The Augmented Sphenocorona
The augmented sphenocorona is the 87th Johnson solid (J87). Its surface consists of 16 equilateral triangles and 1 square, for a total of 17 faces, 26 edges, and 11 vertices.
It is constructed by augmenting one of the square faces of the sphenocorona (J86) with a square pyramid. The triangular face of the augment that touches the remaining square face is very nearly coplanar: their dihedral angle is approximately 171.8°.
Projections
Here are some views of the augmented sphenocorona from various angles:
Projection | Description |
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Top view. |
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Side view. |
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Front view. |
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45° side view. |
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Parallel to line from augment apex to upper left vertex. |
Coordinates
The Cartesian coordinates of the augmented sphenocorona with edge length 2 are:
- (0, 0, ±1)
- (±A, √B, ±1)
- (0, √C, ±D)
- (±1, √E, 0)
- ((A+B√2)/2, (B−A√2)/2, 0)
where A, B, C, D, and E are roots of the following polynomials within the given ranges:
15A4 − 24A3 − 100A2 + 112A + 92 = 0, | 1≤A≤2 |
225B4 − 24B3 − 3176B2 − 96B + 3600 = 0, | 1≤B≤2 |
225C4 − 24C3 − 3176C2 − 96C + 3600 = 0, | 3≤C≤4 |
15D4 − 36D3 − 82D2 + 100D + 95 = 0, | 1≤D≤2 |
E2 − 4E − 20 = 0 | 6≤E≤7 |
Note that B and C are different roots of the same polynomial. E has the closed-form expression 2+2√6.
The approximate numerical values are:
- A = 1.705453885692834
- √B = 1.044713857367277
- √C = 1.914399800381786
- D = 1.578855253321743
- √E = 2.626590848527109