The Hebesphenomegacorona
The hebesphenomegacorona is the 89th Johnson solid (J89). Its surface consists of 21 regular polygons (18 triangles and 3 squares), 33 edges, and 14 vertices.
The hebesphenomegacorona 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. As Norman Johnson explains, a lune is a square with two opposite edges attached to equilateral triangles, and a spheno (Latin for wedge) complex is two lunes joined together to form a wedgelike structure. A hebespheno complex is a larger, blunter complex of three lunes. A corona is a crownlike complex of 8 triangles, whereas a megacorona is a larger corona with 12 triangles. Hebesphenomegacorona, therefore, describes the structure built from a hebespheno complex attached to a megacorona of 12 triangles.
Projections
Here are some views of the hebesphenomegacorona from various angles:
Projection  Description 

Top view. 

Front view. 

Side view. 

45° side view. 
Coordinates
The Cartesian coordinates of the hebesphenomegacorona with edge length 2 are:
 (±1, ±1, 2B)
 (±1, ±(1+2A), 0)
 (±(1 + C/√(1−A)), 0, −(2A^{2}+A−1)/B)
 (0, ±1, −D)
 (±(DC+E)/(FE), 0, ((2A−1)D)/F − C/(FE))
where:
 B = √(1−A^{2})
 C = √(2(1−2A))
 D = √(3−4A^{2})
 E = √(1+A)
 F = 2(1−A)
and A is the root of the following polynomial between 0.2 and 0.22:
26880A^{10} + 35328A^{9} − 25600A^{8} − 39680A^{7} + 6112A^{6} + 13696A^{5} + 2128A^{4} − 1808A^{3} − 1119A^{2} + 494A − 47 = 0
The approximate value of A is 0.216844815713457.
Credits
The above coordinates were adapted from the solution published by
A. V. Timofeenko (2008), The nonPlatonic and nonArchimedean
noncomposite polyhedra
, Fundamentalnaya i Prikladnaya Matematika,
vol.14 (2), pp.179205. (А. В. Тимофеенко (2008)
«Несоставные многогранники, отличные от тел Платона и Архимеда»,
Фундаментальная и Прикладная Математика, том 14й (2),
ст.179205.)
An error in the printed coordinates of the last set of vertices in the paper was corrected: a factor of 2 in the denominator of the first term in the last coordinate was misprinted as 4, thus effectively turning ((2A−1)D)/F into ((2A−1)D)/(2F). This error appears to be a mere typo; the derivation itself is correct, and the correct coordinates can be derived from it. This correction was pointed out by Loïs Mignard.