Generating Polygons with Triangles - Discrete and Computational Geometry and Graphs - page 120

v 1 + v 2 + v 3

v 1 + v 2

v 1

O

Fig. 3. The orthogonal Hill-tetrahedron Q 3 ( ˀ/ 2)

3 by
translations only. Akiyama et al. [ 1 ] proved that for any parallelohedron ʠ, there
is an ane transformation f of

A 3-dimensional convex polytope is called a parallelohedron if it tiles R

3 such that f (ʠ) is generated by the orthogonal

R

Hill-tetrahedron Q 3 ( ˀ/ 2).
The simplicial element numbers of the families of regular d -polytopes for d

≥
2 are determined by Akiyama et al. [ 2 - 4 ] as shown in Table 2 . (Their “element
number” is slightly different from our simplicial element number, but Table 2
follows from their works.)

Table 2. e (the d -dimensional regular polytopes)

d # of regular polytopes simplicial element number

2 ∞ ∞

3 5 4

4 6 4

≥ 5 3 3

Acknowledgment. Many thanks to the referees for valuable comments.

References

1. Akiyama, J., Kobayashi, M., Nakagawa, H., Nakamura, G., Sato, I.: Atoms for par-
allelohedra. Bolyai Society Mathematical Studies 24 (2013); Geometry-Intuitive,
Discrete, and Convex, pp. 1-21
2. Akiyama, J., Hitotumatu, S., Sato, I.: Determination of the element number of the
regular polytopes. Geom. Dedicata 159 , 89-97 (2012)
3. Akiyama, J., Sato, I.: The element number of the convex regular polytopes. Geom.
Didicata 151 , 269-278 (2011)
4. Akiyama, J., Maehara, H., Nakamura, G., Sato, I.: Element number of the platonic
solids. Geom Dedicata 145 , 181-193 (2010)