Regular 4-polytopes Research Articles

Oscillating 4-Polytopal Universe in Regge Calculus

Abstract The discretized closed Friedmann–Lemaître–Robertson–Walker (FLRW) universe with positive cosmological constant is investigated by Regge calculus. According to the Collins–Williams formalism, a hyperspherical Cauchy surface is replaced with regular 4-polytopes. Numerical solutions to the Regge equations approximate well to the continuum solution during the era of small edge length. Unlike the expanding polyhedral universe in three dimensions, the 4-polytopal universes repeat expansions and contractions. To go beyond the approximation using regular 4-polytopes we introduce pseudo-regular 4-polytopes by averaging the dihedral angles of the tessellated regular 600-cell. The degree of precision of the tessellation is called the frequency. Regge equations for the pseudo-regular 4-polytope have simple and unique expressions for any frequency. In the infinite frequency limit, the pseudo-regular 4-polytope model approaches the continuum FLRW universe.

High-dimensional Private Quantum Channels and Regular Polytopes

As the quantum analog of the classical one-time pad, the private quantum channel (PQC) plays a fundamental role in the construction of the maximally mixed state (from any input quantum state), which is very useful for studying secure quantum communications and quantum channel capacity problems. However, the undoubted existence of a relation between the geometric shape of regular polytopes and private quantum channels in the higher dimension has not yet been reported. Recently, it was shown that a one-to-one correspondence exists between single-qubit PQCs and three-dimensional regular polytopes (i.e., regular polyhedra). In this paper, we highlight these connections by exploiting two strategies known as a generalized Gell-Mann matrix and modified quantum Fourier transform. More precisely, we explore the explicit relationship between PQCs over a qutrit system (i.e., a three-level quantum state) and regular 4-polytopes. Finally, we attempt to devise a formula for such connections on higher dimensional cases.

The intersection condition for regular polytopes

The automorphism group of every abstract regular polytope of type {k1|⋯|km} is a smooth finite quotient of the corresponding Coxeter group [k1,…,km] satisfying a particular condition on subgroups known as the intersection condition (or intersection property), and vice versa. In this paper, we consider the question of how many cases of the intersection condition need to be checked, sometimes under additional hypotheses about the finite quotient group or the type of the polytope. We also apply this to the construction of a family of polytopal regular maps of every possible genus, and an infinite family of locally spherical regular 4-polytopes of type {3|5|3}, with all but finitely many alternating groups as automorphism groups.

Regular 4-polytopes from the Livingstone graph of Janko’s first group

The Janko group J 1 has, up to duality, exactly two regular rank four polytopes, of respective Schlafli types {5,3,5} and {5,6,5}. The aim of this paper is to give geometric constructions of these two polytopes, starting from the Livingstone graph.

Modular Reduction in Abstract Polytopes

AbstractThe paper studies modular reduction techniques for abstract regular and chiral polytopes, with two purposes in mind: first, to survey the literature about modular reduction in polytopes; and second, to apply modular reduction, with moduli given by primes in ℤ[τ] (with τ the golden ratio), to construct new regular 4-polytopes of hyperbolic types ﹛3, 5, 3﹜ and ﹛5, 3, 5﹜ with automorphism groups given by finite orthogonal groups.

Hamiltonian Submanifolds of Regular Polytopes

We investigate polyhedral 2k-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex k -Hamiltonian if it contains the full k-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called super-neighborly triangulations), we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the d-dimensional cross polytope. These are the “regular cases” satisfying equality in Sparla’s inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of seven copies of S 2×S 2. By this example all regular cases of n vertices with n<20 or, equivalently, all cases of regular d-polytopes with d≤9 are now decided.

Locally toroidal regular polytopes of rank 4

The paper studies various relationships between locally toroidal regular 4-polytopes of types {6, 3,p} and {3, 6, 3}. These relationships are based on corresponding relationships between the regular honeycombs with the same Schlafli-symbol in hyperbolic 3-space. Also the paper discusses regular tessellations (secretions of rank 3) which are locally inscribed into regular 4-polytopes. In particular, this leads to local criteria for the finiteness of the polytopes.

Regular polytopes of type {4,4,3} and {4,4,4}

Abstract regular polytopes generalize the classical concept of a regular polytope and regular tessellation to more complicated combinatorial structures with a distinctive geometrical and topological flavour. In this paper the authors give an almost complete classification of the (universal) locally toroidal regular 4-polytopes of Schlafli types {4,4,3} and {4,4,4}.

Primary Hypergeodesic Polytopes

In this paper, we introduce the concept of a hypergeodesic polytope, as a four-dimensional analogue of the well-known geodesic polyhdron in 3-space. In particular, we describe twenty-eight types of primary hypergeodesic polytopes, those which are most typical, and which are most likely to be employed in structural design. All of these primary hypergeodesic polytopes are derived from two regular 4-polytopes, the 120- and the 600-cell. They outwardly resemble 3-dimensional geodesic domes having one or two layers of framework, and possess many desirable architectural characteristics. Cartesian coordinates of their vertices and chord factors can be determined from the figures and tables provided in the appendices to this article.

Self-Dual Regular 4-Polytopes and Their Petrie-Coxeter-Polyhedra

Coxeter’s regular skew polyhedra in euclidean 4-space IE4 are intimately related to the self-dual regular 4-polytopes. The same holds for two of the three Petrie-Coxeter-polyhedra and the (self-dual) cubical tessellation in IE3.In this paper we discuss the combinatorial Petrie-Coxeter-polyhedra associated with the self-dual regular 4-incidence-polytopes.