Geometric Numbers

Michael Lahanas

Griechische Mathematik: Geometrische Zahlen

  • Triangular number and sums

Triangular numbers are so called because they can be arranged in a triangular array of dots (elements).

Consider the sum 1+2+3+...+n = n(n+1)/2. This sum can be considered as the number of elements of a triangular number, see example with n = 3.

For n =4 we have a special number for the Pythagoreans, the Tetraktys, that has 10 elements.

I swear by the discoverer of the Tetraktys,Which is the spring of all our wisdom, The perennial root of Nature's fount. (Iambl., VP, 29.162)

  • The sums of the uneven numbers, as reported by Aristotle in Metaphysics, give us the series of squared numbers:

1 + 3 = 22 , 1 + 3 + 5 = 32, 1 + 3 + 5 + 7 = 42, etc.

  • Any pair of adjacent triangular numbers add to a square number

    1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, 45 + 55 = 100

    The Pythagoreans were the first people to discover this relationship.

  • According to Plutarch the square of the nth triangular number equals the sum of the first n cubes.

The image shows the first 3 triangular elements with 1,3 and 6 elements and their square are 1, 9 and 36 respectively. We have:

12 = 13, 32 = 13 + 23, 62 = 13+23+33 , etc.

This is equivalent to say that 13+23+...+ n3 = (n(n+1)/2)2

Square numbers are so called because they can be arranged as a square array of dots.

The first four perfect numbers are: 6, 28, 496, 8128. Euclid was able to find that each of these numbers is of the form 2n(2n+1- 1), where 2n+1-1 is prime. Euclid proved that all numbers of this form were perfect. The Pythagoreans knew that 1 + 2 + 4 + ... + 2k = 2k+1 - 1.

Euclid's perfect numbers are triangular.

  • Odd Square number relation to Triangular number

Divide the quadratic number in triangular numbers. We obtain (2n+1)2 = 8n(n+1)/2 + 1

Divide the quadratic number in two triangular numbers. n(n-1)/2 + n(n+1)/2 = n2. This is a graphic explanation why any pair of adjacent triangular numbers add to a square number.

  • Pythagorean Numbers

Subtracting two quadratic numbers (n+1) and n we obtain, see figure above , (2n+1) = (n+1)2 - n2

Assume that there is a n for which 2n+1 = m2 (i.e. 2n+1 is a quadrat) and m is odd. Then n = (m2-1)/2, n+1 = (m2+1)/2 or

m2 + ( (m2-1)/2)2 = ((m2-1)/2)2.

We have therefore a method to produce integer solutions forming a Pythagorean triple.

Every perfect number is triangular. The number 666, also known as the Number of the Beast, is a triangular number.

  • Pentagonal Numbers

LINKS

On Polygonal Numbers

Triangular Numbers 1 and Triangular Numbers 2

Formulas and their geometric interpretation

Polygonal numbers Patterns of prime distribution

References

There exist triangular numbers that are also square

What is a number? (From Rational , Irrational to Surreal Numbers)

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