My new shape is a very excellent one. It is the surely-soon-to-be-infamous johansendociollagram. It is seventeen-sided, and I will tell you how to construct and tell you some of its traits.
Do you have paper, ruler, protractor, scissors, pencil and glue? Then I’ll begin. Draw a rectangle of substantial size and cut it out. Choose of the smallest ones, an arbitrary side (that instruction work even if you drew a square, yay). Measure that smallest side, and from an arbitrary vertex* draw a mark on the two connecting sides which are that distance away. Connect those marks with pencil. On the currently unmarked longest side (line C) draw a mark in the middle. At ninety degrees, draw an inner line of length greater than a quarter of the length of the shortest side and both less than three times a quarter the length of the shortest and half the length of the longest side. With the end of the current floating line as one vertex, and line C the home of two others, draw the six sides of half of a regular dodecagon again internal to the rectangle and ensuring that you do not cross off onto the marked-off corner.** Cut out your half-dodecagon and stick the long-side to the line of your marked off corner- facing outside and with the long line of the half-dodecagon sticking out of both sides and no other line being at-any-point internal to the rectangle. . You now have your own johansendociollagram.
One interesting trait of the johansendociollagram. Is its area, knowing only the length of its longest side (l) and shortest side (h), the formula for the area is simply lh – (1/16)h^2. Of course, since you put in the effort to construct then you can easily tell why that is.
If I were a nerd, I would tell you its perimeter. But, alas for you, I am not a nerd but just a creative genius creating new shape so you’ll have to work it out yourself.
Another interesting question is about the entire family of ‘johansendociollagram’s. For exmple, what if instead of a regular dodecagon, it had a regular decagon. Then it would have the same area formula, but fifteen sides. Octagon, same area formula; thirteen sides. Heptagon, same area formula; eleven sides. Quadrilateral, same area formula; eleven sides. !. !!. How peculiar, the formula has broken down. What mystery of the mathematical world has the hip ‘johansendociollagram’ family helped shed on the square quadrilateral? It appears that unlike the other even-sided regular polygons which become polygons of half-their-sides-plus-one, when halving the regular quadrilateral you simply get an irregular quadrilateral which can itself be halved into an irregular quadrilateral of the same sort, or even potentially a regular quadrilateral.
Wow. And that is simply one advance by the simple ‘johansendociollagram’.
*’Vertex’ is geometry for ‘corner’
** Alternative method for getting this far. Draw a regular dodecagon, bisect that with a longer line. Make that line the longest side of a rectangle with the other side having length no greater than one and a half times the width of the dodecagon and no less than half the width of the dodecagon.
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