Hypocycloids and

Tuyetdong Phan-Yamada Elysus Gwin Glendale Community College Los Medanos College

Introduction This paper compiles several theorems relating to and epicy- cloids with the aim of providing a project for students interested in a further understanding of polar . Experiments lead to finding connec- tions between hypocycloids, hypotrochoids, and polar curves. Throughout, there is a direct connection between characteristics of the parametric equations and easily identifiable visual aspects of the corresponding curves. GeoGebra is used extensively. Relationships with circulant graphs and ancient Greek models of planetary orbits are also described. Readers can download a set of ready-made GeoGebra files (-.ggb and hypo-.ggb) to explore the concepts discussed in this article. See Phan-Yamada (n.d.). These files along with their step-by-step instruc- tions can be used as a project for calculus students.

Hypocycloids A hypocycloid [a, b, t] is a traced out by a point P on one (red) (B, b) as it rolls around inside another fixed (green) circle (O,a ) for a > 0. If the initial position of P is (a, 0) and the parameter t is chosen as in Figure 1, Figure 1a . a = 3, b = 1, b/a = 1/3 it can be shown that the parametric equations of the hypocycloid are ab− ab− x=−+( ab )cos tb cos t and y=−−( ab )sin tb sin t , where b b a and b are real numbers and a > b > 0. Figures 1a, 1b, and 1c depict some hypocycloids with given values. If the red circle is rolled about the exterior of the fixed green circle, the resulting curve is called an epicycloid. The standard parametric equations of ac+ are x=+−( ac )cos tc cos t and c ac+ y=+−( ac )sin tc sin t , where a and c are real numbers. Here, c c represents the radius of the outer circle that is being moved. Returning to hypocycloids, if we loosen the restrictions on b, allowing it to be greater than a or negative, then the resulting curve is an epicycloid. We notate hypocycloids as [a, b, t], which is an adaptation of the notation presented in Hall (1992).

Theorem 1: Figure 1b. a = 2, b = 1 2,. b/a = 3/5 The hypocycloid [a, b, t] is an epicycloid for b < 0 and b > a.

www.amatyc.org 5 By reversing the argument in theorem 1, it follows that all epicycloids are hypocycloids, hence, they inherit all properties, as found in the following discus- sion of hypocycloids.

Properties of Hypocycloids Theorem 2: ab− Let [a, b, t] be the hypocycloid x=−+( ab )cos tb cos t and b ab− y=−−( ab )sin tb sin  t that is traced by a fixed pointP on the circum- b ference of the circle (B, |b|) rolling around a circle (O, a), where a > 0 and b ≠ 0 are real numbers with a rational ratio. If b/a = r/s, where s and r are relatively prime natural numbers, then

a) The hypo/epicycloid intersects the circle (O, a) exactly s times. These intersections are called cusps of the hypo/epicycloid. Figure 1c .a = 7, b = 2, b/a = 2/7 b) The period of the hypocycloid is 2πr.

The Figure 1 set illustrates examples of 3-, 5- and 7-cusped hypocycloids with s = 3, 5, and 7, respectively. There are no 1- or 2-cusped hypocycloids. If a = 2b, meaning b/a = 1/2, then a – b = b. Thus, the parametric equations of hypocycloid become x = 2bcos t and y = 0. This means point P traces along the x-axis with the interval [–a, a]. If a = b, then a – b = 0. Thus, the parametric equations of the hypocycloid be- come x = a and y = 0, so the graph of g is a point P(a, 0). However, we have 1-cusped epicycloids for b/a = –1 and 2-cusped epicy- cloid for b/a = –1/2, as shown in Figures 2a, 2b, and 2c. Figures 3a and 3b shows incomplete and complete curves for the case b/a = 3/5, meaning r = 3. Using graphing software such as GeoGebra, we observe that with a = 4 hypocycloids, b/a = 2/5 and b/a = 3/5 are exactly the same graphs as shown in Figure 2a . b/a = 1/2; segment Figures 4a and 4b. In general, if b’/a= r’/s and b/a = r/s and r’ + r =s, then they generate the same graph. Such ratios are called co-ratios of s-cusped hypocycloids. However, their directions are opposite. If they both start at (a, 0), one of them will roll clockwise and the other will roll counterclockwise, as shown in Figures 5a, 5b, and 5c. In fact, both the circle radius b and the circle radius (a – b) rolling on the inside of a fixed circle (O,a ) generate the same hypocycloid, but their directions are reversed. In other words, they have the same x-coordinates but their y-coor- dinates are opposite of each other. This property, known as the double generation theorem, was discovered by Daniel Bernoulli in 1725.

