {R}}^{d},\,d\ge 2\) |
\(\begin{array}{l} u_k = \left( \frac{1}{2}^{\{k\}}, 0^{\{d-k\}} \right) , \forall k \in \left\{ 0,\ldots ,d \right\} \\ \end{array}\)
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\({\tilde{D}}_{d}\), given in \({\mathbb {R}}^{d},\,d\ge 4\) |
\(u_0 = \left( 0^{\{d\}} \right) \)
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\(u_{1} = \left( 1, 0^{\{d-1\}} \right) \)
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\(u_k = \left( \frac{1}{2}^{\{k\}}, 0^{\{d-k\}} \right) , \forall k \in \left\{ 2,\ldots ,d-2 \right\} \)
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\(u_{d-1} = \left( \frac{1}{2}^{\{d-1\}}, -\frac{1}{2} \right) \)
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\(u_{d} = \left( \frac{1}{2}^{\{d\}} \right) \)
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\({\tilde{E}}_{6}\), given in \({\mathbb {R}}^{8}\) |
\(\begin{array}{l} u_0 = \left( 0^{\{8\}}\right) \\ u_1 = \left( 0^{\{5\}},-\frac{2}{3}^{\{2\}}, \frac{2}{3} \right) \\ u_2 = \left( \frac{1}{4}^{\{5\}},-\frac{1}{4}^{\{2\}}, \frac{1}{4}\right) \\ u_3 = \left( -\frac{1}{4}, \frac{1}{4}^{\{4\}},-\frac{5}{12}^{\{2\}}, \frac{5}{12} \right) \\ \end{array}\)
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\(\begin{array}{l} u_4 = \left( 0^{\{2\}}, \frac{1}{3}^{\{3\}},-\frac{1}{3}^{\{2\}}, \frac{1}{3} \right) \\ u_5 = \left( 0^{\{3\}}, \frac{1}{2}^{\{2\}},-\frac{1}{3}^{\{2\}}, \frac{1}{3} \right) \\ u_6 = \left( 0^{\{4\}}, 1,-\frac{1}{3}^{\{2\}}, \frac{1}{3} \right) \\ \end{array}\)
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\({\tilde{E}}_{7}\), given in \({\mathbb {R}}^{8}\) |
\(\begin{array}{l} u_0 = \left( 0^{\{8\}}\right) \\ u_1 = \left( 0^{\{6\}},\frac{1}{2}, -\frac{1}{2}\right) \\ u_2 = \left( -\frac{1}{4}^{\{6\}}, \frac{1}{2}, -\frac{1}{2} \right) \\ u_3 = \left( \frac{1}{6}, -\frac{1}{6}^{\{5\}}, \frac{1}{2}, -\frac{1}{2} \right) \\ \end{array}\)
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\(\begin{array}{l} u_4 = \left( 0^{\{2\}}, -\frac{1}{4}^{\{4\}}, \frac{1}{2}, -\frac{1}{2} \right) \\ u_5 = \left( 0^{\{3\}}, -\frac{1}{3}^{\{3\}}, \frac{1}{2}, -\frac{1}{2} \right) \\ u_6 = \left( 0^{\{4\}}, -\frac{1}{2}^{\{2\}}, \frac{1}{2}, -\frac{1}{2} \right) \\ u_7 = \left( 0^{\{5\}}, -1, \frac{1}{2}, -\frac{1}{2} \right) \\ \end{array}\)
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\({\tilde{E}}_{8}\), given in \({\mathbb {R}}^{8}\) |
\(\begin{array}{l} u_0 = \left( 0^{\{8\}}\right) \\ u_1 = \left( 0^{\{7\}}, 1 \right) \\ u_2 = \left( \frac{1}{6}^{\{7\}}, \frac{5}{6} \right) \\ u_3 = \left( -\frac{1}{8},\frac{1}{8}^{\{6\}}, \frac{7}{8} \right) \\ u_4 = \left( 0^{\{2\}}, \frac{1}{6}^{\{5\}}, \frac{5}{6} \right) \\ \end{array}\)
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\(\begin{array}{l} u_5 = \left( 0^{\{3\}}, \frac{1}{5}^{\{4\}}, \frac{4}{5} \right) \\ u_6 = \left( 0^{\{4\}}, \frac{1}{4}^{\{3\}}, \frac{3}{4} \right) \\ u_7 = \left( 0^{\{5\}}, \frac{1}{3}^{\{2\}}, \frac{2}{3} \right) \\ u_8 = \left( 0^{\{6\}},\frac{1}{2}^{\{2\}}\right) \\ \end{array}\)
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\({\tilde{F}}_{4}\), given in \({\mathbb {R}}^{4}\) |
\(\begin{array}{l} u_0 = \left( 0, 0, 0, 0 \right) \\ u_1 = \left( \frac{1}{2}, \frac{1}{2}, 0, 0 \right) \\ u_2 = \left( \frac{2}{3}, \frac{1}{3}, \frac{1}{3}, 0 \right) \\ \end{array}\)
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\(\begin{array}{l} {}\\ u_3 = \left( \frac{3}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \right) \\ u_4 = \left( 1, 0, 0, 0 \right) \\ \end{array}\)
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\({\tilde{G}}_{2}\), given in \({\mathbb {R}}^{2}\) |
\(\begin{array}{l} u_0 = \left( 0, 0 \right) \\ u_1 = \left( 1, 0 \right) \end{array}\)
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\(\begin{array}{l} {}\\ u_2 = \left( 1, \sqrt{3} \right) \\ \end{array}\)
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3 Quality Definitions
- The aspect ratio of \(\sigma \) is the ratio of its minimal height to the diameter of its circumscribed ball: \(\alpha (\sigma ) = \frac{h(\sigma )}{2R(\sigma )}\).
