TOPICS
Search

Snub Disphenoid

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
J84
J84Net

The 12-faced convex deltahedra also known as the Siamese dodecahedron, which is also Johnson solid J_(84).

It is implemented in the Wolfram Language as PolyhedronData["SnubDisphenoid"].

SnubDisphenoidCoords

The coordinates of the polyhedron vertices of a snub disphenoid of unit side length may be found by solving the set of four simultaneous equations

(1/2)^2+x_2^2+z_1^2=1
(1)
(x_2-1/2)^2+(z_3-z_1)^2=1
(2)
(1/2)^2+x_2^2+(z_3-z_2)^2=1
(3)
x_2^2+x_2^2+(z_2-z_1)^2=1
(4)

for the four unknowns x_2, z_1, z_2, and z_3. The analytic solution requires solving the cubic equation, and the solutions are given by the smallest positive real roots of

2x_2^3-3x_2^2-2x_2+2=0
(5)
32z_1^6+64z_1^4-22z_1^2-1=0
(6)
16z_2^6+8z_2^4-15z_2^2-8=0
(7)
2z_3^6-z_3^4-8z_3^2-4=0.
(8)

Numerically,

x_2 approx 0.644584
(9)
z_1 approx 0.578369
(10)
z_2 approx 0.989492
(11)
z_3 approx 1.56786.
(12)

The surface area of the unit snub disphenoid is

S=3sqrt(3),
(13)

and the volume V is given by the positive real root of

5832V^6-1377V^4-2160V^2-4=0,
(14)

approximately V approx 0.859494.

See also

Deltahedron, Disphenoid, Johnson Solid

Explore with Wolfram|Alpha

References

Timofeenko, A. V. "The Non-Platonic and Non-Archimedean Noncomposite Polyhedra." J. Math. Sci. 162, 710-729, 2009.

Cite this as:

Weisstein, Eric W. "Snub Disphenoid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SnubDisphenoid.html

Subject classifications