Isogonal polytope

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An isogonal polytope or vertex-transitive polytope is a polytope (or polytope-like object) whose vertices are identical under its symmetry group. In other words, given any two vertices, there is a symmetry of the polytope that transforms one into the other.

Possibly, the earliest mention of the concept of an isogonal figure is found in the Rigveda (ca. 1500-1000 BCE), an ancient text written in Vedic Sanskrit, where it states tásmin sākáṃ triśatā́ ná śaṅkávo 'rpitā́ḥ ṣaṣṭír ná calācalā́saḥ (Therein are set together spokes three hundred and sixty, which in nowise can be loosened.), referring to the transitivity of the so-called "pegs" (śaṅkú) which connect the vertices of the circle to its center.[1]

Properties and terminology[edit | edit source]

An isogonal polytope has a singular vertex figure. (However, polytopes with a single vertex figure are not necessarily isogonal, such as the pseudo-uniform polyhedra.) The dual of an isogonal polytope is an isotopic polytope, which are made out of one facet type.

Restricting discussion to finite planar polytopes, all vertices of an isogonal n -polytope lie on an (n  - 1)-hypersphere, and every proper element of rank r  is inscribed in an (r  - 1)-hypersphere. The convex hull of an isogonal polytope is also isogonal.

An isogonal polytope does not necessarily have all its elements isogonal. An example is the triangular gyroprism, which in general form has scalene triangles. They do not even need to be abstractly isogonal, such as in the scaliform polychora.

Subsets of isogonal polytopes include:

Isogonality applies directly to abstract polytopes, compounds, tilings, skew polytopes, exotic polytopoids, and certain types of incidence geometries where the notion of a vertex is specified (such as complex polytopes).

Classification[edit | edit source]

Isogonal polytopes are rarely studied as a group on their own, as they become very broad in high dimensions.

Isogonal polygons[edit | edit source]

The isogonal polygons are easily characterized, comprising the regular polygons and the semi-uniform polygons. The latter are formed by taking a regular polygon with an even number of sides and altering every other side length. There are uncountably many semi-uniform polygons due to the continuum of side length ratios.

Isogonal polyhedra[edit | edit source]

The (finite and planar) isogonal polyhedra are a very broad set that has not been fully characterized by any reasonable standard.[2] As almost all such polyhedra can have their edge lengths continuously varied, one question would be to classify all abstract polyhedra that have isogonal realizations. Unfortunately, even the sub-problem of classifying all abstractly distinct noble polyhedra is open.

The convex isogonal polyhedra are fully characterized, and are all formed by varying the edge lengths of the convex uniform polyhedra.

In 1997, Grunbaum investigated the 3D isogonal prismatoids, which include the gyroprisms and crown polyhedra.

Isogonal polychora[edit | edit source]

Isogonality is an even weaker condition for 4D and above. There are convex isogonal polychora that cannot be formed by edge length modification of the convex uniform polychora. Most notably, many but not all convex isogonal swirlchora fit this description.

Types of convex isogonal polytopes[edit | edit source]

Polygons[edit | edit source]

  • Regular polygons (infinite, half symmetry variants exist for even-sided polygons with two alternating edge lengths)

Polyhedra[edit | edit source]

Polychora[edit | edit source]

Polytera[edit | edit source]

Types of convex isogonal elementary tessellations[edit | edit source]

Tilings[edit | edit source]

Honeycombs[edit | edit source]

Optimization[edit | edit source]

Many isogonal polytopes can be continuously deformed until they have a single edge length and therefore become scaliform or uniform. However, not all isogonal polytopes have this property, such as the snub decachoron, due to having fewer variables than edge length types. The grand antiprism is the only known convex exception which can be made uniform, having two variables and three edge length types.

Most isogonal polytopes can be optimized in the sense of minimizing the edge length variations. However, some isogonal polytopes cannot be optimized in any meaningful way, such as the antiditetragoltriates, because their optimized forms are not topologically identical to these shapes. If an isogonal polytope has only one variable (its size) in its highest symmetry, such as the triangular duoantiprism, it is considered to be optimized.

There are two methods of optimization: the absolute-value method and the ratio method, which may yield different solutions. Both methods rely on a set variable, defined by a constant. All regular, uniform and scaliform polytopes are optimized polytopes in both of the two methods.

Absolute-value method[edit | edit source]

This method uses all possible sums of absolute values of differences between two edge length types, and then computing the minimum value of the resulting function. The number of sums required for an isogonal polytope with n edge types is n(n-1)/2. The feasibility of this method depends on the number of variables and the number of edge lengths.

As an example, a truncated octahedron in B3 symmetry can be defined by all permutations and sign changes of (0, a, b), where a and b are nonzero and distinct from each other. The variable a can be set to a constant value such as 1, yielding (0, 1, b). The two edge lengths are given by:

  • d1 = 2 (distance between (0, 1, b) and (1, 0, b))
  • d2 = 2|b-1| (distance between (0, 1, b) and (0, b, 1))

Since there are only two edge lengths, only one absolute value sum is needed: f(x) = |2-2|b-1||. The minimum of this function is attained when b is equal to 2, which evaluates to 0. Therefore, any optimized truncated octahedron in this sense is equilateral and hence uniform.

Ratio method[edit | edit source]

This method is dependent on the polytope's largest and smallest edge lengths. It involves dividing the largest edge length by the smallest edge length, giving a value equal to or greater than 1, and then finding the lowest possible value. Due to the unpredictability of edge lengths per given variables, a convenient method is to enumerate all divisions between two edge types and their reciprocals, and then finding the lowest possible value greater than (or equal to) 1 that is not within the "area" of the functions. For an isogonal polytope with n edge lengths, n(n-1) divisions are required. For topologically similar isogonal polytopes, the largest and smallest edge lengths may vary between edge length types depending on the variable used.

We can use the truncated octahedron example from earlier. For b > 2, 2|b-1| is larger than 2, and for 1 < b < 2, 2|b-1| is smaller than 2. Dividing the appropriate edge lengths results in a minimum ratio of 1:1 at b = 2, implying that any optimized truncated octahedron in this sense is equilateral and hence uniform.

It is more useful in most cases than the absolute-value method, especially when an isogonal polytope has no uniform realization. The optimized form in this case is the closest to what a "uniform" variant would look like, whereas the result obtained through the absolute-value method has a bigger ratio in some cases. Notable examples are the step prisms, which are more reliably optimized generally than the absolute-value method.

References[edit | edit source]