Theorem 3: The Double Generation Theorem For any hypocycloids [a, b, t] and [a, b’, w], where a, b, b’ are positive integers, if b’ + b = a then the two hypocycloids coincide. We note in passing that certain kinds of hypocycloids can be made to cor-

Figure 2b. b/a = –1 or b/a = 2; (1-cusped ) respond to cyclic circulant graphs (Weisstein, n.d.a.). In particular, cusps may be labeled 0, 1, 2, …, (s – 1) counterclockwise about the hypocycloid with each representing a vertex. Then two vertices are adjacent if the corresponding cusps

6 MathAMATYC Educator ~ Vol. 6, No. 1 ~ September 2014 Figure 2c . b/a = –1/2 or b/a = 3/2; (2-cusped) Figure 3a. Incomplete curve g as t ∈ (0, 2π )

Figure 3b. Complete curve g as t ∈ (0, 6π ) Figure 4a . b/a = 2/5

www.amatyc.org 7 Figure 4b. b/a = 3/5 Figure 5a . b/a = 1/7; direction: (0123456)

Figure 5b. b/a = 6/7; direction: (0654321) Figure 5c. Red: b/a = 1/7 or 6/7; blue: b/a = 2/7 or 5/7; black: b/a = 3/7 or 4/7

8 MathAMATYC Educator ~ Vol. 6, No. 1 ~ September 2014 are connected by the curve. This will naturally produce a circulant graph with s vertices, where vertices i and (i + r) mod s are adjacent, i = 0, 1, 2, …, (s – 1). For example, if b/a = 2/7 then r = 2 and s = 7. Hence, the hypocycloid has 7 points and the corresponding graph will have 7 vertices. As t grows, the curve traces out a path through the cusps 2 to 4 to 6 and so on. We abbreviate this as (0246135). The hypocycloid generated by b/a = 5/7 will have the same shape but is traced out in the other direction, with the order of vertices being (0531642). (See the blue curve in Figure 5c.) By superimposing hypocycloids with the same cusps but different r values, we can further represent every other circulant graph on s vertices. Up to isomor- phism, there are (s – 1)/2 such hypocycloids for each prime s. Hypocycloids or epicycloids without intersections of any arcs are called s-cusped simple hypocycloids (or epicycloids). Weisstein (n.d.b.) gives the names for some special hypocycloids (or epicycloids), as shown in Figure 6a, Figure 6a . b/a = 1/3 or b/a = 2/3; 6b, and 6c. deltoid: 3-cusped hypocycloid

Theorem 4: Let [a, b, t] be a hypocycloid (or epicycloid). If b/a = ±1/s, then the curve is an s-cusped simple hypocycloid (or epicycloid). b 1 We will see the curve cross itself when ≠± , as shown in Figures 7a, as 7b, and 7c. Note that all cusps lie on green lines an even π/s from the positive x-axis whereas the intersections between arcs are on either blue lines an odd π/s from the positive x-axis.

Hypotrochoids A [a, b, d, t] is a curve traced out by a point D on the that coincides with a diameter of the circle (B, b) as it rolls around along another (fixed) circle (O,a ). Let the distance between D and B be d, and let a, b, and d Figure 6b. b/a = 1/4 or b/a = 3/4; : 4-cusped hypocycloid be non-zero rational numbers such that a > 0 and d > 0. The parametric equa- ab− tions for a hypotrochoid are x=−+( ab )cos td cos t and b ab− y=−−( ab )sin td sin t . As with hypocycloids, an is a b hypotrochoid when b < 0 or b > a. Figures 8a, 8b, and 8c show some examples hypocycloids or epicycloid (blue curve) and hypotrochoids or (black curve). Davis (1993) includes a discussion of the use of to describe the motion of planets by ancient Greek mathematicians (pp. 55–58). Briefly, Ptolemy used the epicycle (literally, on the circle in Greek) to explain the motion of planets around the Earth. Archimedes and Plutarch relate an earlier theory of planetary motion around the sun put forward by Aristarchus. Orbits centered on the sun conform to epitrochoids. Figure 9a shows the motion of Saturn (the outer curve), Jupiter (the middle), and Mars (the inner) as viewed from the Earth and assuming a circular orbit. Jupiter’s path has 11 loops, curled outward with no intersections between the big loops. We can guess its parametric equations to be b/a = –1/11. Using Figure 6c . b/a = –1/2 or b/a = 3/2; nephroid: 2-cusped epicycloid GeoGebra, we can create a similar curve (Figure 9b). www.amatyc.org 9 Figure 7a. 5-cusped hypocycloid, b/a = 3/5 Figure 7b. 8-cusped epicycloid, b/a = –3/8

