Puzzle Description:

A famous medieval exercise from China (or Japan): Connect all four corner points of a square to two opposite median points! It makes a central polygon. Is it a regular octogon or not? Why or why not?

#1: Joe (infrapinklizzard) on Jul 23, 2013 [SPOILER]

Given that you're starting with a symmetrical shape and using points that are on axes of symmetry, yes, the octagon formed will be regular.

(Four of its vertices will be on the horizontal & vertical axes of symmetry and four on the diagonal ones.)

You're wrong, Joe. It's true what you wrote about symmetry, but it's not enough to regularity. Think about differences of a square and a rombus. Both are symmetrical but not equal.
So the real question is: WHY regular or WHY NOT?

Derp. Of course. The diagonal axes are stretched compared to the orthogonal ones, so the internal angle of their rays is smaller than those of the horiz&vert ones. This pushes the crossings farther toward the center than the vertices on the horizontal & vertical axes. (The diagonal radii are shorter than the orthogonal ones.)

Pls answer the question: why regular or why not? :)
You mentioned "internal angle". It is determined in a regular octogon: definetely 135°, so that is no question...

Our octogon has got the same edges (cause of symmetries) but the internal angles are
[180° - (arctg2-arctg0.5) =] 143,13° and 126,87°. Four of this, four of that (cause of symmetries).
So this is a very pretty, symmetric, but NON-REGULAR octogon.
Q.E.D.

I love math...but these puzzles are too easy to be much fun to solve, Thomas. I've solved several, but at this point I don't think I'll be doing any more...and I really WANTED to like your puzzles.

Goto next topic

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