folding beauties.pdf

Folding BeautiesAuthor(s): Leah Wrenn BermanSource: The College Mathematics Journal, Vol. 37, No. 3 (May, 2006), pp. 176-186Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/27646327 .

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Folding Beauties

Leah Wrenn Berman

Leah Berman ([email protected]) attended the Illinois Mathematics and Science Academy and Lewis & Clark

College. She received her Ph.D. from the University of

Washington in 2002. Since then, she has worked at Ursinus College in Collegeville, PA. She is interested in

geometry, cooking, and singing. She lives with her

husband, their daughter liana, who was born this past August, and Jasper their dog.

Conies formed by folding At a minicourse on origami at the 2004 Joint Mathematics Meetings led by Thomas

Hull, I was introduced to an amusing method for constructing a parabola.

1. Identify a point P in the interior of a square of paper. (Any paper with a straight bottom would work, but origami traditionally uses square paper.)

2. Choose a point on the bottom edge of the paper, and fold that chosen point onto

P.

3. Repeat as many times as you like, with different points on the bottom edge folded onto P.

The collection of creases thus created outlines a parabola (see Figure 1).

Figure 1. Folding the edge of a square onto a point outlines a parabola

This method for constructing a parabola turned out to be quite well known. The book by Yates [11] discusses it and has a wealth of information about many kinds

of curves. The short book by Row [9], first published early in the 20th century, has

many interesting sections on geometrical paper folding, including a discussion of this

construction method. Another approach was given by Pohl [8, p. 30] in her book on

string designs. To convince yourself that the curve thus outlined is a parabola, consider the follow

ing (see Figure 2). Suppose Q is the chosen point on the edge of the paper, the crease

?i is the perpendicular bisector of PQ, ?2 is the line through Q perpendicular to the bottom edge of the paper, and i\ and i2 intersect at R. Since R lies on the perpendicu

176 ? THE MATHEMATICAL ASSOCIATION OF AMERICA



Figure 2. Repeated folding forms lines tangent to the parabola that is the locus of points

equidistant from P and the bottom edge of the square.

lar bisector of P Q, it is equidistant from P and the edge of the paper, so P is the focus

of a parabola V and the edge of the paper is its directrix.

It remains to show that line ?\ is actually tangent to V at R. A convenient fact

about parabolas is that if ? is any line that is neither equal to nor parallel to the axis

of the parabola V, then ? is tangent to V if and only if the intersection of ? and V is a single point. To see this, suppose that line ?\ intersects V at points R and R\ and

suppose Q! is the closest point on the directrix of V to R'\ by construction, Q is the

closest point on the directrix to R. Since Rf is on the parabola V, it must be equidistant from P and Q'. But R' is also on ?\, the perpendicular bisector of PQ, so R' must be

equidistant from P and Q as well. Therefore, the lengths \R'Q'\ = \R'P\ = \R'Q\. Because Q' is the (unique) closest point to R' on the directrix, it follows that Q = Q! and therefore R! = R. Hence R is the only point on ?\ that lies on the parabola, and so

?\ is tangent to the parabola at R.

Ellipses and hyperbolas can also be outlined by repeatedly folding points on a circle

onto a point P (see Yates [11, p. 50]): if P lies inside the circle, the creases outline an ellipse, while if P lies outside the circle, the creases outline a hyperbola. In both

cases, the center of the circle is one focus and P is the other focus (see Figure 3). If P

lies on the circle, all of the creases pass through the center of the circle.

(a) (b)

Figure 3. Folding a circle onto a point P. The point Q lies on the circle that is folded onto P, and A is the center of the circle, (a) If P is inside the circle, an ellipse is outlined, (b) If P is outside the circle, a hyperbola is outlined.

VOL. 37, NO. 3, MAY 2006 THE COLLEGE MATHEMATICS JOURNAL 177



Envelopes The following discussion of the construction of envelopes is adapted from Rutter [10,

pp. 244-269], Cox, Little and O'Shea [3, pp. 139-144], and Bruce and Giblin [2, Ch. 5]. The last reference has especially detailed information, examples, and exercises on envelopes, and the last two references also describe how to compute envelopes

given an implicit, rather than parametric, function.

Definition 1 (Intuitive Definition). Given a one-parameter family of curves Qt, an envelope curve (or simply, an envelope) is a curve where every point is a point of

tangency to some member of the family.

One problem with this intuitive definition is that an envelope curve may consist of several disconnected pieces. For example (see also Bruce and Giblin [2, pp. 73-75]), consider the family of circles of radius 1 centered at points along the x-axis: there are two curves that are tangent to each member of this family, the lines x = 1 and x = ? 1. An envelope of the family of circles may consist of either one of the lines, or both lines; see Figure 4. Other problems and delicacies with defining envelopes are

addressed by Bruce and Giblin [2, Ch. 5].