Figure 7c. 4-cusped epicycloid, b/a = 7/4 Figure 8a . b/a = –1/7, d = 2; 7-looped epitrochoid

10 MathAMATYC Educator ~ Vol. 6, No. 1 ~ September 2014 Figure 8b. b/a = 3/8, d = 4.5; 8-looped hypotrochoid

Figure 9a: The motion of Saturn, Jupiter (in middle loop) and Mars in respect of the Earth. Adapted from “Astronomy.” (1823) Encyclopaedia Britannica, Vol. III, Plate LXXIV, Fig. 120.

Mars’ orbit is similar to a cardioid, so we take b/a = –1. Saturn’s orbit is nearly a 28-looped simple epitrochoid, so we take b/a = –1/28 with d < |b|. If b/a = 1/2 and d ≠ b, then the parametric equations are x = (b + d) cos t xy22 and y = (b – d) sin t. We then have an ellipse of the form +=1. Figure 8c . b/a = 1/2, d = 2; (bd+− )(22 bd ) ellipse: 2-looped hypotrochoid This is a circle if d = 0. Now we examine a special case of hypotrochoids. The 8-looped hypotro- choid in Figure 8b almost resembles a rose curve in the polar coordinates if it passes through the origin O eight times. So we will set d = |a – b|; then the hypotrochoid [a, b, d, t] with b/a = r/s will pass through the origin s times. Its ab− parametric equations are x=−( ab )cos t +− ( ab )cos t and b ab− y=−( ab )sin t −− ( ab )sin t . When we factor out a – b and apply the b trigonometric identities for the sum of cosines and the difference of sines, we a 2 ba−  have x=2( ab − )cos t  cos t  and 22bb   a 2 ba−  y=2( ab − )cos t  sin t  . These equations look like polar equations 22bb   2ba− 2b aa if we substitute wt= . We then have tw= and tw= . 2b 2ba− 22b ba− Figure 9b. b/a = –1/11, d = 2 .5 a This implies that x=2( ab − )cos w cos w and 2ba− a y=2( ab − )cos w sin w . These are polar equations of polar curves 2ba− www.amatyc.org 11 a R=2( ab − )cos w with parameter w. It turns out that the hypotrochoid 2ba− is a rose curve in some cases, as described in the following theorem.

Theorem 5: a For any hypotrochoid x=2( ab − )cos w cos w and 2ba− a y=2( ab − )cos w sin w , where a and b are non-zero rational numbers, 2ba− b/a = r/s, and a > 0, we have the following:

a) If s = n(2r – s) for n ≠ ±1, then it is a rose curve with n petals when n is odd and 2n petals when n is even. Otherwise, the curve is called Figure 10a . a = 3, b = –2 .25, b/a = –3/4; a rosette with the intersection at (x, y), where rosette with 4-petal inner loops a x=2( ab − )cos w cos w and 2rs− a y=2( ab − )cos w sin w , and w = riπ/s for 2rs− i = 0, 1, 2, …, (s – 1).

b) All intersections lie on concentric (O, Ri) where iπ Ri =2( ab − )cos for i = 1, 2, 3, …, N. For any such 2rs− hypotrochoid, the number N of concentric circles are determined as follows: 21rs−− i) When s is odd, N = . 2 2rs− ii) When s is even, N = −1. 2 Figure 10b. a = 2 .6, b = 0 .7, b/a = 7/25; Figures 10a, 10b, and 10c show some examples of rose curves and rosettes. rosette with 25-petal inter loops Figures 11a and 11b show some examples of rosette hypotrochoids with the calculation of the number of concentric circles. We have presented a computational methodology for guiding student exploration of the properties of hypotrochoids. The methodology includes visually-guided parametric experimentation, theoretical proof, and historical context—thus offering ample opportunity for students to learn the interplay between theoretical mathematics, applied mathematics, and science. We have included properties that are discussed in standard textbooks; however, we unify the discussion. For example, we show that hypocycloids and rosettes are special cases of hypotrochoids, where for any ratio r/s, they each have the same number of petals located in the same orientation on the plane. Figure 12 is an example of a family of such curves for the ratio r/s = 15/32. Most students do not have access to advanced visual mathematic software, thus, we make available free downloadable GeoGebra files that make the exploration discussed in this paper easy and fun. Download the files and explore!