- The fatness of \(\sigma \) is the ratio of its volume to its maximal edge length taken to the power \(d\): \(\varTheta (\sigma ) = \frac{vol(\sigma )}{L(\sigma )^{d}}\).
- The thickness of \(\sigma \) is the ratio of its minimal height to its maximal edge length: \(\theta (\sigma ) = \frac{h(\sigma )}{L(\sigma )}\).
- The radius ratio of \(\sigma \) is the ratio of its inradius to its circumradius: \(\rho (\sigma ) = \frac{r(\sigma )}{R(\sigma )}\).
4 Delaunay Criterion for Coxeter Triangulations
- If a simplex in \({\mathcal {T}}\) is not self-centred, then \({\mathcal {T}}\) is not Delaunay.
- If a simplex in \({\mathcal {T}}\) contains its circumcentre on the boundary, then \({\mathcal {T}}\) is Delaunay with zero protection.
- If a simplex in \({\mathcal {T}}\) contains its circumcentre strictly inside, then \({\mathcal {T}}\) is Delaunay with non-zero protection.
5 Main Result
Fatness \({\hat{\varTheta }}^{1/d}\) | Aspect Ratio \({\hat{\alpha }}\) | Thickness \({\hat{\theta }}\) | Radius Ratio \({\hat{\rho }}\) | |
---|---|---|---|---|
\({\tilde{A}}_{d},\,d\) odd | \(\frac{2^{3/2}}{\left( \sqrt{d+1}\right) ^{1+2/d}}\) | \(\sqrt{\frac{6d}{(d+1)(d+2)}}\) | \(\frac{2\sqrt{d}}{d+1}\) | \(\sqrt{\frac{6d}{(d+1)(d+2)}}\) |
\({\tilde{A}}_{d},\,d\) even | \(\frac{2^{3/2}(\sqrt{d+1})^{1-2/d}}{\sqrt{d(d+2)}}\) | \(\frac{2}{\sqrt{d+2}}\) | ||
\({\tilde{B}}_{d}\) | \(\frac{2^{1/2+1/d}}{\sqrt{d}\left( \sqrt{d+1}\right) ^{1/d}}\) | \(\frac{d\sqrt{2}}{(d+1)\sqrt{d+2}}\) | \(\frac{1}{\sqrt{d+1}}\) | \(\frac{2d}{\sqrt{d+2}(1+(d-1)\sqrt{2})}\) |
\({\tilde{C}}_{d}\) | \(\frac{\sqrt{2}}{\sqrt{d}\left( \sqrt{d+1}\right) ^{1/d}}\) | \(\frac{\sqrt{2d}}{d+1}\) | \(\frac{1}{\sqrt{d+1}}\) | \(\frac{2\sqrt{d}}{2+(d-1)\sqrt{2}}\) |
\({\tilde{D}}_{d}\) | \(\frac{2^{1/2+2/d}}{\sqrt{d}\left( \sqrt{d+1}\right) ^{1/d}}\) | \(\frac{d\sqrt{2}}{(d+1)\sqrt{d+4}}\) | \(\frac{1}{\sqrt{d+1}}\) | \(\frac{d\sqrt{2}}{(d-1)\sqrt{d+4}}\) |
\({\tilde{E}}_{6}\) | \(\root 12 \of {\frac{64}{137781}}\) | \(\frac{2}{7}\) | \(\frac{1}{\sqrt{14}}\) | \(\frac{1}{2}\) |
\({\tilde{E}}_{7}\) | \(\root 14 \of {\frac{1}{177147}}\) | \(\frac{7\sqrt{13}}{104}\) | \(\frac{\sqrt{21}}{24}\) | \(\frac{14\sqrt{13}}{117}\) |
\({\tilde{E}}_{8}\) | \(\root 8 \of {\frac{1}{3240}}\) | \(\frac{8\sqrt{19}}{171}\) | \(\frac{2\sqrt{19}}{57}\) | \(\frac{8\sqrt{19}}{95}\) |
\({\tilde{F}}_{4}\) | \(\root 8 \of {\frac{1}{405}}\) | \(\frac{4\sqrt{2}}{15}\) | \(\frac{2\sqrt{5}}{15}\) | \(\frac{4\sqrt{2}}{3(2+\sqrt{2})}\) |
