Sly Stormy Snowy Snowflake

The snowflake can be tiled with $6$ unit rhombi with an interior angle of $\cfrac{\pi}{9}$ (purple), $18$ unit rhombi with an interior angle of $\cfrac{2\pi}{9}$ (light blue), $24$ unit rhombi with an interior angle of $\cfrac{\pi}{3}$ (dark blue), and $12$ unit rhombi with an interior angle of $\cfrac{4\pi}{9}$ (green).

So one way the area can be expressed is $6 \sin \cfrac{\pi}{9} + 18 \sin \cfrac{2\pi}{9} + 24 \sin \cfrac{\pi}{3} + 12 \sin \cfrac{4\pi}{9}$ , where $A = 1$ , $B = 3$ , $C = 4$ , and $D = 2$ .

Knowing the total area, the Python computer program below can be used to brute force other possible non-negative integer values for $A$ , $B$ , $C$ , and $D$ . Out of all the possibilities, C only has a unique value of $24$ .

(We can also observe that there are always solutions for $B = A + 12$ and $D = 18 - A$ because of the identity $\sin \cfrac{\pi}{9} + \sin \cfrac{2\pi}{9} = \sin \cfrac{4\pi}{9}$ .)

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import math
G = 6 * (math.sin(1.0 * math.pi / 9) + 3 * math.sin(2.0 * math.pi / 9) + 4 * math.sin(
    1.0 * math.pi / 3) + 2 * math.sin(4.0 * math.pi / 9))
for A in range(0, 6 * 24):
    for B in range(0, 6 * 13):
        for C in range(0, 6 * 10):
            for D in range(0, 6 * 9):
                V = A * math.sin(1.0 * math.pi / 9) + B * math.sin(2.0 * math.pi / 9) + C * math.sin(
                    1.0 * math.pi / 3) + D * math.sin(4.0 * math.pi / 9)
                if abs(V - G) < 0.00001:
                    print(A, B, C, D)

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(0, 12, 24, 18)
(1, 13, 24, 17)
(2, 14, 24, 16)
(3, 15, 24, 15)
(4, 16, 24, 14)
(5, 17, 24, 13)
(6, 18, 24, 12)
(7, 19, 24, 11)
(8, 20, 24, 10)
(9, 21, 24, 9)
(10, 22, 24, 8)
(11, 23, 24, 7)
(12, 24, 24, 6)
(13, 25, 24, 5)
(14, 26, 24, 4)
(15, 27, 24, 3)
(16, 28, 24, 2)
(17, 29, 24, 1)
(18, 30, 24, 0)