Diagonals checkingSearch for edges and face diagonals, satisfying the system
a² + b² = d²
a² + c² = e²
b² + c² = f²
a² + f² = g²
With the system it follows that
g² = a² + f²
g² = b² + e²
g² = c² + d²
We'd obtained all possible decompositions of the square of a space diagonal g² to sums of two squares.
The problem is to find three pairs (a, f), (b, e), (c, d), satisfying the conditions of the system.
First, note that
a < b < c
d < e < f
a < b < d < e < f
How are related to each other c and d - is not known.
Maybe c < d, maybe d < c
Secondly, knowing the 2 pairs (a, f), (b, e), then the pair (c, d) should not look for,
c² = e² - a²
c² = f² - b²
So edge c can be deducted on one of formulas like this:
? = sqrt(f² - b²) = sqrt(f - b) * sqrt(f + b)
Well diagonal d can either be deducted from the formula
d = sqrt(g - c) * sqrt (g + c),
or you can simply search for a pair between g decompositions,
just c² + d² = g².
Please note, if you know c, you have to search and pair (c, d) and pair (d, c)
If all decompositions of g² are stored in some table TASK,
so we can search target a, b, c, d, e, f from the query:
WITH t(A, B, F, E, FBextC, C) AS
(SELECT
a.X AS A
,b.X AS B
,a.Y AS F
,b.Y AS E
,CAST(SQRT( F-B ) AS DECIMAL(25,4)) * CAST(SQRT( F+B ) AS DECIMAL(25,4)) AS FBextC
,CAST(FBextC AS BIGINT) AS C
FROM TASK a, TASK b
WHERE A > 0
AND A < B
AND B < F / SQRT(2)
AND E > A * SQRT(2)
AND ABS(FBextC-C) < 0.001
AND ((A MOD 2) + (B MOD 2) <> 2)
AND ((A MOD 2) + (B MOD 2) + (C MOD 2) = 1)
AND (((A MOD 5) = 0) OR ((B MOD 5) = 0) OR ((C MOD 5) = 0))
AND (((A MOD 7) = 0) OR ((B MOD 7) = 0) OR ((C MOD 7) = 0))
AND (((A MOD 9) = 0) OR ((B MOD 9) = 0) OR ((C MOD 9) = 0))
AND (((A MOD 11) = 0) OR ((B MOD 11) = 0) OR ((C MOD 11) = 0))
AND (((A MOD 16) = 0) OR ((B MOD 16) = 0) OR ((C MOD 16) = 0))
AND (((A MOD 19) = 0) OR ((B MOD 19) = 0) OR ((C MOD 19) = 0))
AND ((CASE WHEN ((A MOD 3) = 0) THEN 1 ELSE 0 END) + (CASE WHEN ((B MOD 3) = 0) THEN 1 ELSE 0 END) + (CASE WHEN ((C MOD 3) = 0) THEN 1 ELSE 0 END)) >= 2
AND ((CASE WHEN ((A MOD 4) = 0) THEN 1 ELSE 0 END) + (CASE WHEN ((B MOD 4) = 0) THEN 1 ELSE 0 END) + (CASE WHEN ((C MOD 4) = 0) THEN 1 ELSE 0 END)) >= 2
)
SELECT A, B, C, d.Y AS D, E, F
FROM t, TASK AS d
WHERE C = d.X
UNION ALL
SELECT A, B, C, d.X AS D, E, F
FROM t, TASK AS d
WHERE C = d.Y;
The query check the next conditions:
1. A > 0 for discarding a trivial decomposition g² = 0² + g²
2. A < B
3. as far as b < c ? b² + c² = f², thus b < f / sqrt(2)
4. as far as a < c ? a² + c² = e², thus e > a * sqrt(2)
5. a and b can't be odd simultaneously
6. 2 of the edges a, b, c must be even and 1 edge must be odd
7. 1 of the edges a, b, c must be divisible by 5
8. 1 of the edges a, b, c must be divisible by 7
9. 1 of the edges a, b, c must be divisible by 9
10. 1 of the edges a, b, c must be divisible by 11
11. 1 of the edges a, b, c must be divisible by 16
12. 1 of the edges a, b, c must be divisible by 19
13. at least two of of the edges a, b, c must be divisible by 3
14. at least two of of the edges a, b, c must be divisible by 4
All these conditions are described here:
en.wikipedia.org/wiki/Perfect_cuboid