How to Draw a Special Spiral

Geometry Level 3

Given the central square which is divided into four congruent triangles, a square spiral has been formed by successively circumscribing bigger and bigger squares in such a way that all the triangles have the same area.

We could circumscribe even more squares while satisfying that condition.

Now, starting with a square, can we also draw a square spiral by successively inscribing infinitely many distinct squares in a similar manner?

Yes No

1 solution

Michael Huang
Oct 28, 2018

Let $A_n$ denote the area of $n$ -th square, where $A_0$ is the area of the smallest square at the start of the circumscribing procedure. Since the triangles are congruent to each other, the area sum of four identical triangles set can be represented as $A_0 = \left(A_{n + 1} - A_{n}\right)$ . In this case, the recurrence relation of the square areas is $A_{n + 1} = (n + 2)A_0$ where $A_n \neq A_m > 0$ and $n \neq m$ . But since $A_n$ increases linearly, it strictly increases throughout the values of $n$ , which makes inscribing procedure impossible.

How to Draw a Special Spiral

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