Figure 10c . a = 3, b = 1 .25, b/a = 5/12; 12-petal rose

12 MathAMATYC Educator ~ Vol. 6, No. 1 ~ September 2014 21rs−− Figure 11a . R = 10, s = 9 (odd); N = = 5; 2 all intersections lie on 5 concentric circles .

Figure 12: Hypocyloid (blue), rose (green) and hypotrochoid (black) with d = 1 .3, a = 3 .2 and b = 1 5. .

References Davis, D. M. (1993). The Nature and Power of Mathematics. New York: Dover Publications. Hall, L. (1992) , Roses, and Thorns – Beyond the . The College Mathematical Journal, 23, 20-35. Phan-Yamada (n.d.) Graphing with GeoGebra. Retrieved from https://sites.google.com/site/ phanyamada/interactive-graphing 2rs− Weisstein. (n.d.a) In Circulant graph. Retrieved from http://mathworld.wolfram.com/CirculantGraph. Figure 11b. r = –1, s = 6 (even); N = −=1 3; html 2 Weisstein. (n.d.b) In Hypocycloid. Retrieved from http://mathworld.wolfram.com/Hypocycloid.html all intersections lie on 3 concentric circles .

Lucky Lucy

Radicals are really exponents, and negative exponents 24 2466− =×=− 6 46 mean division, so… −6

George Alexander Madison Area Technical College Madison, WI Appendix Proof of Theorems Theorem 1: For any non-zero rational numbers a > 0, b, and c, the hypocycloid [a, b, t] is an epicycloid if b < 0 or b > a.

Proof: ab− ab− We will first show that x=−+( ab )cos tb cos t and y=−−( ab )sin tb sin  t with b < 0 is the same as b b ac+ ac+ x=+−( ac )cos tc cos t and y=+−( ac )sin tc sin t . c c ab− ab− The parametric equations of hypocycloids are x=−+( ab )cos tb cos t and y=−−( ab )sin tb sin t . When we b b ac+ ac+ replace c with 1 – b, we have x=+−( ac )cos tc cos t and y=++( ac )sin tc sin t . Since cos(−=xx ) cos and −c −c ac+ ac+ sin(−=−xx ) sin( ), we can write x=+−( ac )cos tc cos t and y=+−( ac )sin tc sin t . c c Now we will show that the same hypocycloid equations also generate epicycloids when b > a > 0. We denote the hypocycloid by [a, b, t] and the epicycloid by [a, c, w] where t and w are parameters for each curve, respectively. Since b > a > 0, let c = b – a −c so that b = c + a. We replace c in the parametric equations of [a, b, t] to get x=− ccos t ++ ( ac )cos t and ac+ −c y=− csin t −+ ( ac )sin t . We now define the parameterw such that (b – a)t = bw. ac+ b ac+ ba− c ac+ This implies tww= = and wtt= = . Thus x=− ccos w ++ ( ac )cos( − w ) and ba− c b ac+ c ac+ ac+ y=− csin w −+ ( ac )sin( − w ). Since cos (–w) = cos (w) and sin (–w) = –sin (w), we get x=+−( ac )cos wc cos w c c ac+ and y=+−( ac )sin wc sin w , which are the parametric equations of epicycloids. QED c Theorem 2: ab− ab− Let [a, b, t] be the hypocycloid x=−+( ab )cos tb cos t and y=−−( ab )sin tb sin  t that is traced by a fixed point b b P on the circumference of the circle (B, |b|) rolling around a circle (O, a), where a > 0 and b ≠ 0 are real numbers with a rational ratio. If b/a = r/s, where s and r are relatively prime natural numbers, then

a) The hypo/epicycloid intersects the circle (O, a) exactly s times. These intersections are called cusps of the hypo/ epicycloid.

b) The period of the hypocycloid is 2πr.

