## Sunday 6 July 2008

### Guess The Pattern: Win a Prize

The first person who can tell me how I made this pattern* wins a special prize. It`s a set of geometrical relationships, but which?. I`m wondering if there is some music here (time will tell, I`m working on it)..

It`s not designed to look nice, infact you could say it isn`t designed at all really, it`s just as it should be, as the proportions dictate. It has a sort of (ancient) Egyptian or Art Deco look to it, a bit.

*Don`t try anything cute like....'you made it with a pencil, a ruler, a compass and some paper'

EDIT: I just made this one from the same proportions..(why wasn`t geometry this interesting at school?)

Anonymous said...

Hi ed,

To start, here's how I think you made the second picture:

First you drew a square with the two diagonals (you might have derived the square just from the initial line but I can't see any working there). Then you made the two arcs with a compass placed at the corner with the radius set to the width of the square. Then you drew the horizontal line where the arcs and diagonals intersect. A second diagonal is then drawn from the new corner defined by this horizontal line to the original corner, which gives you a new intersection point for the next horizontal line and so on. Overall it gives you a nice perspective effect.

In the main picture, you start with the smallest square at the bottom (which also has vertical lines drawn to the top of the big square) and draw diagonals. This gives you the radius of the first arcs. These arcs then intersect your big vertical lines, from which you draw more diagonals and thus get the radius for the next biggest arcs. You then drew regular vertical lines to the heights of your intersection points, and then connected up these points with a ruler to get that nice big parabola shape.

The overall geometic relationship you are interested in depends on which sections of the diagram you are comparing but the most obvious interesting relationship is between the lengths of all those diagonal lines that you use to get the big parabolas. Each vertical line is the same length as the diagonal meeting the vertical line below it, and both are identical to the radius of the arc that produced them.

The radius of each arc forms one side of a right angled triangle on the verticle line, where the next biggest arc radius forms the hypotenuse. If we define that initial square as having area 1, with sides square root of 1, then its diagonal is the square root of 2 (since it forms the side of a square double the size). This square root of 2 becomes one side of the triangle, which when squared and added to 1 squared = the next diagonal squared (pythagoras theorem): so 2 + 1 = the diagonal squared, i.e. the diagonal is the square root of 3. In general, you always get the sqaure of the previous diagonal + 1, which will give you an ascending series of square roots. So that parabola is actually defining the series of square roots.

It took me a while to work this out, including having to use algebra for the first time in about 10 years.

Finally since I see you reference that book by Robert Lawlor in one of your other posts, I'd guess you picked up some inspiration from there.

So do I get my special prize?? It should be my own weight in diamonds or a trolley dash through a heroin depository at least I think.

E.L. said...

[same text as on the patterns society forum]

Congrats, the genesis is not quite correct for either picture as they were basic asymmetric figures originally (you suppose I drew them symmetrically from the beginning), for the larger picture I worked from a basic square roots figure but then made it symmetric and extended the parabolas.

They both derive from figures in the book The Elements of Dynamic Symmetry by Jay Hambidge (quite an old book).

The Lawlor book seems to derive square roots from the vesica pisces from which you can get the square roots of 2, 3 and 5 (as that book is concerned with the history of geometry, whereas the Hambidge one has some history but is concerned with communicating the concepts to designers and artists of the day)

The larger picture was originally just a root 5 rectangle on its side, with the square, root 2, 3 and 4 rectangles within it, with the diagonals and parabolas only drawn on one side and the horizontal lines at the bottom showing the linear decimals (root 2= 1.4142, root 3 1.732, root 4 = 2, root 5 = 2.236). I basically mirrored the pattern, extended the parabolas and continued the sequence up to the square root of 12 (it could go on infinitely).

The more basic square figure was again only drawn on one side, I mirrored it again continued the progression further than root 5 (root 5 seems to be some sort of practical limit in Hambidge`s book, and most geometry books I have read so far including the Lawlor one).

I`ll think of a special prize and will award it to you when you return from die Schweiz soon, well done.

Anonymous said...

its spoctigon, everyone knows that pffft!