Polygonal Numbers and Pie

November 21st, 2007

Four mathematical illustrations from Wells’ Dictionary of Curious and Interesting Numbers.

“The maximum number of pieces into which a pancake [or pie] can be cut with 6 slices” is 22 (p78):

pie slices

22 is also the 4th pentagonal number — whereas “the 4th centered hexagonal number, obtained by arranging hexagonal layers around a central point,” is 37 (p99):

hexagonal numbers

“By a different division [of the above] the nth centered hexagonal number is equal to 6Tn-1 + 1, where Tn is the nth triangular number” (p100). Here we see a composition using the 3rd triangular number:

hexagonal as triangular numbers

And the 4th triangular number? — 10, the tetractys, a figuration “so holy [to the Pythagoreans] that they even swore oaths by it” (p61, c.f. the obverse seal):

tetractys

2 Responses to “Polygonal Numbers and Pie”

  1. Mirco Says:

    Nice diagrams.

    Prophecy: this century will rediscover the central role of figurate numbers, particularly in the physical sciences (Pythagoras vindicated!).

    Till then, we can already use them as Nature’s built-in Yantras

  2. Mirco2 Says:

    Nice diagrams.

    Prophecy: this century will rediscover the central role of figurate numbers, particularly in the physical sciences (Pythagoras vindicated!).

    Till then, we can already use them as Nature’s built-in Yantras

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