Proof: Part (a). Let P be the intersection of the curve g and the circle (O, a). Then the set of coordinates of P is the solution set of ab− ab− x=−+( ab )cos tb cos t and y=−−( ab )sin tb sin  t and x2 + y2 = a2. Then we have b b

14 MathAMATYC Educator ~ Vol. 6, No. 1 ~ September 2014 22  ab− ab− 2 (abtb− )cos + cos t+−( abtb )sin − sin  t= a  bb 

22 ab−−22  ab ⇒−(ab ) cos t + 2( abb − ) cos t cos t+ bcos  t bb 

22 ab−−22  ab 2 +−(ab ) sin t − 2( abbt − ) sin sin  t+ bsin  t= a. bb  When we apply the Pythagorean Identity, we have the following:

22 ab −  ab−2 (abbbabt− ) ++ 2( − )coscos t− sinsin t  ta= bb 

2 22 ab −  ab−2 ⇒−a2 abb +++ b2( bab − )coscos t t− sinsin t  t= a bb 

22 ab −  ab− ⇒−+++2abb b2 bab ( − ) cos t cos t− sin t sin  t =0. bb 

Using the cosine sum identity, we get Proof: Part (b). ab − The x-coordinate of the hypocycloid repeats itself when the 2(ba−+ b )cos t t =−2( ba b ) b periods of both cosines in its formula coincide. One term is just cos (t) with period 2π. For the other term, since ab − ⇒+cos tt =1 ba/= rs /, (abb−=− )/ ( sr )/ r , it follows that b ab−−  sr  ab− costt = cos  . Let p be the period of this ⇒+t tk =2 π br   b function. Then a ⇒=tk2 π sr−− sr  b cos (tp+= )  cos t  rr   br ⇒=tk2ππ = 2 k for positive integers k . sr−− sr  sr −  as ⇒costp += cos t  . rr  r  Next, notice that when we substitute this value of t into the formula for x we have Since cosine is a periodic function with period 2π, then sr− 22ππkr  s− r kr  p has to be an even multiple of π. That is, x=−+⋅( ab )cosb cos   r s  rs  sr− 222πππkr kr kr pm= 2 π for some whole number m. =−+abbcos cos  cos  r sss   This will first happen when the periodp = 2πr since r 2π kr and s are relatively prime. A similar reasoning follows for the = a cos . s y-coordinate and sine. Thus, the period is 2rπ. QED Similarly, we can show the y-coordinate of the intersec- Theorem 3: The Double Generation Theorem: 2π kr tion is ya= sin . So the points of intersection are the s For any hypocycloids [a, b, t] and [a, b’, w] where a, b, b’ s are positive integers, if b’ + b = a then the two hypocycloids evenly spaced points about the circle, resembling the s roots of coincide. unity. Thus there are s intersections of the curve g and the circle (O, a), and hence the curve has s cusps. QED

www.amatyc.org 15 Proof: We denote the first hypocycloid by a,[ b, t] and the second by [a, b’, w] where t and w are parameters. Let b’ = a – b. Replace b’ b′ b′ in the parametric equations of [a, b, t], we have x= b′′cos t +− ( ab )cos t and y= b′′sin t −− ( ab )sin t . Since ab− ′ ab− ′ both circles (B, b) and (B, a – b) start from the same point (a, 0), we get ()a−= b t bw . This implies that w=−=−( abtbbtab ) /′′ /( ) and t= bwab/( −=− ) ( abwb′′ ) / . Therefore, ab−−′′ab x= b′cos w+−( ab ′ )cos w =− ( ab ′′ )cos wb + cos w bb ab−′′ ab− and yb=′sin  wabw−−( ′ )sin =−− ( abwb ′′ )sin + sin w . bb Thus the x-coordinate of the second hypocycloid [a, b’, w] is the same as the first hypocycloid a,[ b, t] and their y-coordinates are opposite each other. QED

