Regular 3-polytopes of order 2^𝑛𝑝

Abstract

In a paper from 2006, Schulte and Weiss initiated the problem of characterizing the regular polytopes of orders 2 n ⁢ p for 𝑛 a positive integer and 𝑝 an odd prime. In this paper, we first prove that if a 3-polytope of order 2 n ⁢ p has Schläfli type { k 1 , k 2 } , then p ∣ k 1 or p ∣ k 2 . This leads to two classes, up to duality, for the Schläfli type, namely Type (1) where k 1 = 2 s ⁢ p and k 2 = 2 t and Type (2) where k 1 = 2 s ⁢ p and k 2 = 2 t ⁢ p . We then show that there exists a regular 3-polytope of order 2 n ⁢ p with Type (1) when s ≥ 2 , t ≥ 2 and n ≥ s + t + 1 coming from a general construction of regular 3-polytopes of order 2 n ⁢ ℓ 1 ⁢ ℓ 2 with Schläfli type { 2 s ⁢ ℓ 1 , 2 t ⁢ ℓ 2 } , where both ℓ 1 and ℓ 2 are odd. Furthermore, for p = 3 and n ≥ 7 , we show that there exists a regular 3-polytope of order 3 ⋅ 2 n with type { 6 , 2 s } if and only if 2 ≤ s ≤ n − 2 and s ≠ n − 3 . For Type (2), we prove that there exists a regular 3-polytope of order 2 n ⋅ 3 with Schläfli type { 6 , 6 } when n ≥ 5 coming from a general construction of regular 3-polytopes of Schläfli type { 6 , 6 } with orders 192 ⁢ m 3 , 384 ⁢ m 3 or 768 ⁢ m 3 for any positive integer 𝑚.