Today is π-day. Investigating the properties of a circle is one of my favorite hobbies, so this is an appropriate event for some discussion on the circle. I love drawing circles and lines on a piece of paper. It's fascinating to discover natural laws in the circle. At each of my drawings, I gain some new insghts. Recently I came across a Cut The Knot post presenting an inventive way to divide a circular area into any number of equal parts, with compass and ruler. I googled a little more over it and curiously, it seems that the presented yin and yang-like solution is the only one that has been presented (if you know of other published references, I would be eager to hear about). It appeared in a Mathematical Association book in 1995 but I didn't find any other reference to solutions of this problem. This is surprising, because area division is a fundamental problem with broad applications in this world where sharing physical resources equally is a necessary guarantee for peace. Playing a bit with the elementary components of the yin and yang pattern gives related solutions, of which a random one is sketched in Figure 1.

I convinced myself that the most general solution is in fact a quite natural and intuitive one, because it figures the way how circular patterns often grow in nature, like the annual rings in wood. Other patterns also give some insight in various ways to divide a circular area into parts, like the beautiful quintuple emulsions formed with microfluidics of the figured image (courtesy of A. Abate/ Harvard University).

The general rule to have a circular area divided equally into N of such nested circles is that, for a starting circle of radius 1, the surrounding Nth circle radius is the square root of N (noted √N).

The first circle has radius 1, area π.

The second circle has radius √2, area 2π.

The third circle has radius √3, area 3π.

The fourth circle has radius √4 (= 2), area 4π.

And so on...

Square roots can be easily drawn with compass and ruler, so the same for nested circles with radius equal to √N, where N is incremented from 1 to any positive integer value. Figure 4 shows a convenient procedure for N=1 to 9:

Step 1. Draw nested circles with diameter 1 to 9 whose left diameter endpoints coincide.

Step 2. Draw a perpendicular to the diameter direction at the intersection of the right endpoint of the circle with diameter 1.

Step 3. Draw circles of radius equal to √N thanks to intersections of first circles with perpendicular at N=1.

Step 4. Erase lines and circles of intermediate steps. The obtained outer circle is divided into N equal parts each of area π.

Now, from this pattern, it is possible to start a set of variations on circular area division, because the nested circles are whole, contrarily to the yin and yang pattern. We may for example start to divide the circle of area 2π into two equal half-circles and shift those components randomly (see Figure 5). Or we may start to divide the circle of area 3π into equal parts, etc. I'm sure anyone can find original new ways to arrange and transform all those equal parts, in order to gain some new intuitions on the circle.

Happy π-day!

what would be the application of the above problem? i am intrigued by it.

ReplyDeletethanks

Interesting question, Avinash. I'll have to think over that for a while. I have no idea for a direct technical application. But it could help to gain some insight in physical laws where equal quantities of matter are distributed over a surface. By the way, I think we could do the same for equal spherical volumes. Maybe my next challenge;-)

ReplyDeleteHi Arjen,

ReplyDeleteNice piece on the circle., Of course there is two ways to look at things in the world , one being to quantify them an another to qualify them. Here you qualified acircle which is what Archimedes did when he made some of the first pure mathematical approximations of Pi when he considered a circle as a polygon with infinity as its limit. However looking at the qualities of things also has proven useful as tolend insight, particularly in physics.

That is rather then to think of a circle as is commonly done, being an infinite number of points of equal distance from a central radius, which requires action to construct, if only in one’s mind, rather we could say qualitatively that it is the boundary of least length that can enclose the greatest area and thus form a line being a continuum in having no beginning or end, has no need to be referenced to infinity.

This of course is to express a circle by its character, which recognizes its quality of conservation. The interesting thing is this conservation has the circle also to have a corresponding symmetry which displays yet another aspect of its character in terms of quality; which Emmy Nether was to prove true in the more general sense. I’ve often thought that one of the problems in physics is that the qualities of things are not often looked to enough as the more fundamental aspects of things, rather then how they are quantified. So while PI is a ratio needing infinity to have it quantified, in terms of quality it is simply something both finite and whole that requires no such abstraction, which can in turn often leads to contradiction and paradox .

“From man or angel the great ArchitectDid wisely to conceal, and not divulge

His secrets to be scanned by them who ought

Rather admire; or if they list to try

Conjecture, he his fabric of the heav'ns

Hath left to their disputes, perhaps to move

His laughter at their quaint opinions wide

Hereafter, when they come to model heav'n

And calculate the stars, how they will wield

The mighty frame, how build, unbuild, contrive

To save appearances, how gird the sphere

With centric and eccentric scribbled o'er,

Cycle and epicycle, orb in orb.”

-Milton- Paradise Lost

Best,

Phil

Hello Phil,

ReplyDeleteThanks for these ideas on quantity and quality, and for Milton's lost paradise. I'm not too well acquainted with the history and ideas of mathematics, but these drawings with circles made me look closer at it. It seems that since the time of Wallis's semi-geometric derivation of a value of pi, men chose to investigate the quantity (through infinite series) rather than the quality and lost a paradise. I find it rather awkward that proofs of analytical approximations on pi are today so much disconnected from the physics. My thoughts aren't crystallized on that point, so I'm still searching and will post some more on it the coming weeks.

Best wishes,

Arjen

Hi Arjen,

ReplyDeleteI guess what I’m simply trying to express is how it’s so widely believed within physics that all can be boiled down to a mathematical description, as to have the reason for things. So as Emmy Noether was able to takes two aspects of quality being conservation and symmetry to prove that one can never or does exist without the other it doesn’t to me seem to have had the impact on scientific thinking that one would have thought it should; despite the emphasis drawn to it by both Einstein and Hilbert.

That is one could say that what she demonstrated mathematically is what Plato so long ago understood, being that truth and beauty were one and inseparable; that’s here finding conservation as the truth and symmetry as beauty. That’s to say with this demonstrated it is still widely ignored that truth and beauty on their own being not representing reason, yet rather symptoms or indicators of it. Plato would state the reason being “the good”, which I express here as what serves to be its quality. So I’m not actually saying to ignore mathematics or minimize it’s importance, yet rather it aids solely to predict outcome, rather then to be what’s responsible for it, which for me is the root of it all,with that being reason itself.

Best,

Phil

Avinash, Phil,

ReplyDeleteThanks for your comments which stimulated continued insight in the next post.

I think the most interesting application might be in analog computing: applying various algorithms to simple analogs to reproduce the computing power of more complex ones.

ReplyDeleteHi Jason,

ReplyDeleteDo you mean that the nested circle areas may be used to model more complex algorithms?