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Other tests involve determining how much area overlaps with a circle of the same area [2] or a reflection of the shape itself. [1] Compactness measures can be defined for three-dimensional shapes as well, typically as functions of volume and surface area. One example of a compactness measure is sphericity.
If the shape is considered as a union of five cubes yielding a simple nonconvex solid without self-intersecting surfaces, then it has 360 faces (all triangles), 182 vertices (60 with degree 3, 30 with degree 4, 12 with degree 5, 60 with degree 8, and 20 with degree 12), and 540 edges, yielding an Euler characteristic of 182 − 540 + 360 = 2.
A compound of two spherical tetrahedra can be constructed, as illustrated. The two tetrahedra of the compound view of the stellated octahedron are "desmic", meaning that (when interpreted as a line in projective space) each edge of one tetrahedron crosses two opposite edges of the other tetrahedron. One of these two crossings is visible in the ...
The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. Its interior is an octahedron, and correspondingly, a regular octahedron is the result of cutting off, from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e., rectifying the tetrahedron).
However, there are other shapes of constant width. A standard example is the Reuleaux triangle , the intersection of three circles, each centered where the other two circles cross. [ 2 ] Its boundary curve consists of three arcs of these circles, meeting at 120° angles, so it is not smooth , and in fact these angles are the sharpest possible ...
Regular polygrams {n/d}, with red lines showing constant d, and blue lines showing compound sequences k{n/d} In geometry, a generalized polygon can be called a polygram, and named specifically by its number of sides. All polygons are polygrams, but they can also include disconnected sets of edges, called a compound polygon.
This definition rules out, for example, the square pyramid (since although all the faces are regular, the square base is not congruent to the triangular sides), or the shape formed by joining two tetrahedra together (since although all faces of that triangular bipyramid would be equilateral triangles, that is, congruent and regular, some ...
The polytopes of rank 2 (2-polytopes) are called polygons.Regular polygons are equilateral and cyclic.A p-gonal regular polygon is represented by Schläfli symbol {p}.. Many sources only consider convex polygons, but star polygons, like the pentagram, when considered, can also be regular.