Wheels

A Surprising Symmetry

An examination of an article by Frank Farris
by Michael J. Pawlisheck ,     Fall 1996

This notebook  has been slightly edited by Ken Levasseur - Jan  1997.
Graphics were removed to save time in tranfering file.

A review of the article published in Mathematics Magazine

Frank Farris was designing a computer laboratory excersise for his calculus students when he happended to sketch the curve defined in the following vector equation:

(x,y) = (cos(t), sin(t)) + 1/2(cos(7t), sin(7t)) + 1/3(sin(-17t), cos(-17t))

This equation traces the the curve of particle mounted on a wheel mounted mounted on a wheel.
A visual explanation of the wheels can be seen in the figure below.  The smallest wheel rotates in the clockwise direction, while the the other two wheels are rotating in a counter clockwise direction.  The smallest wheel is also out of phase with the other two.

Show[Graphics[{Circle[{0,0},1],
				Circle[{1,0},1/2],Circle[{1.5,0},1/3]}]]

The graph generated by  the equation is shown below:

Clear[F]
F[t_]:=E^(I t) + 0.5 E^(7 I t) + I/3 E^(-17 I t)
ParametricPlot[{Re[F[t]],Im[F[t]]},{t,0, 2Pi},
Axes ->False,AspectRatio ->Automatic]
Clear[F]

Frank Farris goes on to explain how the symmetry of this curve is illuminated by by introducing complex notation (as I did above) in which the terms represent a terminating complex notation.
This particular curve has a symmetry of 6, because the frequencies used, 1, 7, and -17 are all congruent to 1 mod 6. He  then provides further examples of k mod m to show varying frequencies.  The next curve has a symmetry of 7 since the frequencies are congruent to 5 mod 7.

Clear[F]
F[x_]:= E^(-2 I x) + (E^(5 I x))/2 + (E^(19 I x))/4
ParametricPlot[{Re[F[x]],Im[F[x]]},{x,-2Pi,Pi},Axes->False,AspectRatio ->Automatic]
Clear[F]

The frequencies in this example are -2, 5, and 19 with the same circle coefficients (sizes) as the first example.  Farris examines one last curve before beginning to form a proof.  This last one has four circles and a symmetry of 9.  The frequencies, 1, -16, -7 and 29 are all congruent to 2 mod 9.  He has also adjusted the coefficients to 1, i/2, 1/5, and i/5, to make the pattern more pleasing.

Clear[F]
F[y_]:= E^(2 I y) + I/2 E^(-16 I y) + 1/5 E^(-7 I y) + I/5 E^(29 I y)
ParametricPlot[{Re[F[y]],Im[F[y]]},{y,0, 2Pi},
Axes ->False,AspectRatio->Automatic]

The next section examines how various frequencies can be generated and what rules can be used to generate them.

An examination of the 'rules' conjectured by Frank Farris

In the third section of Frank Farris's article he mentions that discovering the criteria for symmetry would be an interesting calculus problem.  This section examines how to generate the frequencies.

The calculations below generate tables of frequencies that can be examined.  
The first step clears the variable k of any previous values.  
The second step establishes the range which will be examined, of course the range could be larger or smaller.  So it is entirely subjective.
The third step generates a table of y=k mod m.
The fourth step creates a table, which lists the position from the prevoius table, that is equal to the value shown.
The fifth step lists the actual value, in the range of k, that was generated in the first step.   It is a rather convuluted way to get the value listed, and there is likely a more efficient way.  So it is the fifth step that finally generates what could be acceptable frequencies.
The sixth step is a title for the table that is actually output to the screen and it goes at the bottom of the table.
The first set of tables generated are for an unacceptable set of frequencies.  The are unaccetable because they aren't prime to m which is the symmetry to be expected.  An examination of the graphs generated from unacceptable frequencies will help explain why they are unacceptable.

Text["Unacceptable Frequency Table"]
Clear[k];
k=Range[-10,100];

y=Mod[k,4]
y1=Flatten[Position[y,2]]
m14=k[[y1]]
Text["m14"]
GCD[m14,4]

TextClear[k];
k=Range[-10,100];

y=Mod[k,4];
y1=Flatten[Position[y,1]];
m14=k[[y1]]
Text["m14"]
GCD[m14,4];

y=Mod[k,4];
y1=Flatten[Position[y,3]];
m34=k[[y1]]
Text["m34"]
GCD[m34,4];

y=Mod[k,5];
y1=Flatten[Position[y,2]];
m25=k[[y1]]
Text["m25"]
GCD[m25,5];

y=Mod[k,5];
y1=Flatten[Position[y,3]];
m35=k[[y1]]
Text["m35"]
GCD[m35,5];

y=Mod[k,6];
y1=Flatten[Position[y,1]];
m16=k[[y1]]
Text["m16"]
GCD[m16,6];

y=Mod[k,6];
y1=Flatten[Position[y,5]];
m56=k[[y1]]
Text["m56"]
GCD[m56,6];

k1=7;
y=Mod[k,k1];
y1=Flatten[Position[y,1]];
m17=k[[y1]]
Text["m17"]
GCD[m17,7];

k1=7;
y=Mod[k,k1];
y1=Flatten[Position[y,3]];
m37=k[[y1]]
Text["m37"]
GCD[m37,7];

k1=7;
y=Mod[k,k1];
y1=Flatten[Position[y,5]];
m57=k[[y1]]
Text["m57"]
GCD[m57,7];