Theorem 4: number N of concentric circles are determined as Let [a, b, t] be a hypocycloid (or epicycloid) . If b/a = ±1/s, follows: then the hypocycloid (epicycloid) is an s-cusped simple hypo- 21rs−− i) When s is odd, N = . (or epicycloid). 2 2rs− ii) When s is even, N = −1. Proof: 2 From the proof of theorem 2, we know that the curve have 2krπ 2kπ Proof: t = . t = cusps at s If r = 1, then s for k = 0, 1, 2, …, If b/a = then the hypotrochoids equations be- (s – 1). Thus t < 2π. Therefore, the cusps lie 2π/s apart from s come x=2( ab − )cos w cos w and one to the next counterclockwise. If r = –1, then t = –2kπ/s. So 2rs− the cusps are 2π/s from one to the next but in clockwise s y=2( ab − )cos w sin w . direction. Therefore, there is no intersecting arc. QED 2rs− Theorem 5: Part (a). a If s= nrs(2 − ), then x=2( a − b )cos( nw )cos w and For any hypotrochoid x=2( ab − )cos w cos w and 2ba− y=2( a − b )cos( nw )sin w . The parametric equations are the a y=2( ab − )cos w sin w , where a and b are non-zero same as the polar equation R=2( a − b )cos( nw ). Thus, the 2ba− curve is a rose curve with |n| petals if |n| is odd and 2n petals if rational numbers, b/a = r/s, and a > 0, we have the following: n is even. Since (rs , )= 1, n ≠±1. a) If s= nrs(2 − ) for n ≠ ±1, then it is a rose curve Since the polar equation of the hypotrochoid is with n petals when n is odd and 2n petals when n a R=2( ab − )cos w , the length of each petal would be is even. Otherwise, the curve is called a rosette 2ba− sw sw with the intersection at (x, y), where 2(a – b). This implies that cos= 1. So = 2iπ a 2rs− 2rs− x=2( ab − )cos w cos w and 2rs− for some integer i. Hence, 2iπ (2 r− s ) 4 ir ππ 4 ir a wi= = −=2,π since every 2π is one y=2( ab − )cos w sin w , and w = riπ/s 2rs− ss s for i = 0, 1, 2, …, (s – 1). revolution. If we roll back i rounds, we return to the same point. When we rewrite w = 2r(2iπ/s), the tip of s petals would b) All intersections lie on concentric circles (O, Ri) be distributed evenly in one revolution, 2π. Thus, w = 2riπ/s, iπ where i = 0, 1, 2, …, s − 1. where Ri =2( ab − )cos for 2rs− i = 1, 2, 3, …, N. For any such hypotrochoid, the

16 MathAMATYC Educator ~ Vol. 6, No. 1 ~ September 2014 Part (b). If s is even, then 2r – s is even. So when i=(2 rs − ) / 2, Since cos t decreases from 0 to π/2, for each w = iπ/s or the radius R = 0. After that, all intersections are on the same swrs/ (2−= ) iπ / (2 rs − ) the radius R is getting smaller and circles. There is an intersect circle for each value of i, and we R = 0 as sw/ (2 r−= s )π / 2. do not count at i = 0 as there is no intersection. Thus, there are If s is odd, then 2r – s is odd. It implies that the smallest (2rs− ) / 2 circles. Note that the origin is an intersection but circle is at i < (2r – s), so we must subtract 1 before taking R = 0 and there are no circles. Thus there are N intersection 2rs− half. Note that there is no integer i such that the curve crosses circles where N = −1, and the radius of each circle is 21rs−− 2 N = and radius of each circle is the origin. Thus, 2 iπ Ri =2( ab − )cos. QED iπ 2rs− Ri =2( ab − )cos . 2rs−

Tuyetdong Phan-Yamada ([email protected]) is a mathematics instructor at Glendale College, California. She earned her bachelor’s degree from UC Irvine and master’s degree from CSU Los Angeles. She is interested in writing projects for community college mathematics courses. She created several GeoGebra files as projects for trigonometry, , statistics, and calculus. Her GeoGebra projects were published in Math Teacher Journal as well as presented at CMC-South Conference and STEM Conference.

Elysus Gwin ([email protected]) was born and raised in California. He earned his bachelor’s degree from UC Berkeley and master’s degree from UC Irvine. His interests range from set theory to cryptography and geometry. He teaches at Los Medanos College, and is the faculty adviser for the campus math club. When not teaching or reading books, he can usually be found hiking.

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The Student Mathematics League was founded in 1970 by Nassau Community College in New York. In 1981 the American Mathematical Association of Two-Year Colleges assumed sponsorship with Terry Shell of Santa Rosa Junior College as its first director. The League has grown to more than 165 colleges in more than thirty-five states as well as Bermuda involving over eight thousand community college math students. New registrations are welcome anytime during the testing year. However, to insure receiving materials in time to compete in the first round and to avoid the late fee, register online by September 30, 2014 to participate in the 2014–15 academic year. For more information, visit http://www.amatyc.org/SML/.