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A skew zig-zag dodecagon has vertices alternating between two parallel planes. A regular skew dodecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew dodecagon and can be seen in the vertices and side edges of a hexagonal antiprism with the same D 5d, [2 +,10] symmetry, order 20. The dodecagrammic ...
The area density (also known as areal density, surface density, superficial density, areic density, mass thickness, column density, or density thickness) of a two-dimensional object is calculated as the mass per unit area. The SI derived unit is the "kilogram per square metre" (kg·m −2).
If the edge length of a regular dodecahedron is , the radius of a circumscribed sphere (one that touches the regular dodecahedron at all vertices), the radius of an inscribed sphere (tangent to each of the regular dodecahedron's faces), and the midradius (one that touches the middle of each edge) are: [21] =, =, =. Given a regular dodecahedron ...
The volume is computed as F times the volume of the pyramid whose base is a regular p-gon and whose height is the inradius r. That is, =. The following table lists the various radii of the Platonic solids together with their surface area and volume.
The area of an ellipse is proportional to a rectangle having sides equal to its major and minor axes; The volume of a sphere is 4 times that of a cone having a base of the same radius and height equal to this radius; The volume of a cylinder having a height equal to its diameter is 3/2 that of a sphere having the same diameter;
The surface area A and the volume V of the rhombic dodecahedron with edge length a are: [4] =, =. The rhombic dodecahedron can be viewed as the convex hull of the union of the vertices of a cube and an octahedron where the edges intersect perpendicularly.
The 120-cell whose coordinates are given above of long radius √ 8 = 2 √ 2 ≈ 2.828 and edge-length 2 / φ 2 = 3− √ 5 ≈ 0.764 can be constructed in this manner just outside a 600-cell of long radius φ 2, which is smaller than √ 8 in the same ratio of ≈ 0.926; it is in the golden ratio to the edge length of the 600-cell ...
For example, a truncated pentagon {5 ⁄ 1} becomes a decagon {10 ⁄ 1}, so truncating a pentagram {5 ⁄ 2} becomes a doubly-wound pentagon {10 ⁄ 2} (the common factor between 10 and 2 mean we visit each vertex twice to complete the polygon).