One of the properties of the number associated with my diurnal age today is that it represents the surface area (rounded to the nearest whole number) of a dodecahedron with an edge of 37 units. The formula and result is shown below where \(n\) represents edge length:$$ \begin{align} \text{Surface Area } &= 3 \times n^2 \times \sqrt{{25} + 10 \times \sqrt{5}} \\ &= 3 \times 37^2 \times \sqrt{25 + 10 \times \sqrt{5}} \text{ when }n=37\\ \\&= 28264.0027368755 \dots \\ \\ &\approx 28264 \end{align}$$What's special about the dodecahedron with an edge of 37 units is how close its exact surface area approaches a whole number. The table below shows the results for edges from 1 to 44:
Edge With Decimal Rounded 1 20.645729 21 2 82.582915 83 3 185.81156 186 4 330.33166 330 5 516.14322 516 6 743.24624 743 7 1011.6407 1012 8 1321.3266 1321 9 1672.3040 1672 10 2064.5729 2065 11 2498.1332 2498 12 2972.9849 2973 13 3489.1282 3489 14 4046.5628 4047 15 4645.2890 4645 16 5285.3066 5285 17 5966.6156 5967 18 6689.2161 6689 19 7453.1081 7453 20 8258.2915 8258 21 9104.7664 9105 22 9992.5327 9993 23 10921.591 10922 24 11891.940 11892 25 12903.581 12904 26 13956.513 13957 27 15050.736 15051 28 16186.251 16186 29 17363.058 17363 30 18581.156 18581 31 19840.545 19841 32 21141.226 21141 33 22483.199 22483 34 23866.462 23866 35 25291.018 25291 36 26756.865 26757 37 28264.003 28264 38 29812.432 29812 39 31402.154 31402 40 33033.166 33033 41 34705.470 34705 42 36419.066 36419 43 38173.953 38174 44 39970.131 39970
Clearly the dodecahedron with edge of 37 units is a clear winner as it has a surface area that differs by only about 0.003 from its surface area when rounded to the nearest whole number.
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