Exploring the geometric ratios in Rick’s paradox

Making curves out of lines

Rick’s paradox in a nutshell is a pattern of nested triangles that appear to create create curves that spiral toward the center. This pattern is commonly applied in art, an example being zentangles.

The pattern is made by first drawing a single triangle, and then choosing three points on each side that are of an equal distance from a certain vertex. Afterward, these three points are connected to form a new triangle. and the process is repeated.

Properties of the pattern

The pattern is constructed such that the triangles are similar to one another. What therefore defines a specific case of the pattern is its scale factor. The square of said scale factor will also give the ratio of one triangle’s area to another. These properties are outlined as follows:

Let AB = BC = AC (equilateral triangle), then

AD = BE = CF and AF = BD = CE

by extent, AD/BD = BE/CE = CF/AF (ratios are also equal)

Scale factor = DE ÷ AB = AB:DE

Area factor = (Scale factor)² = DE² ÷ AB² = AB²:DE²

Variables and prerequisites

The subject of this article is to generalize the area factor of a given Rick’s paradox pattern. To emphasize certain concepts, special cases of Rick’s paradox will be used, specifically one where AD:DB can be expressed as 1:d, where d is an integer, and d+1 will be the length of the larger triangles. As mentioned, all of these cases will be equilateral by definition. And only the first recursion of the pattern will be calculated for convenience.

The actual math

d = 1

The simplest pattern would be when the vertices of the new triangle bisect the previous one. Upon illustrating such a figure, it can be directly seen that the new triangle makes up 1/4 of the triangle it is nested in.

d = 2

Although this example may seem more daunting than the last, it is still straightforward to calculate. In Rick’s paradox, whatever the case or scale factor, nesting a new triangle will also create three congruent triangles. Since the given two sides are known, trigonometry can be used to obtain the third. But that can be saved for later. In such case, the three outer triangles are actually special 30–60–90 triangles, as proved by SAS. Given this, the third side, and also the side length of the triangle is found to be √3. The area factor of this pattern can be obtained simply by taking the square of the scale factor:

√³²:(1+2)² = 1:3 or 1/3

d = 3

This time, there are no convenient right triangles that can be directly found. Fortunately, as previously mentioned, trigonometry can be applied given that two sides of the outer triangle are known along with the angle opposite the side in question. Using the cosine rule:

c² = a² + b² − 2ab*cos(C)

x² = ¹²+ ³² − 2(1)(3)*cos(60)

x² = 10 − 6*0.5

x² = 7

Since all that is needed is the area factor, 7 can already be substituted into the scale factor:

area factor = 7:(1+3)² = 7:16 or 7/16

d = 4

The process is exactly the same as the previous case: simply apply the cosine rule

c² = a² + b² − 2ab*cos(C)

x² = ¹²+⁴²− 2(1)(4)*cos(60)

x² = 17−4

x² = 13

area factor = 13:(1+4)² = 13:25

With this case, it can clearly be seen that the area of the inner triangle is now more than half of the area of the outer triangle (0.52>0.5).

Generalization for all ‘d’

Given a ratio of 1:d, we can generalize the ratio of the area of an inner triangle to the are of the triangle it is directly nested in as follows:

x² = 1 + d²− 2(1)(d)*(0.5)

x² = 1 + d²− d

x² = d²− d+ 1

The ratio would therefore be:

d²−d+1 / (d+1)²

or

d²−d+1 / d²+2d+1

Just to see how intriguing the entire paradox is, the following visual better outlines the curves the triangles form: