Origami imperfection 2

Geometry Level 4

Make 6 surface modules for the sides of a cube. Pictures 1 and 2.

To three of these modules add an inward corner so the cube will have an indentation. Pictures 3 and 4. (Note the equilateral triangle in picture 4.)

Assemble 5 indented cubes into a ring (glue required) and 20 cubes into a dodecahedron. Final picture.

When two indented cubes are attached they don't form the exact interior angle required for a regular pentagon. The error is small enough to not be noticeable and in reality, the paper bends to hide the imperfection. Find the error in degrees.

Source of this model

The answer is 1.471220634.

1 solution

Jeremy Galvagni
Nov 19, 2018

This is a top view of three cubes. The three sides of the pentagon should be apparent. $ABCD$ is the square side of a cube but is seen edge-on, it is foreshortened. $AB:BC$ appears to be $\sqrt{2}$ .

$F$ and $E$ are midpoints of their respective sides, so this ratio is preserved. $\tan{\angle FED} = \sqrt{2}$

$\angle AEH$ is twice this angle: $2 \cdot \tan ^{-1} \sqrt{2} \approx 109.4712^{\circ}$ . So the error is $109.4172-108 = \boxed{1.4712}^{\circ}$

A new book for my Christmas wish list. Great problem!

David Vreken - 2 years, 6 months ago

Origami imperfection 2

The answer is 1.471220634.

1 solution

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