How To Really Tile Rep-Tile Triangles?

In my essay on the rep-tile puzzle, I introduced the concept of the rep-tile, covered how Solomon W. Golomb defined a rep-k polygon and demonstrated what it means to construct a rep-3 triangle. If you haven’t read that essay, I would recommend you to do so. This is because I will be building on top of those concepts in this essay.

You see, the rep-3 triangle is a special case for which I couldn’t find any structured approach. However, I’m happy to report that Golomb figured out structured approaches to tiling higher-order rep-k triangles. I will be demonstrating how we could tile the various possible higher-order rep-k triangles in this essay. Let us begin.

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The Conditions for Tiling Rep-Tile Triangles

In his 1964 paper titled “Replicating Figures in the Plane” (linked in the references at the end of this essay), Golomb stated three specific conditions for tiling rep-tile triangles. In other words, if these conditions are not met, it would not be possible to tile the corresponding triangle.

Given the intention to tile an arbitrary triangle into n-tiles, Golomb’s conditions are as follows:

1. A triangle can be tiled if ’n’ is of the form ‘k²’.

2. A triangle can be tiled if ’n’ is of the form ‘3k²’.

3. A triangle can be tiled if ’n’ is of the form ‘k² + l²’.

In the above set of conditions, ’n’, ‘k’, and ‘l’ refer to positive integers. Accordingly, any other form of ’n’ would not be tile-able as far as triangles are concerned. This came to be known as Golomb’s theorem in this field.

After Golomb, Snover et al. proved Golomb’s theorem in their 1989 paper titled “Rep-tiling for triangles” (also linked in the references section at the end of the essay). Now that we have covered the relevant background, let us see how we can tile triangles that conform to each of the above-stated conditions.

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How to Rep-tile a Triangle into n tiles, where n = k²?

The procedure to tile an arbitrary triangle into ’n’ tiles where ’n’ is of the form ‘k²’ is pretty straightforward. Firstly, we divide each of the sides into ‘k’ equal lengths. Then, we interconnect all of these points. Since this might be challenging to imagine, let us work with an example.

Say that we are interested in tiling a triangle into 9 tiles. Now, 9 = 3². So, n = 9 and k = 3. Knowing this, we can divide each side of the triangle into three equal parts. Then, we interconnect the resulting points using straight lines as shown below.

How to Rep-tile a Triangle into n tiles, where n = 3k²?

Rep-tiling a triangle into ‘n = 3k²’ tiles appears challenging on the surface, but with a little bit of knowledge application, it is simple enough. In my essay on the rep-tile puzzle, I demonstrated how you could tile a rep-3 triangle.

Similar to that approach, you will need start with a 90–60–30 triangle with side length proportions of x, 2x and √3x respectively. Once you construct a rep-3 triangle based on the given specification, you can apply the ‘k²’ algorithm that we just saw to each one of these triangles.

By doing this, you would end up with ‘n = 3k²’ tiles.

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How to Rep-tile a Triangle into n tiles, where n = (k² + l²)?

This is the most complex one out of all three conditions. When tiling a triangle into ‘n = (k² + l²)’, you will first need to sort ‘k’ and ‘l’ in terms of magnitude. For our purposes, let us say that k > l.

Next, you will need to construct a right-triangle such that two sides of length ‘k’ and ‘l’ are perpendicular to each other. Consequently, the hypotenuse would have a length of ‘√ (k² + l²)’ or √n. Now, let us visualize this triangle such that the hypotenuse appears horizontal on the bottom.

Next, drop a perpendicular line to the hypotenuse from the opposite vertex. Then, divide the side that is ‘k’ units long into ‘k’ equal proportions and the side that is ‘l’ units long into ‘l’ equal proportions. As a final step, just inter-connect the resulting points.

Let us say that we would like to tile a triangle using 5 tiles. We know that 5 = 2² + 1². So, n = 5, k = 2, and l = 1. If we execute the above discussed algorithm, the resulting triangle would look like this:

As you may have noted, this algorithm results in tiles that don’t match vertices at the boundary between the k-triangles and the l-triangle(s). Still, the triangle is considered tiled.

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Final Remarks

You might be wondering what the purpose of a rep-tile is. Well, in two words: infinite scalability.

In the work done by Martin Gardner on this topic, I came across a beautiful little poem by British mathematician Augustus De Morgan on the scalability of rep-tiles (De Morgan had been researching this topic much before Golomb).

To conclude this essay with a poetic touch, I leave you with a quote from De Morgan’s own words. Enjoy!

Great fleas have little fleas

Upon their backs to bite ‘em,

And little fleas have lesser fleas,

And so ad infinitum.

The great fleas themselves, in turn,

Have greater fleas to go on;

While these again have greater still,

And greater still, and so on.

— Augustus De Morgan

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References and Credit: Solomon W. Golomb (scientific paper), Snover et. al (scientific paper), and Martin Gardner.

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Further reading that might interest you: How To Really Solve The Monkey And The Coconuts Puzzle? and How To Really Benefit From Curves Of Constant Width?

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