Weird numbers: Geometric numbers

In the primary school, we learn the first numbers. We learn how to count from 1 to 10 with our fingers, and perhaps up to 20 if we use our toes.

A child can count, surely, as he decides, not necessarily in order. Children have fun with numbers at first time but in some cases, teaching methods and certain teachers may cause that at some point, children who have fun with numbers in the beginning, suddenly start to hate them ... and later they also hate everything related to mathematics.

There are one day to celebrate the Children's Day in every country. In Mexico, this day is celebrated on April 30. Well, for those days, I share some "geometric numbers" so that you can have fun again... like children.

If you play with objects shaped close to the sphere you can form certain geometric arrangements. If you place 3 marbles, for example, you have a shape that is close to that of an equilateral triangle. The same thing happens if you place 6 marbles in the arrangement of 3, 2, 1 or 10 marbles in the arrangement of 4, 3, 2, 1. Well, it is easy to see that you can get similar triangular arrangements by adding rows of 5 and 6 marbles and so on.

What you get are the “triangular numbers” like 3, 6, 10, 15… (the number 15, for example, appears whenever a game of pool is started). By the way, the number 1 is also included in this classification.

Are there other geometric numbers?

Yes, indeed, it is easy to imagine the "square numbers". Think, for example, of golf ball arrangements of 4, 9, 16, 25, ... Interestingly, these square numbers are the same as the squares of all the numbers that we could count: 1, 2, 3, ... (the natural numbers) and, of course, 1 included.

You can even build centered hexagonal numbers. Remember that hexagons are the shapes of the cells of the honeycombs. If fruits are placed in that arrangement, there we have the numbers 7, 19, 37... and, of course, 1 is also included.

And speaking about polygons, there are centered pentagonal numbers as well. A little more difficult to imagine them but with enough spheres, let us say tennis balls, we can experiment with the arrangements that give us the shape of a 5-sided polygon like 6, curiously and as you can guess, 1 is also included.

Thinking in a geometric way we can begin to imagine arrangements of spheres, let us use ping-pong balls, which can be placed so that they form other figures such as rectangles. The "pronic numbers" are arrangements of 1x2, 2x3, 3x4, ... that form, precisely, rectangles which, if a row of spheres is eliminated, they become square numbers.

But let us not think about anything else in two dimensions, let us go to the third dimension like the orange sellers do and arrange the fruits in pyramids like when we place a base of 3 oranges and 1 on top. In this case, we obtain the number 4 which is one of the "tetrahedral numbers". The other ones would be (try to construct them!): 4, 10, 20, 35, ...

It is easy to imagine that there are also "cubic numbers" although they are no longer so easy to place in the real world, but with a cubic box everything is possible! We would then have 1, of course, and then: 8, 27, 64, ..., which, by the way, coincide with the cubes of the natural numbers.

Are there more?

Yes.

There are "dodecahedral numbers" whose formation with fruits, marbles or balls requires more imaginative techniques, but what about with magnetized spheres? Considering that 1 is again included, if you try with these spheres, you can form the dodecahedral numbers: 1, 20, 84, 220, 455, ...

And by the way, there are more numbers of this type.