Figure 4. A family of circles, with an envelope

A more precise definition is needed to compute envelopes explicitly; our approach is to construct an envelope curve using the following procedure.

Given a family of parametric curves, so that for each t, Qt is a parametric curve

parametrized by s as

Gt(?) = (x(t,s),y(t,s)),

first find a solution to

dx dy dx dy --=0 (1)

ds dt dt ds

for s as a function of t. A solution to (1) usually will be written as s0(t), although at

this point there is no claim either of existence or of uniqueness of such a solution.

Using such a solution so(t), an envelope curve s is defined as a parametric function

of t by the formula

s{t) = Gt(so(t)) = (*(*, *>(*)), y(t, jo(0)). (2)

It is clear from equation (2), since s(t) = Gt(so(t)), that each point on the curve ?

is also a point on some Gt, and it follows from equation (1) and the Chain Rule that the line tangent to s(t) at a point s (to) is also tangent to the curve GtQ(so(to)), at least at points where the tangent lines are well-defined.




Folding parametrized curves onto a point In the remainder of the paper, a generic parametric curve in the plane is represented

by c(0 = if it), git)). The term "the perpendicular vector" of a vector v = (v\, v2) refers to the vector (?v2, v\). The point P = (a, b) in the plane is arbitrary but fixed.

A family of lines, the folding creases J^tis), is constructed as each point Q = c(0 is

folded onto P; in this case, P is called the folding point. Such a crease passes through the midpoint \{a + fit), b + git)) of segment PQ and is perpendicular to PQ. It is

convenient to parametrize the crease using a slope vector that is "the perpendicular

vector" to ?(P ?

c(0). The resulting parametrization is

Ft{s) = {\ia + fit) + si-b + git))), \ib + git) + sia - fit))))

. (3)

Definition 2. The folding curve ofc(t) generated by P is the envelope of the family

of folding creases {Ft is)}.

Note that since each Tt is linear in s, equation (1), which we need to solve to

compute an envelope curve for the family of folding creases, is also linear in s. That

is, we need to solve

\ii-a + fit)) fit) + i-b + git))g\t)

+ siib - git)) fit) + i-a + fit))g\t))) = 0 (4)

for 5" as a function of t, and the unique solution is

,? (-a + fiO) fit) + i-b + g(r))g'(Q s0(n =-. (5)

(-6 + ^(0)^(0 + (a - /(0)^(0

Following our procedure for computing envelope curves, the envelope to {Tt} is there

fore the parametric curve Ttisît)) using the s o it) given in (5).

Theorem 1. Given an arbitrary parametrized curve c(i) and a point P = ia, b), each point on the envelope sito) of the family of folding creases {J-tis)} is the inter section of the folding crease J-tQis) and the normal to the curve c(/) at cito).

Proof The normal line to c(i) at cito) is given by

ntoiq) =

ifito) -

qg'ito), g(t0) + qfih)). (6)

Solving the system of equations ntQiq) =

TtQis) for s yields s0it0), where s0it) is

precisely the quantity found in equation (5). That is, Tîsoito)) is the point on Tîs) that also lies on ntn.

Conies, revisited

In the opening section, it was asserted that the folding curve of a circle generated by a

point P is an ellipse if P is inside the circle and a hyperbola if P is outside the circle. Here we explore this in more detail.

Let c(0 = (rcos(/), r sin(O); that is, c(0 is a circle of radius r centered at the

origin. It suffices to consider the point P = (a, 0) (if P is a generic point ia,b), simply




rotate the plane as necessary). The family of folding creases is

Ft(s) = (I(a + rcos(Y) + s(r sin(i))), \(s(a

- rcos(O) + r

sin(f)))

and the normal lines are

nt0(q) = (r cos(io)

- qr cos(f0), r sin(i0)

- qr sin(i0)).

Solving ntQ(q) =

!FtQ(s) for s yields

asin(t0) s?(to) =-?-77T' ?r + a cos(io)

so the folding curve is

2 2

?W = o r

o ?

, x(cos(0> sin(O). (7) 2r ? 2a cos(i)

It turns out that (7) represents a parametrized conic section! To see this, recall that every conic section may be constructed as the locus of

points Q so that if P is a fixed point (a focus) and if ? is a fixed line (the directrix), then

d(Q,P) d(Q,C)

is a constant, called the eccentricity and denoted e (for more information, see Brannan et al. [1, pp. 11-19]). If 0 < e < 1, the conic is an ellipse; if e = 1, it is a parabola; and if e > 1, it is a hyperbola.

Figure 5. An ellipse with foci at (0, 0) and (a, 0) and semi-major axis a.

