Surface Area of a Regular Dodecadhedron

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.

A Mathematical Look At 2024

Well, 2024 is almost upon us and so it's time to look at some of the mathematical properties of that number. First and foremost is its factorisation which is:$$2024 = 2^3 \times 11 \times 23$$It can be noted that this factorisation involves only the digits 1, 2 and 3. The final day of 2023 can be written in MM-DD-YY format as 12-31-23 or 123123 which also contains only the digits 1, 2 and 3.

FIRST FUN FACT

The first entry in the OEIS is for A000292:

A000292

Tetrahedral (or triangular pyramidal) numbers:$$\text{a}(n) = \text{C}(n+2,3) = \frac{n \times (n+1) \times (n+2)}{6}$$

Figure 1 illustrates the triangular pyramidal numbers as a sum of triangular numbers stacked upon each other. In the case of 2024, \(n=22\) and this number represents the number of balls in the triangular pyramid in which each edge contains 22 balls. The sequence progresses as follows:

0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341, 13244, 14190, 15180

Figure 1: source

SECOND FUN FACT

The tetrahedron is one of the Platonic Solids and therefore a shape of great significance.

However, 2024 is also connected with the dodecahedron because it is a member of OEIS A006566 with \(n=8\):

A006566

Dodecahedral numbers: $$ \text{a}(n) = \text{C}(3n,3)  =\frac{n \times (3n - 1) \times (3n - 2}{2} $$

The initial members of the sequence are:

0, 1, 20, 84, 220, 455, 816, 1330, 2024, 2925, 4060, 5456, 7140, 9139, 11480, 14190, 17296, 20825, 24804, 29260, 34220, 39711

So 2024 represents the number of balls in the triangular pyramid in which each edge contains 8 balls. The following video shows how to construct a dodecahedron from nanodots:

The connection between the tetrahedron and the dodecahedron is visible in the GIF below:

Dodecahedron with five tetrahedra inside (source)

THIRD FUN FACT

 The next sequence for 2024 listed in the OEIS is A003242:

A0032425

Number of compositions of \(n\) such that no two adjacent parts are equal (Carlitz compositions).

In the case of 2024, \(n=15\) and here a few examples of the 2024 possible Carlitz compositions (permalink):

• 1, 2, 1, 2, 1, 2, 1, 2, 1, 2
• 1, 4, 2, 1, 4, 3
• 2, 1, 6, 1, 2, 1, 2
• 3, 2, 1, 4, 5
• 6, 2, 1, 2, 3, 1

FOURTH FUN FACT

Watching this video on YouTube, I learned that 2024 is also the sum of consecutive cubes beginning with \(2^3\) and ending with \(9^3\). Thus we have:$$2024=2^3+3^3 + \dots + 8^3+9^3$$This means that next year, 2025, can be represented as:$$2025=1^3+2^3 + \dots +8^3+9^3$$