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Similarly, a k-isohedral tiling has k separate symmetry orbits (it may contain m different face shapes, for m = k, or only for some m < k). [ 6 ] ("1-isohedral" is the same as "isohedral".) A monohedral polyhedron or monohedral tiling ( m = 1) has congruent faces, either directly or reflectively, which occur in one or more symmetry positions.
(Definition varies among authors; e.g. some exclude solids with dihedral symmetry, or nonconvex solids.) Uniform if every face is a regular polygon, i.e. it is regular, quasiregular or semi-regular. Semi-uniform if its elements are also isogonal. Scaliform if all the edges are the same length. Noble if it is also isohedral (face-transitive).
Gray's Anatomy is a reference book of human anatomy written by Henry Gray, illustrated by Henry Vandyke Carter and first published in London in 1858. It has had multiple revised editions, and the current edition, the 42nd (October 2020), remains a standard reference, often considered "the doctors' bible ".
Each Catalan solid has constant dihedral angles, meaning the angle between any two adjacent faces is the same. [1] Additionally, two Catalan solids, the rhombic dodecahedron and rhombic triacontahedron, are edge-transitive, meaning their edges are symmetric to each other.
This shape is called a plesiohedron. The tiling generated in this way is isohedral, meaning that it not only has a single prototile ("monohedral") but also that any copy of this tile can be taken to any other copy by a symmetry of the tiling. [1] As with any space-filling polyhedron, the Dehn invariant of a plesiohedron is necessarily zero. [3]
When depicted in Leonardo's form, with equilateral triangle faces, it is an example of a non-convex deltahedron, one of the few known deltahedra that are isohedral (meaning that all faces are symmetric to each other). [4]
The dual of a non-convex polyhedron is also a non-convex polyhedron. [2] ( By contraposition.) There are ten non-convex isotoxal polyhedra based on the quasiregular octahedron, cuboctahedron, and icosidodecahedron: the five (quasiregular) hemipolyhedra based on the quasiregular octahedron, cuboctahedron, and icosidodecahedron, and their five (infinite) duals:
In their book The Fifty-Nine Icosahedra, Coxeter et al. enumerated 59 such stellations of the regular icosahedron. Of these, many have a single face in each of the 20 face planes and so are also icosahedra. The great icosahedron is among them. Other stellations have more than one face in each plane or form compounds of simpler polyhedra.
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