\({\tilde{G}}_{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{1}{\sqrt{3}}\) | \(\frac{1}{2}\) | \(\frac{2}{1+\sqrt{3}}\) |
Fatness \(\varTheta \) | Aspect Ratio \(\alpha \) | Thickness \(\theta \) | Radius Ratio \(\rho \) | |
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\(\varDelta \) | \(\frac{1}{d!}\sqrt{\frac{d+1}{2^d}}\) | \(\frac{d+1}{2d}\) | \(\sqrt{\frac{d+1}{2d}}\) | \(\frac{1}{d}\) |
\(d\) | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|
\({\tilde{A}}_{d}\) | 1.000 | 0.949 | 0.894 | 0.845 | 0.802 | 0.764 | 0.730 |
\({\tilde{B}}_{d}\) | – | 0.474 | 0.462 | 0.445 | 0.429 | 0.412 | 0.398 |
\({\tilde{C}}_{d}\) | 0.666 | 0.612 | 0.566 | 0.527 | 0.495 | 0.468 | 0.444 |
\({\tilde{D}}_{d}\) | – | – | 0.400 | 0.393 | 0.383 | 0.371 | 0.363 |
\({\tilde{E}}_{d}\) | – | – | – | – | 0.286 | 0.243 | 0.204 |
\({\tilde{F}}_{d}\) | – | – | 0.377 | – | – | – | – |
\({\tilde{G}}_{d}\) | 0.577 | – | – | – | – | – | – |
\(d\)
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|
\({\tilde{A}}_{d}\)
| 1.000 | 0.949 | 0.894 | 0.845 | 0.802 | 0.764 | 0.730 |
\({\tilde{B}}_{d}\)
| – | 0.701 | 0.623 | 0.568 | 0.526 | 0.492 | 0.464 |
\({\tilde{C}}_{d}\)
| 0.828 | 0.717 | 0.641 | 0.584 | 0.540 | 0.505 | 0.475 |
\({\tilde{D}}_{d}\)
| – | – | 0.667 | 0.589 | 0.537 | 0.497 | 0.467 |
\({\tilde{E}}_{d}\)
| – | – | – | – | 0.500 | 0.431 | 0.367 |
\({\tilde{F}}_{d}\)
| – | – | 0.553 | – | – | – | – |
\({\tilde{G}}_{d}\)
| 0.732 | – | – | – | – | – | – |
\(d\)
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|
\({\tilde{A}}_{d}\)
| 1.000 | 0.891 | 0.864 | 0.807 | 0.781 | 0.743 | 0.721 |
\({\tilde{B}}_{d}\)
| – | 0.816 | 0.688 | 0.607 | 0.551 | 0.509 | 0.475 |
\({\tilde{C}}_{d}\)
| 0.760 | 0.648 | 0.579 | 0.529 | 0.491 | 0.461 | 0.436 |
\({\tilde{D}}_{d}\)
| – | – | 0.818 | 0.697 | 0.619 | 0.562 | 0.518 |
\({\tilde{E}}_{d}\)
| – | – | – | – | 0.528 | 0.422 | 0.363 |
\({\tilde{F}}_{d}\)
| – | – | 0.472 | – | – | – | – |
\({\tilde{G}}_{d}\)
| 0.707 | – | – | – | – | – | – |
\(d\)
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|
\({\tilde{A}}_{d}\)
| 1.000 | 0.866 | 0.816 | 0.745 | 0.707 | 0.661 | 0.632 |
\({\tilde{B}}_{d}\)
| – | 0.500 | 0.447 | 0.408 | 0.378 | 0.354 | 0.333 |
\({\tilde{C}}_{d}\)
| 0.577 | 0.500 | 0.447 | 0.408 | 0.378 | 0.354 | 0.333 |
\({\tilde{D}}_{d}\)
| – | – | 0.447 | 0.408 | 0.378 | 0.354 | 0.333 |
\({\tilde{E}}_{d}\)
| – | – | – | – | 0.268 | 0.191 | 0.153 |
\({\tilde{F}}_{d}\)
| – | – | 0.298 | – | – | – | – |
\({\tilde{G}}_{d}\)
| 0.500 | – | – | – | – | – | – |