Suppose a < r, and consider an ellipse with foci at (0, 0) and (a, 0) and eccentric

ity e = ". Choose an arbitrary point Q on the ellipse, and let d(Q, (0, 0)) = p and

d(Q, (a, 0)) = p' (see Figure 5). It is well known (see Brannan et al. [1, p. 17]) that if a is the length of the semi-major axis of the ellipse, then the sum of the focal dis tances p + p' is 2a and the distance between the two foci is lote. Combining these

facts shows that




a a = 2a - ?

r

r

a=2

P + p' = r. (8)

By applying the the law of cosines, we see that

(p')2 =

p2 + a2 - 2pa cos(f);

substituting from equation (8) and simplifying produces

2r ? 2a cos(t)

That is, the parametric equation of an ellipse with foci at (0, 0) and (a, 0) and

eccentricity e = a/r is

2 2

(cos(i), sin(O), 2r ? 2a cos(i)

which is precisely the folding curve s(t) given in (7). The situation if (a, 0) is outside the circle, so that a > r, proceeds similarly, us

ing the fact that r ? p

= ?p\ and yields the same parametrization. When a > r, the

eccentricity ? > 1, so the conic section is a hyperbola.

Example: folding an ellipse onto a point Consider the ellipse parametrized by e(t) = (5 cos(0, 2 sin(f )). If P = (a,b), then the

family of folding creases is given by

Ft(s) = (\(a + 5cos(f) + s(-6 + 2sin(0)), 5(^(0

- 5cos(?)) + ?+ 2sin(r))),

and the normal lines are given by

nt(q) = (5 cos(t)

? 2q cos(t), 2 sin(t)

? 5q sin(O).

Solving nt(q) = Tt(s) for s yields

2bcos(t) ? 5a sin(i) -f 21 cos(i) sin(i)

io(0 = 10 ? 2a cos(t)

? 5Z? sin(r)

In the special case where P is the origin, s0(t) = y? cos(?) sin(?). Substituting this

into J-t(s) and simplifying produces the folding curve

s(t) = (?(121 cos(0

- 21 cos(3i)), ^(-5 sin(i) - 21 sin(3i))).

The ellipse and the folding curve generated by P = (0, 0) are shown in Figure 6.

Several other folding curves of the ellipse are shown in Figure 7; in each case, P is indicated by a black dot.




Figure 6. The ellipse (5 cos(?), 2 sin(r)) and its folding curve generated by P = (0, 0).

(a) (b) (c)

Figure 7. The ellipse (5 cos(i), 2 sin(O) and three more folding curves, generated by (a) P =

(?, 0), (b) P = (4, 0), and (c) P = (0, 3).

Folding curves and negative pedal curves

The following definition is adapted from Yates [11, pp. 77-80,160-165] and Lawrence

[6, p. 46].

Definition 3. Given a curve cit) and a point P, the negative pedal curve /x(0 (gen erated by P) is the envelope of the family of lines J\fVtis), where if Q

? c(i0) is a

particular point on cit), MVtQis) is the line perpendicular to segment PQ passing through Q.

The family of lines enveloping the negative pedal curve of a circle is shown in

Figure 8(a); Figure 8(b) shows the negative pedal curve along with the folding curve, both generated by P.

Since every member of the family of lines that generates the negative pedal curve

is parallel to a folding crease of cit) generated by P, it is reasonable that the folding curve and the negative pedal curve should be related.

For the following discussion, the parametrization

MVtQis) = ifito) + si-b + gito)), sia

- fito)) + gito)) (9)

is used. Note that the slope vector for each line in the family is "the perpendicular vector" to PQ, where Q = c(i0).

Theorem 2. Let cit) be a parametrized curve, let P = ia, b) be a point, and let

sit) be the folding curve of cit) generated by P. A point on the negative pedal curve




(a) (b)

Figure 8. (a) Lines forming the negative pedal curve of a circle, (b) The negative pedal curve

(thick curve) of a circle, along with the folding curve (thin curve).

ix(t) is the point of intersection of the line passing between s(t) and P with the line

MVt(s).

Proof Because the negative pedal curve is the envelope of the family of lines

{J\fVt(s)}, using equation (2) and constructing an envelope for the family {MVt(s)} shows that the negative pedal curve ?i(t) is given parametrically as J\fVt(so(t)), where

s0(t) is a solution to

(-a + f(t))f(t) + (~b + g(t))g'(t)

+ s((b -

g(t))f(t) + (-a + f(t))g'(t)) = 0. (10)

By construction, for a fixed t, AfVt(s) and ̂ (s) are parallel, since both are per

pendicular to segment Pc(t). Moreover, the points P, ^(s^t)), and J\fVt(so(t)) are

collinear, where s0(t) is as given in (5). To see this, note that for any point Q and lines

ix(t) = M + ?w, l2(t) = TV + i(?w),

where M, Af, and Q are collinear, k e M is a scaling factor, \QN\ = k\QM\ and w

is perpendicular to QM, similarity of triangles implies that the points Q, i\(t0), and

