Circloid

Copyright 2005 by Paul Niquette, all rights researved.

n the right you see an animated figure called an "epicycloid."  It is produced by a point on the circumference of a circle rotating inside the circumference of a larger circle without slipping.  The outside circle has a radius a and the rotating circle has a radius b.  For the case you see here, b = a/3, and the figure in red is called a "deltoid."

Animation by Weisstein, Eric W. "Deltoid." From MathWorld -- A Wolfram Web Resource.

Examine the figure on the left.  You will see that as the inside circle rotates clockwise it is orbiting counter-clockwise, while its center makes the angle Φ with the x axis.  Epicycloids are best described mathematically by writing a pair of equations using Φ as a parameter... 

x = (a - b) cos Φ - b cos [(a - b) Φ/ b]
y = (a - b) sin Φ + b sin [(a - b) Φ/ b]

The radii a and b can take on any values.  Integers produce closed figures.  For the deltoid, b = a / 3, and for simplicity, we can let a = 3 so that b = 1.  The equations then read as follows:

x = 2 cos Φ cos (2 Φ)
y = 2 sin Φ -  sin (2 Φ)

Perhaps you are asking the question, "What does all this trigonometric stuff have to do with sweeping a line segment around inside a figure as posed in the Circloid puzzle?"  Read on.

n the right, you see a point P on the deltoid produced by the inside circle having rolled so that its center makes an angle  with the x axis.   You also see a point Q on the deltoid produced by the inside circle having rolled on around so that its center makes an angle  radians with the x axis. 

Don't you now want to know what the length of the line segment PQ might be?  Prepare yourself for a surprise.

The equations for the deltoid tell us the exactly locations of P and Q as follows:

xP = 2 cos  + cos (2)
yP = 2 sin  -sin (2)

xQ = 2 cos () +  cos (2+ 2)
yQ = 2 sin () -  sin (2+ 2)

Trigonometry students know that cos () = - cos  and sin () = - sin  and that adding  2 to any angle is the same as adding zero.  So our equations for point Q simply read...

xQ = - 2 cos Φ cos (2 Φ
yQ = - 2 sin Φ -  sin (2 Φ

For calculating the length of PQ, we first take the following differences:

xP - xQ = 2 cos Φ + cos (2 Φ) + 2 cos Φ cos (2 Φ) = 4 cos Φ
yP - yQ = 2 sin Φ -sin (2 Φ) + 2 sin Φ + sin (2 Φ)  = 4 sin 

...and applying Pythagoras Theorem, we find that...

Length of PQ = [(xP - xQ)2 + (yP - yQ)2]1/2 = [16 cos2 (Φ) + 16 sin2(Φ)]1/2 = [16]1/2  = 4.

Therefore, PQ = 4 units in length.  Notice, please, that PQ = 4 no matter what that angle Φ might be!  Exclamation point intentional.  By inspection you will see how the longest line segment that will fit inside the deltoid is indeed 4.  So the solution to the puzzle is...
 

Deltoid

h, but wait.  Is that the only solution?  Try letting b = a / 5.  The figure will have five cusps, making it a "pentoid."   Do the math and you will find that for a = 5 and b = 1. PQ will be 8 units in length -- again independent of 

Indeed, if b = a / (2k + 1) where k = 1, 2, 3..., then PQ will have the same properties no matter how large k gets to be.  How about an exclamation point for that!  The length of PQ will be given by [4k/(2k + 1)] a.  Thus, as k increases without bound, the length of PQ approaches 2a.  Hey, that's exactly the diameter of the outer circle.  Which brings us all the way around to where the puzzle began. 

Our efforts have produced an epicycloid with an infinite number of cusps. If there is not a name for that, let's call it a circloid

Don't bother to look it up in a dictionary, circloid appears here first.

 
ome puzzle-solvers may be disappointed that the Circloid puzzle gives away the solution without showing how it was found.  That may not be possible in this case.  Here's why.  One day, the author, who has an abiding interest in lighter-than-air craft (see "Sun Ship"), launched a search on the web that turned up a book entitled The Deltoid Pumpkinseed.  That stimulated curiosity about the term "deltoid," and a subsequent search produced the Mathworld link you see above. 

While watching that hypnotic animation, the author gradually became distracted by an irrelevant question:  "What is the longest line segment that will fit inside a deltoid?"  There was absolutely no reason to ask it.  That idle query set off an investigation which resulted in a discovery. 

But the discovery was not the solution to a puzzle at all.

Producing the Circloid puzzle was merely a matter of turning the discovery inside out. That left a huge gap in reasoning, though.  Other puzzles have been created about the same way (see for example "Reaman Numeral" or "Fermat's Really Last Theorem") -- but the pathways to their solutions were gap free.

So, then, what is the purpose of the Circloid puzzle?  That will be found not in the success of one discovery but in the failure of another.
ake a closer look at the deltoid.  Does it not appear to be made up of three circular arcs?  We can confirm that hypothesis with a few mathematical operations.  If so, we may be on the brink of another discovery here.  It takes only three points to define a circle, so let's get started.  This diagram will help.

Hypothetical arc ABC is shown in green.  Its unknown radius is...
c = AE = BE = CE.
By inspection, you will easily see that AO = 3, that BO = 3 - 2 = 1, and that angle DAO = 30o
( π/6 radians).  Thus,
DO = 3 sin( π/6) = 3/2.
Now, BD = 3/2 - 1 = 1/2, which gives us right triangle ADE with hypotenuse c and legs (27/4)1/2 and c - 1/2. 

Pythagoras' Theorem and a little algebra will do the rest: c2 = [(27/4)1/2]2 + (c - 1/2)2 = 27/4 + c2 - c + 1/4, so that
c = 28/4 = 7.

he three points A, B, C are each exactly 7 units distant from E, which seems to finish the job for us.  In mathematics, though, "seems" won't do.  It only takes one exception to demolish our hypothesis.  A simple spreadsheet shows that for any point F on the deltoid other than A or B or C, the distance EF will always be greater than 7 units!  An exclamation point has been reluctantly placed at the end of the previous sentence.  So much for that discovery.  The "error function" for our hypothesis appears abpve.


 
 


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