In an earlier article (Math Is Your Friend) I said that most of the time we use mathematics to “translate the world”. That is, we look at a complicated problem out there, take its most essential parts, and then state those it in very structured and unambiguous language, which allows us to solve this idealization of the problem, and then take the solution and apply it (“re-translate”) it back to the real world.
I also said that of course that is not all we do. Mathematics being a language, we use it in our moments of leisure and heightened imagination, the same way poets or storytellers do regular language. That is, we take math words and rules, and start playing with them, mixing concepts, exploring relations, and creating the math equivalent of unicorns.
In general, we love to take some thing or concept that we are familiar with and say, “let's see, what if...?” And in this way sometimes we come across unexpected things, which are the best of all things because they push us to ask: “why did this happen?”
Well, today we are going to do just that.
How do we start playing?
Let’s look at those two classic things we study in mathematics: numbers and shapes. In number theory surely one of the most interesting topics are prime numbers, those fascinating “building blocks” of all integers. And in geometry we have the circle, that beautiful queen of shapes. Let’s start with these two venerable concepts.
So what if we count prime numbers in a circle?
Let’s do that! And we start by defining a decagon inside a circle, and putting the 10 digits on each of its vertices:
And now, because we are in a playful mood, let’s start seeking out primes (by their last digit) and every time we find the next one, we draw a line from the one immediately before. Let’s count, say up to 300 and see what our first few sequences look like, say up to 300:
That
last circle shows the movement of our lines through 14 primes in a non-overlapping way, to better appreciate its trips. Now, since the primes cannot have more than 6 possible ending digits, we quickly obtain a quasi-symmetric polygon that we will call, for lack of brighter imagination, the P-Polygon. If we rotate our circle a bit to appreciate its vertical symmetry, we have:
Note that the irregular part of this figure emerges from the fact that we include the 2, the only even prime, and that it only gets into our sequence once, but that one time is enough to create an extra area in an otherwise symmetric, generic figure. Let’s see what peculiarities this extra bulk confers to our shape.
Exploring a new shape
First of all, we notice that we can define three different medium heights in our P-Polygon:
Where:
h/2 is the real medium height,
hV is the height from the base to vertex V,
hI is the height from the base to intersection point O,
l/2 is the medium length, and
∠DVC is 1440.
Three different heights of interest! Seeing this, we can ask ourselves idle but intriguing questions like:
1. What is the area of the P-Polygon as a ratio of the unit circle in which it is inscribed?
2. What are the areas of inner triangles ABO, ADO and CDV?
3. What are the ratios between the inner triangles and the total area of the P-Polygon?
4. What are the areas if the polygons that we get if we horizontally cut the P-Polygon at the 3 different heights of interest?
5. Do all those areas and rates have a discernible pattern?
All these questions are more or less simple, but since they are asked about an unusual shape, they provide us with a good chance to exercise our knowledge of geometry. Another nice thing is that oftentimes a geometric puzzle can be solved in more than one way and we can use our imagination to look for unusual methods, or combinations of methods, to arrive at the answer.
On to wackier questions
Having a polygonal shape, what happens if we start drawing circles on it? In the case of the triangle, the results are well-known and have provided us with centuries of experiments and surprising findings. So we can go ahead and try drawing circles on our P-Polygon P to see if some interesting relation or pattern appears.
Let’s look at the circles that emerge if we take point O as their origin. Five circles appear, in this order:
- on touching segment AD
- on touching segment BC
- on touching vertices A and D
- on touching vertex V, and
- on touching vertices B and C.
We can work in a similar fashion and get circles with origins at OV and Oh:
All three options seem equivalent, but we soon realize that this cannot be the case: if we take Oh as the origin we only generate four circles instead of five, because being the real mean height, the same circle touches AD and BC at the same time. By overlaying all three sets of circles in a single image and putting them on a single origin for clarity, we can see that a ripple effect is actually created:
All the circles are different except for the circles that touch vertex V, so we get 12 circles in total, and not the 14 we would get by adding 5 + 5 + 4. Also, their “undulation” is irregular, with two areas of concentration of circles with very similar diameters, and two areas with circles with a greater difference between their diameters. Even more wacky questions:
What is the ratio of all the diameters of the resulting 12 circles?
If we create twelve sine waves from the 12 circles, how do they behave and when do they coincide?
There are many more things that can spring up if we keep thinking about our P-Polygon, and just for the joy of doing it, we keep asking, “what if…?”
The image at the beginning of this article shows three different rotations of the P-Polygon, using different fixed points for the rotation.
The beauty of all this is that the least we get, is having hours of fun and of discovery of beautiful shapes and symmetries, as well exercise our knowledge to solve these problems. And in the best of cases, finding a solution of one of them can lead us to an unexpected result, a relationship, or the best of all: a pattern that we did not know existed.
This is a small sample of the poetry of mathematical inquiry.