?2(to) are collinear for any choice of t0 (see Figure 9). Therefore, because P, Tt (s0(t)), and J\fVt(so(t)) are collinear and s(t) = ^Ft(s0(t)), J\fVt(s0(t)) must be the intersec

tion point of the line Ps(t) with J\fVt(s).

k\w\

t0(k\N)

Figure 9. Relationships between collinear points and perpendicular lines




However, substituting s0it) for s in the left hand side of (10) yields 0, so soit) satisfies (10); thus, the point J\fVtisoit)) is also on the negative pedal curve ?i(t).

Corollary. The negative pedal curve of cit) with pedal point P is the dilation by a

factor of two of the folding curve of cit) generated by folding point P.

(a) (b)

(c) (d)

Figure 10. Some hypocycloids and associated folding curves; the folding point is indicated

with a black dot, and the folding curve is the thick curve, (a) A hypocycloid with 3-fold

symmetry, (| cos(0 + | cos(2i), ? sin(i) ?

| sin(2r)), folded onto the origin, (b) An astroid,

(^(3 cos(0 + cos(30), \0 sin(i) ?

sin(3?))), folded onto the origin, (c) A hypocycloid with

5-fold symmetry, (^(cos(y) + 2cos(0), \(? sin(y) + 2sin(0)), folded onto the origin,

(d) A hypocycloid with 3-fold symmetry, (|(2cos(i) + 3cos(2i)), ^(2sin(i) -

3sin(2i))), folded onto a point on a line of reflective symmetry.

1 84 ? THE MATHEMATICAL ASSOCIATION OF AMERICA



(There is some difference of opinion on what the "proper" name of the scaling or dilation transformation is: Some, such as Eves [5, p. 124] and Coxeter [4, pp. 67

69]), think it should be called a dilatation or homothety/homothecy, while others (for example, Lay [7, pp. 77-78]) call it a dilation.)

Proof The proof of Theorem 2 showed that each point on ??it) is of the form

NVtisoit)), where P, J7,Cso(O). and MVt(so(f)) are all collinear. Since line Tt per

(c) (d)

Figure 11. Some interesting curves and their associated folding curves; the folding point is indicated with a black dot, and the folding curve is the thick curve, (a) The Lissajous curve

(cos(3?), sin(2i)) folded onto the origin, (b) The Lissajous curve (cos(?), sin(2i)) folded onto the point (1, 0). (c) A conchoid of Nicomedes, (2 + 8cos(i), (2 + 8cos(i))tan(0), folded onto (5, 0). (d) A cardioid, (2cos(/)(l + cos(O), 2sin(i)(l + cos(i))), folded onto its cusp (the origin).




pendicularly bisects segment Pc(t) and line J\fVt passes through point c(t) and is

perpendicular to segment Pc(t), similarity of triangles shows that J7*(so(0) must bi sect the segment joinig P and J\fVt(so(t)). Therefore, MVt(so(t)) may be constructed as the dilation by a factor of 2 of the point s(t) = Tt (so(t)) along the line Pe(t).

Pretty pictures Some lovely curves can be generated by folding parametrized curves onto their points of symmetry. A few especially attractive and interesting examples are shown in Fig ures 10 and 11.

Acknowledgments. The author would like to thank the anonymous referee for his or her clear,

thoughtful, and extremely helpful comments.

References

1. David Brannan, Matthew Esplin and Jeremy Gray, Geometry, Cambridge University Press, 1999.

2. J. W. Bruce and P. Giblin, Curves and Singularities, Cambridge University Press, 1984.

3. David Cox, John Little, and Donal O'Shea, Ideals, Varieties, and Algorithms, Springer-Verlag, 1992.

4. H. S. M. Coxeter, Introduction to Geometry, 2nd ed., Wiley, 1969.

5. Howard Eves, A Survey of Geometry, Vol. 1, Allyn and Bacon, 1963.

6. J. Dennis Lawrence, A Catalog of Special Plane Curves, Dover, 1972.

7. David C. Lay, Linear Algebra and Its Applications, 3rd ed., Addison-Wesley, 2003.

8. Victoria Pohl, How to Enrich Geometry Using String Designs, National Council of Teachers of Mathematics, 1986.

9. T. Sundara Row, Geometric Exercises in Paper Folding, W. W. Beman and D. E. Smith, eds., Open Court, 1917.

10. John W. Rutter, Geometry of Curves, Chapman & Hall/CRC, 2000.

11. Robert C. Yates, Curves and Their Properties, National Council of Teachers of Mathematics, 1952 (reprinted

1974).




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