The Helix Puzzle - A Simple Geometric Challenge - An illustration of a black outlined cylinder that is 16 centimetres high and has a pink helix wrapped around it. The helix makes the constant angle of 60° with vertical lines drawn on the cylinder's surface. Below this figure, the following text is written: "Length of helix = ??"

The helix puzzle is a fun and simple geometric challenge that tests your geometric reasoning skills. To be precise, this puzzle features a circular helix.

Since helixes are not an everyday discussion topic, I’ll start the essay by giving a brief introduction to the topic. Following this, we will cover an interesting property of helixes and then jump straight into the puzzle. So, stay sharp; let’s begin.

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What is a Helix?

Before we answer that question, let us first establish the concept of self-congruent curves. Let us say that there exist two circular arcs that feature the same curvature. In this case, we could take one of those arcs and slide it along the other arc, and the two will fit snugly. This property of curves (and lines) is known as self-congruent curves.

When we extend this concept to 3 dimensions, we end up with applications such as curved sheaths for curved walking sticks, curved scabbards for curved swords, etc. Along these lines, another 3-dimensional application of self-congruent curves is the helix.

Imagine a cylinder with vertical lines drawn on its surface parallel to its axis. A circular helix is a curve that wraps around this cylinder in such a manner that the curve makes the same constant angle with each of the vertical lines on the cylinder’s surface.

The Helix Puzzle — A Simple Geometric Challenge — An illustration of a black outlined cylinder that has a pink helix wrapped around it. The helix makes a constant angle of θ with vertical lines drawn on the cylinder’s surface.
The circular helix makes a constant angle with projections of the cylindrical axis (illustration created by the author)

Other kinds of helixes involve other cross-sections and parameters, but for the context of this essay, this definition should suffice. Before we jump into the puzzle, it is also worth covering an interesting property of helixes.

An Interesting Property of the Helix

Remember that I told you that a helix is a self-congruent curve? Well, if we play around with the helix’s angle a little bit, we could establish that lower dimensional self-congruent curves emerge at the limits of a circular helix’s angle.

Consider (again)the fact that a helix makes a constant angle with the vertical lines drawn on a cylinder’s surface parallel to the cylinder’s axis.

If this angle approaches zero, the circular helix would approach a straight line. Similarly, if this angle approaches 90°, then the circular helix would approach a circle. In both the cases, the property of self-congruency would remain intact.

The Helix Puzzle — Problem Statement

The problem statement of the helix puzzle is pretty simple. Consider a cylinder that is 16 centimetres (cm) high. Now, imagine a circular helix wrapped around this cylinder. The constant angle that this helix makes with the vertical lines drawn on the cylinder’s surface is 60°.

Given these conditions, calculate the length of the helix.

That is it! It might feel that I have provided you with insufficient information. But this is not the case. Try to think outside the box and be creative about your approach.

Spoiler Alert:

The approach to solving this puzzle can be made challenging or simple. If you wish to figure everything out on your own (the more challenging option), then I recommend that you tune off of this essay now and give the puzzle a try.

However, if you wish to make the approach simpler, check out the hint right below. Once you are done with your trial, you could continue reading this essay. Beyond this section, I will be explicitly discussing solutions to the puzzle.

Hint:

Try and use the Pythagorean theorem to solve this puzzle.

How to solve the Helix Puzzle

I will start with the hint that I have provided above. How can I involve the Pythagorean theorem with this problem? I know that involves a right-angled triangle.

So, what do right-angled triangles and helixes have in common?

The answer to these questions becomes clear when we use a little bit of imagination. Visualise a right-angled paper triangle that has its base aligned with the base of the cylinder we have. Now, wrap this paper triangle around the cylinder. Naturally, the height of this triangle needs to be the same as that of the cylinder (which is 16 cm).

By doing this, we learn that the hypotenuse of the right-angled triangle is the helix we are looking for. The only condition is that this hypotenuse makes an angle of 60° with the vertical lines on the cylinder’ surface.

This is all that we need to proceed toward solving our puzzle. The corresponding triangle will look as follows:

The Helix Puzzle — A Simple Geometric Challenge — An illustration with a right-angled triangle on the left and a cylinder with a helix wrapped around it on the right. The triangle seems to be wrapped to form the helix. Consequently, the hypotenuse forms an angle of 60° with the vertical projections of the cylinder’s axis and 30° with the base.
A right-triangle wrapped around the cylinder (illustration created by the author)

Consequently, when we apply the Pythagorean theorem, by calculating the length of the hypotenuse, we are also calculating the length of the circular helix:

The Helix Puzzle — A Simple Geometric Challenge — The triangle from before is featured on the left. On the right, the following mathematical calculation is shown: Sin(30°) = 16cm/h; h = 16cm/Sin(30°); h = 32 cm.
Pythagorean theorem applied to the right-triangle (math illustrated by the author)

Therefore, the length of the circular helix is twice the height of the cylinder. Since the height of the cylinder is 16 cm, the length of the helix is 32 cm.

Reference and credit: Martin Gardner.

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Further reading that might interest you: Can You Really Solve This Rep-Tile Puzzle? and How To Casually Guess Numbers After Dice Throws?

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