Geometric Epiphanies

A road trip contemplation about prime numbers leads to a family mathematical discovery—complete with teenage fact-checking and a triumphant 'Eureka!' moment at the dinner table

Posted: 2025-Jul-21

Prime Numbers and Geometric Arrangements

Several weeks ago, I had read about advances in the mathematics of identifying prime numbers. The article explained that the dots representing a prime number could be arranged only in single line and not in a rectangle, while non-prime numbers could be arranged in rectangles. For example, seven is a prime number, and seven dots cannot be arranged into a rectangle; however, eight is a non-prime number, and eight dots can be arranged into a 2x4 rectangle. Perhaps this is a concept that I had simply missed in algebra classes, but it was intriguing to grasp how geometry could illuminate such an abstract concept as prime numbers.

Contemplating Geometric Patterns in Mathematical Operations

The idea of geometry applying in new and unexpected ways has remained on my mind. While driving the other day, I began wondering whether geometry applied to other mathematical principles, such as exponents. I am reluctant to admit that either I had never grasped or, more charitably, had simply forgotten that to "square" or "cube" a number literally describes the geometric configuration of dots. For example, 2^2 is literally a 2D square geometric shape of two rows of two dots per row, and 3^3 is three layers of a 3 rows by 3 dots square geometric shape, forming a 3D cube of 27 dots.

As my mind meandered, I began working through the geometric implications of multiplication vs. exponents. For simplicity's sake, I was working with the number two. As I visualized the dots resulting from multiplications vs exponents, I realized that with multiplication, the next product was simply the previous product plus the base number, e.g. 2 x 3 was (2 x 2) + 2, or 4 + 2, or 6. With exponents, however, the next exponential value was the previous exponential value plus the previous exponential value, e.g., 2^3 was 2^2 + 2^2, or 4 + 4, or 8.

A Family That Maths Together

When I reached home the next evening, I was rather excitedly sharing this insight with my two teenagers. To my consternation, when describing and illustrating the epiphany, the older teenager pointed out that my formula for exponents did not work for a base of 3, i.e. 3^3, or 27 does NOT equal 3^2 (9) + 3^2 (9), or 18. Working together, we corrected the algorithm, tested it, and loudly celebrated our triumph, to the dismay of the others in the house who were settling in for the night's rest.

As refined, the algorithm for calculating the Next Exponential Value (NEV) from the Previous Exponential Value (PEV) is as follows: NEV = PEV + ([N-1] x PEV), where N is the base number. Using eight as the base number and keeping in mind that any base raised to the 0 power is 1, the algorithm yields the following:

  • 8^1 = 1 + ([8-1] x 1), or 1 + (7 x 1), or 1 + 7, or 8
  • 8^2 = 8 + ([8-1] x 8), or 8 + (7 x 8), or 8 + 56, or 64
  • 8^3 = 64 + ([8-1] x 64), or 64 + (7 x 64), or 64 + 448, or 512

Celebrating Mathing

Our triumphal exuberation was not diminished in the least by the fact that surely some mathematician in the past had derived this same algorithm. Our celebration was justified solely and sufficiently on the grounds that we had independently derived the algorithm to calculate the next exponential value on the basis of the preceding exponential value. Had we been in an Athens bath house rather than at our dining table, we might have run naked through the streets shouting "Eureka"!