Author: johansenindustries

It’s A Lot Easier When You Know The Answer

What mathematical term is used to describe the kettle upon the tall stove? Hypotenuse!

I was reading a joke quiz where there were several mathematical terms and cryptic fake definitions. Such as the one in the first paragraph, or ‘Tangent: A professor after a week at the beach’. And one had to match the definition to the word. The ‘tangent’, I found easy. The ‘hypotenuse’ took until everything else was eliminated.

But once everything else was eliminated, then I got it. Like every good puzzle, once you know the answer it is obvious.

So the question that I ask is: Can one simply pretend to know the answer, and then let the ‘aha’ muscles do the work.

As Penny Lane says in Almost Famous: ‘The truth just sounds different’.

My New Shape – Meta

There are discoveries to be made everywhere. There is newness everywhere. The question is if it is a discovery worth being made. Well, ‘discoveries’ are fun so as a simply leisure activity then quite possibly. But there is also the question of general importance.

One quality of the great mathematician, or even the merely competent, is an understanding of what facts or discoveries are interesting. Now, the johansendociollagram is not interesting, and what might be interesting certainly isn’t original. So I could never be a competent mathematician, but then again since I realise the johansendociollagram is not interesting then perhaps it is simply my incompetence at everything past multiplication* that gets in the way.

It is unknown, for it is perhaps most of interest to those in mathematical cycles but it is hardly conductive to mathematical interest, the true interaction between interesting mathematics an physcs and the whys of that, although there are numerous figures. One has to observe, however, that if it turned out that johansendociollagrams were the answer to dark matter (let’s say) then they would have to be interesting in some sense.

But if we think about that example for even a second then it is obvious that the johansendociollagram cannot be the answer to that mystery. For if it were to be found in the simple manipulations of two-dimensional shapes then it would have been solved, with the johansendociollagram coming as an a technical improvement to better explain and summarise the basic d-shape insight. It is perhaps that fact which is the proof of its uninterestingness, but no mathematicians will grab torches at the suggestion of physics being the proof of interestingness.

*But, cor, I used to be good at multiplication. Apart from 7 * 8, that was my Achilles heel.

I Invented a New Shape

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.