What Is a Rep-Tile?

The term "rep-tile" refers to any geometric shape that can completely cover a plane without the need for any additional shapes. For example, think of square tiles; by using identical square shapes, which exhibit both reflective and rotational symmetry, one can tile an entire surface.

Interestingly, among regular polygons (those with equal sides and angles), only the square, equilateral triangle, and regular hexagon can achieve this full plane coverage. Irregular polygons, however, offer a wider array of possibilities.

What Is a Rep-k Polygon?

Golomb defines a rep-tile of order 'k' as a geometric figure that can be divided into 'k' smaller, congruent, and similar versions of itself. For brevity, these are referred to as rep-k polygons.

Take, for instance, the rep-4 polygon depicted below. It consists of a large trapezoid that subdivides into four smaller, congruent trapezoids, each resembling the larger one (if you can overlook the imperfect illustration).

When discussing rep-2 polygons, we only know of two examples: the isosceles right triangle with a hypotenuse-to-side ratio of √2, and the parallelogram with a length-to-breadth ratio of √2. Notably, the properties of the parallelogram's rep-2 nature are unaffected by its angles.

At a right angle, the rep-2 parallelogram becomes a rep-2 rectangle, which has historically influenced art. To discover more about the practical applications of this shape, check out my essay on how mathematics shapes the world of paper.

Can You Solve This Complex Rep-Tile Puzzle?

Now that we've introduced the concepts of rep-tiles and rep-k polygons, let’s dive into the puzzle. The challenge is straightforward:

Construct a rep-3 triangle.

It’s important to note that not every arbitrary rep-k polygon exists; however, a rep-3 triangle does.

To clarify the task, you need to design a triangle that can be divided into three smaller, congruent, and similar triangles.

Hint:

Begin by considering the different types of triangles (like equilateral or isosceles) and eliminate the ones that would make constructing a rep-3 triangle impossible. This process should help you pinpoint the specific triangle that allows for a rep-3 configuration.

Spoiler Alert:

If you wish to solve this puzzle independently, I recommend stopping here. Once you've attempted to tackle it on your own, feel free to return and continue reading.

My Initial Attempts at Solving the Rep-Tile Puzzle

When I first approached this puzzle, I compiled all the relevant knowledge I had about triangles along with the necessary conditions for solving it. I then engaged in a series of trial-and-error attempts.

One of my initial promising solutions involved an equilateral triangle with a side length of 1 unit. I utilized the angle bisectors and incenters to create three triangles within the larger triangle, leveraging the circular symmetry around the incentre.

While this approach was close, it ultimately fell short. The three smaller triangles were indeed congruent but did not maintain similarity to the original triangle; they were not equilateral.

This experience taught me that relying on circular symmetry with an equilateral triangle would hinder the creation of a rep-3 triangle. Following this realization, I invested a lot of time, trial, and even some luck to finally discover the solution.

The Solution to the Rep-Tile Puzzle

I would have loved to present a systematic method for solving this puzzle. Unfortunately, my approach was quite chaotic, characterized by wild idea generation and trial and error.

I began to doubt my ability to find a structured solution, prompting me to search for literature on the topic. Regrettably, I found no structured methodologies addressing this specific puzzle.

However, I did come across resources discussing structured approaches for higher-order rep-k triangles. I plan to explore this subject in a future essay (now available here). For now, here’s the solution to our puzzle:

Initially, I constructed a 90–60–30 triangle with side lengths of 1, 2 (the hypotenuse), and √3 units. After establishing this triangle, I divided the √3 side at one-third of its length and constructed a triangle from the top vertex.

From that point on the bottom edge, I erected a perpendicular line to the opposite side, completing the two additional triangles. This process resulted in three smaller triangles that were both congruent and similar to the original triangle.

References and Acknowledgments: Solomon W. Golomb and Martin Gardner.

If you appreciate my work as an author, consider clapping, following, and subscribing. For further reading, you might enjoy "The Shipwreck Puzzle — A Fun Linear Math Challenge" and "How To Really Benefit From Curves Of Constant Width."

You can read the original essay here.

The video titled "Why this puzzle is impossible" explores the complexities and challenges behind solving such geometric puzzles, providing insights that could enhance your understanding of the topic.