The Snub Disphenoid
The snub disphenoid is the 84th Johnson solid (J84). Its surface consists of 12 equilateral triangles.
It is also known as the Siamese dodecahedron
because it has 12
faces and can be formed by conjoining two regular
octahedra each with a pair of triangles removed and the remaining 6 faces
slightly deformed, like Siamese twins (conjoined twins).
It is one of the special Johnson solids at the end of Norman Johnson's list that are not directly derived from the uniform polyhedra by cutandpaste operations.
Projections
Here are some views of the snub disphenoid from various angles:
Projection  Envelope  Description 

Square  Top view. 

Irregular pentagon  Side view. 

Irregular octagon  Oblique side view at 45° angle. 
Coordinates
The Cartesian coordinates of the snub disphenoid, centered on the origin with edge length 2, are:
 (0, √A, ±1)
 (±√C, √B, 0)
 (0, √B, ±√C)
 (±1, √A, 0)
where A, B, and C are the roots of the following polynomials within the indicated ranges:
2A^{3}  A^{2}  8A  4 = 0,  2 ≤ A ≤ 3 
2B^{3} + 11B^{2} + 4B  1 = 0,  0 ≤ B ≤ 1 
C^{3}  17C^{2} + 64C  64 = 0,  1 ≤ C ≤ 2 
Their numerical values are approximately:
 A = 2.458190775872486
 B = 0.169022229424176
 C = 1.661955541151648
Their square roots are approximately:
 √A = 1.567861848465127
 √B = 0.411123131706519
 √C = 1.289168546448310