Conditions for Tiling Rep-Tile Triangles

Golomb's 1964 paper, "Replicating Figures in the Plane" (referenced at the end), enumerates three essential conditions for successfully tiling rep-tile triangles. If these criteria are not satisfied, tiling the triangle is impossible.

To tile any arbitrary triangle into n tiles, Golomb outlined the following conditions:

1. A triangle can be tiled if 'n' takes the form of 'k²'.
2. A triangle can be tiled if 'n' is expressed as '3k²'.
3. A triangle can be tiled if 'n' is represented as 'k² + l²'.

In these conditions, 'n', 'k', and 'l' are positive integers. Any other formulation of 'n' would render the triangle untileable, a principle known as Golomb's theorem.

Following Golomb's work, Snover et al. validated this theorem in their 1989 publication, "Rep-tiling for triangles" (also referenced below). Now, let’s explore how to tile triangles that meet each of these specified conditions.

Tiling a Triangle into n Tiles, where n = k²

The method for tiling a triangle into 'n' tiles, where 'n' follows the form 'k²', is relatively simple. Start by dividing each side of the triangle into 'k' equal segments. Then, connect these points with straight lines.

For instance, let’s say we want to tile a triangle into 9 segments. Since 9 = 3², we have n = 9 and k = 3. Thus, we divide each side of the triangle into three equal sections and connect the resulting points with straight lines.

Tiling a Triangle into n Tiles, where n = 3k²

At first glance, tiling a triangle into 'n = 3k²' might seem complex, but it can be simplified with some foundational knowledge. In my earlier essay on the rep-tile puzzle, I illustrated how to tile a rep-3 triangle.

To start, use a 90–60–30 triangle with side length ratios of x, 2x, and √3x. Once you construct the rep-3 triangle according to these specifications, apply the 'k²' algorithm we just discussed to each of these triangles.

This will yield 'n = 3k²' tiles.

Tiling a Triangle into n Tiles, where n = (k² + l²)

This scenario is the most intricate of the three. When tiling a triangle into 'n = (k² + l²)', the first step is to arrange 'k' and 'l' in order of size, assuming k > l.

Next, construct a right triangle where the two sides of length 'k' and 'l' are perpendicular. Consequently, the hypotenuse will measure '√(k² + l²)' or √n. Visualize this triangle with the hypotenuse placed horizontally at the bottom.

Then, drop a perpendicular line from the opposite vertex to the hypotenuse. Divide the side measuring 'k' into 'k' equal parts and the side measuring 'l' into 'l' equal parts. Finally, connect these resulting points.

For example, if we wish to tile a triangle using 5 tiles, knowing that 5 = 2² + 1², we identify n = 5, k = 2, and l = 1. Executing the previously discussed method results in the following configuration:

It’s important to note that this approach may lead to tiles that do not align perfectly at the junction between the k-triangles and l-triangle(s), yet the triangle is still considered tiled.

Final Thoughts

You might be curious about the significance of rep-tiles. In essence, they offer infinite scalability.

In the works of Martin Gardner on this subject, I discovered a delightful poem by British mathematician Augustus De Morgan that encapsulates the scalability of rep-tiles (long before Golomb’s time).

To conclude this exploration, here’s a quote from De Morgan that beautifully captures the idea:

"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

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

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For further reading, you might enjoy: How To Really Solve The Monkey And The Coconuts Puzzle? and How To Really Benefit From Curves Of Constant Width?

Read the original essay [here](insert_link).