Derivation of snub disphenoid coordinates (no ontology)

From Hi.gher. Space

The following shows how the coordinates of the snub disphenoid are derived.

Initial setup and constraints

First, we write the general form of the coordinates based on the hemitetragonal symmetry of the snub disphenoid, centered on the origin, as follows. We will assume an edge length of 2.

<0, A, ±1>

<±C, B, 0>

<0, -B, ±C>

<±1, -A, 0>

where A, B, C > 0, and A > B.

Based on the fact that edge lengths must be uniformly 2, we have the following constraints:

1. ||<C, B, 0> - <0, A, 1>|| = ||<C, B-A, -1>|| = 2
2. ||<0, -B, C> - <0, A, 1>|| = ||<0, -B-A, C-1>|| = 2
3. ||<0, -B, C> - <C, B, 0>|| = ||-C, -2B, C>|| = 2

where ||x|| denotes the vector length of x, that is, √(x₁² + x₂² + x₃²), where x = <x₁, x₂, x₃>.

For convenience, we also note that the angle between the edge from <0, A, 1> to <C, B, 0> and the edge from <0, A, 1> to <0, -B, -C> must be 60°, since they form part of one of the equilateral triangles of the snub disphenoid. Thus, their dot product should equal 2*2*cos(60°) (since all edges have length 2):

(<C, B, 0> - <0, A, 1>)∙(<0, -B, -C> - <0, A, 1>) = 4*cos(60°) = 4*(1/2) = 2

Expanding the left-hand side and applying a little algebra, we arrive at:

A² - B² = C + 1

Strictly speaking, we don't need this equation, because it can be derived from the preceding three length constraints described above. However, it is in a convenient form for the algebra that will follow. It is also possible to derive more constraints by requiring the angle between the other edges to be 60°, but again, these are already subsumed by the three length constraints above, and they are not needed for the following derivation.

In summary, then, after applying some algebraic simplifications of the above constraints, our initial constraints are:

A2 + B2 + C2 - 2AB = 3	[Equation 1]
A2 + B2 + C2 + 2AB - 2C = 3	[Equation 2]
C2 + 2B2 = 2	[Equation 3]
A2 - B2 = C + 1	[Equation 4]

subject to the inequalities:

A > B > 0

C > 0

While these inequalities are pretty obvious, it's useful to state them explicitly because they will come in handy later when we need to decide which of multiple possible polynomial roots are admissible.

Solving for B

The first thing we notice about the initial system of equations is that equation 1 and equation 2 are very similar, and we can eliminate many of the squared terms by subtracting these two equations. Then applying a little algebraic rearrangement, we obtain the following relation between A, B, and C:

C = 2AB [Equation 7]

Eliminating C

Now we can use equation 7 to eliminate C from equations 3 and 4. First, we substitute equation 7 into equation 4:

A² - B² = 2AB + 1

A² - 2AB - B² = 1

In order to separate the variables A and B, we complete the square for (A² - 2AB) by adding the term +B²:

A² - 2AB + (B² - B²) - B² = 1

(A² - 2AB + B²) - B² - B² = 1

(A - B)² - 2B² = 1

(A - B)² = 1 + 2B²

A - B = ±√(1 + 2B²)

But since A > B according to our initial constraints, (A - B) must be positive, so we select the positive root:

A - B = √(1 + 2B²)

A = √(1 + 2B²) + B [Equation 8]

Eliminating A

Now, we try to eliminate A by deriving a different expression for A in terms of B by substituting equation 7 into equation 3:

(2AB)² + 2B² = 2

Solving for A and simplifying, we eventually obtain:

A² = (1 - B²) / (2B²)

A = ±√((1 - B²) / (2B²))

But since we require A > 0, we again select the positive root:

A = √((1 - B²) / (2B²)) [Equation 9]

... TBD