Welcome back to Numberholic! I’m your host, Sejari, and it’s always a pleasure to have you here. Today, we’re diving into the ninth episode of Proven Yet Hard to Believe, a series where we uncover the fascinating stories and directly explore the core ideas behind mathematical discoveries that have stood the test of time.
In this episode, we turn our attention to the mystical elegance of regular polyhedra—geometric structures that fascinated the greatest minds of ancient Greece. From Euclid’s Elements to Euler’s groundbreaking formula, we’ll follow the journey of these remarkable shapes and see how mathematical reasoning unveils their hidden properties.
Are you ready to explore the beauty of symmetry and logic? Let’s get started!
A regular polyhedron is a polyhedron in which all faces are regular polygons and exhibit perfect symmetry.
Euclid, a student at Plato’s Academy, stated in Elements that there are only five regular polyhedra.
Let’s begin with the approach we learned in middle school, considering the number of regular polygons that can meet at a single vertex.
For triangles, three, four, or five triangles can meet at a vertex.
For squares, only three can meet.
For pentagons, again, only three can meet. From hexagons onward, it becomes impossible.
Thus, only five cases are possible.
Euler discovered something new:
Vertices – Edges + Faces = 2. This marked the beginning of topology.
With this formula, we can make precise calculations.
Let’s consider a regular polyhedron composed of regular pentagons.
Let the number of faces be f. How many vertices does it have?
Each face has 5 vertices, totaling 5 times f vertices, but since three pentagons meet at each vertex, we must divide by 3. Hence, the number of vertices is 5 times f divided by 3.
Similarly, each face has 5 edges, totaling 5 times f edges, but since each edge is shared by two faces, the number of edges is 5 times f divided by 2.
Substituting these values into Euler’s formula and solving for f, we find f equals 12.
Thus, the regular polyhedron composed of pentagons is the dodecahedron.
However, here we will explore regular polyhedra in a slightly different way.
Canadian number theorist and Queen’s University professor Ram Murty once said, “Solving old problems using new methods is a great source of mathematical inspiration.”
Now, let’s try to visualize regular polyhedra by drawing two-dimensional figures.
Before that, let’s introduce a new perspective on regular polygons: the concept of inversion, which reverses the interior and exterior.
If we disregard size and length, swapping the interior and exterior of a regular polygon makes no fundamental difference.
More precisely, one extra point is needed for this transformation.
Thinking in this way makes it easier to understand. On a spherical surface rather than a plane, the distinction between interior and exterior becomes meaningless.
Now, let’s reconsider the regular polyhedron composed of pentagons.
According to Euler’s formula, it consists of twelve faces.
Let’s verify this for ourselves.
First, we draw a pentagon and mark the number of faces meeting at each vertex—currently, it is just one.
We then add two more pentagons around each vertex, resulting in five pentagons meeting at each point.
There is no need to worry if the shape does not appear to be a perfect pentagon; at this stage, the relative positioning of the pentagons is what matters.
Naturally, five more pentagons are drawn around the existing ones.
By labeling each vertex with the number of adjacent faces, we complete the structure.
So what comes next? We can cover the shape using a large regular pentagon, but instead of just overlaying it, we apply an inverted pentagon.
Ah, and we must not forget the extra point!
This way, we complete the twelve faces needed.
In fact, applying inversion at this stage allows for an even simpler understanding.
We invert, rotate slightly, and fit everything together precisely.
But… a problem remains.
Is this structure actually possible?
Even if we can form a three-dimensional shape with twelve pentagons, can it truly be a regular polyhedron?
Euclid demonstrates this through geometric construction. Let’s examine the case of the dodecahedron.
Euclid constructed a cube and built regular pentagons on it to create a dodecahedron. Let’s follow his method.
We start with a pentagon and use its diagonal as an edge to construct a cube.
At this stage, let’s take a moment to think about the golden ratio, as it is a key element in the construction of a regular pentagon.
Using modern notation, let’s denote the golden ratio as φ(phi) to 1.
From this, we can derive an equation involving φ.
We connect the midpoints of each edge and divide these segments according to the golden ratio.
We then construct vertical segments based on the golden ratio.
The length of these segments is 2 times φ—at least in proportional terms.
Next, we create a roof-like structure.
Can we rediscover the original pentagon within this roof?
We use right triangles to calculate the hypotenuse.
Using the Pythagorean theorem, we find that the hypotenuse length is 2 times φ—identical to the previously determined segment length.
Since in a regular pentagon, the ratio of the side length to the diagonal is the golden ratio, this confirms that our quadrilateral is an isosceles trapezoid, precisely matching the expected shape.
Additionally, the triangles within the structure are congruent to the triangle at the upper part of the regular pentagon, with all three sides equal.
We then construct the same configuration on the front face of the cube.
Observing the pentagons formed, one might worry that they are warped.
However, by comparing the triangles within the shape, we see that they are similar.
This confirms that the pentagons are perfectly planar rather than warped.
Thus, by placing pentagons on each edge of the cube, we achieve 12 faces, completing the dodecahedron.
That’s all for today’s episode of Proven Yet Hard to Believe! If you enjoyed this deep dive, don’t forget to subscribe and stay tuned for our next journey into the world of mathematics. Until next time, keep exploring, keep questioning, and stay Numberholic!