k1=8;
y=Mod[k,k1];
y1=Flatten[Position[y,1]];
m18=k[[y1]]
Text["m18"]
GCD[m18,8];

k1=8;
y=Mod[k,k1];
y1=Flatten[Position[y,3]];
m38=k[[y1]]
Text["m38"]
GCD[m38,8];

k1=8;
y=Mod[k,k1];
y1=Flatten[Position[y,5]];
m58=k[[y1]]
Text["m58"]
GCD[m58,k1];

k1=8;
y=Mod[k,k1];
y1=Flatten[Position[y,7]];
m78=k[[y1]]
Text["m78"]
GCD[m78,k1];

k1=9;
y=Mod[k,k1];
y1=Flatten[Position[y,1]];
m19=k[[y1]]
Text["m19"]
GCD[m19,k1];

k1=9;
y=Mod[k,k1];
y1=Flatten[Position[y,2]];
m29=k[[y1]]
Text["m29"]
GCD[m29,k1];

k1=9;
y=Mod[k,k1];
y1=Flatten[Position[y,5]];
m59=k[[y1]]
Text["m59"]
GCD[m59,k1];

k1=9;
y=Mod[k,k1];
y1=Flatten[Position[y,7]];
m79=k[[y1]]
Text["m79"]
GCD[m79,k1];

Create graphs of other symmetries with various frequencies

The first graph generated in this section is from the set of unacceptable frequencies.  The expected symmetry is 4, since the values are from the table generated by 2 mod 4 over the range of -10 to 100.  In addition all of these graph use the 1, 1/2, 1/3 circle size to minimize the total number of graph generated.  An examination of other coefficients might produce interesting graphs.

Clear[F]
F[t_]:=E^(6 I t) + 0.5 E^(22 I t) + I/3 E^(-10 I t)
ParametricPlot[{Re[F[t]],Im[F[t]]},{t,0, 2Pi},
Axes ->False,AspectRatio ->Automatic]
Clear[F]

As you can see in the graph above the symmetry is not 4 it is 8.  The graph below shows a symmetry of 4 and the frequencies were taken from the first table of 1 mod 4.

Clear[F]
F[t_]:=E^(I t) + 0.5 E^(13 I t) + I/3 E^(21 I t)
ParametricPlot[{Re[F[t]],Im[F[t]]},{t,0, 2Pi},
Axes ->False,AspectRatio ->Automatic]
Clear[F]

The graph below has a symmetry of 5, it is generated from the frequencies 2 mod 5.

Clear[F]
F[t_]:=E^(3 I t) + 0.5 E^(18 I t) + I/3 E^(-7 I t)
ParametricPlot[{Re[F[t]],Im[F[t]]},{t,0, 2Pi},
Axes ->False,AspectRatio ->Automatic]
Clear[F]

There exists the possibility of selected frequencies from one set that is also part of another set, the symmetries are different but the frequencies are the same.  In that case the larger symmetry is graphed, as shown below.  The frequencies are from both 1 mod 4 and 1 mod 8, which of course makes sense, since 8 is a multiple of 4.  To counteract this at least one frequency must be unique.

Clear[F]
F[t_]:=E^(I t) + 0.5 E^(-7 I t) + I/3 E^(9 I t)
ParametricPlot[{Re[F[t]],Im[F[t]]},{t,0, 2Pi},
Axes ->False,AspectRatio ->Automatic]
Clear[F]

Examine other Conic Sections

One of the objectives of this review was to examine other conic sections.  In this section I looked at ellipses.  I tried to arrange the ellipses with different axis, but for graphing I was only able to use the same axis for each ellipse.  First I will re-draw the first graph and then examine it with ellipses.

Clear[F]
F[t_]:=E^(I t)+ 0.5 E^(7 I t) + I/3 E^(-17 I t)
ParametricPlot[{Re[F[t]],Im[F[t]]},{t, 0, 2Pi},
AspectRatio ->Automatic]
Clear[F]

Clear[F]
F[t_]:=E^(I t)+ 0.5 E^(7 I t) + I/3 E^(-17 I t)
ParametricPlot[{Re[F[t]*4],Im[F[t]*2]},{t, 0, 2Pi},
AspectRatio ->Automatic]
Clear[F]

As you can see the graph flattened out.  It seems to have lost its symmetry of 6 and now seems to have a symmetry of 3.  I think any increase in the difference between the foci will just flatten the curve out further.
I suspect that any conic section that is not a circle will not produce symmetrical results.  I have not listed results from a parabola because it is not possible for that to produce a symmetrical graph because it is only a portion of closed section.

Comments

There are portions of Mathematica that I found difficult to master, and the student manual that came with the program was somewhat indequate, due to its incomplete listing of all the functions available.  I perused the manual at the book store and was impressed with its completeness and weight.  I kind of regret not becoming familiar with this program sooner.
 I was positively impressed with this program and I recommend that it be introduced sooner or perhaps the instructors should mention it as an aid to students in Calculus.

Bibliography

Mathematics Magazine, Vol. 69, No. 3, June 1996
Wheels on Wheels on Wheels
 -Surprising Symmetry
   by Frank A. Farris
	Santa Clara University
	Santa Clara, CA 95053

Fundamentals of Complex Analysis for Mathematics, Science, and Engineering
 by E.B.Saff and A.D.Snider
 Pub. 1993 by Prentice-Hall, Inc.