poem’s and newton’s aerodynamic frustum - imelda trejo · poem’s and newton’s aerodynamic...

POEM’s and Newton’s AerodynamicFrustumJaime Cruz-Sampedro and Margarita Tetlalmatzi-Montiel

doi:10.4169/074683410X480258

Jaime Cruz-Sampedro ([email protected]) is aprofessor of mathematics at Universidad AutonomaMetropolitana in Mexico City. He received hisundergraduate degree from the Instituto PolitecnicoNacional in Mexico and his M.S. and Ph.D. in mathematicsfrom the University of Virginia. In addition to the usualuniversity duties, he enjoys the challenge of teachingremedial courses in mathematics as well as that of trainingstudents taking part in mathematical contests.

Margarita Tetlalmatzi-Montiel ([email protected])received her M.S. in mathematics from the University ofVirginia. She is currently an assistant professor ofmathematics at Universidad Autonoma del Estado deHidalgo, in Pachuca, Mexico. Her interests includeteaching, the applications of mathematics and history ofmathematics.

Several years ago, while looking at the Apollo 11 Command Module Columbia (seeFigure 1) at the Smithsonian National Air and Space Museum, a nine year old boyasked us: ‘Why does the Columbia have that shape?’

Figure 1. Apollo 11 Command Module Columbia [1].

We gave the quick response, ‘Because it looks nice!’ The kid seemed quite happywith this answer.

Some years later, we found out that each of us had been trying separately to find abetter answer. We were led to the following problem, once considered by Isaac New-ton:

Problem of the frustum of a cone. Given the altitude and base radius, determinethe dimensions of the frustum of a cone (see Figure 2) subject to least resistance whenmoving in a rare medium with constant velocity.

VOL. 41, NO. 2, MARCH 2010 THE COLLEGE MATHEMATICS JOURNAL 145

It turns out that there is a link between the minimizing frustum, which we call New-ton’s frustum, and a family of means called POEM’s (for pth Order Extreme Means).The best known member of this family is the golden mean. The main goal of this paperis to explain this link. The golden mean (also called the golden ratio) turns out to belinked to a special case of Newton’s problem.

There are numerous situations in the literature in which the golden mean has beennaively interpreted as an indicator of optimal beauty; but only a few in which this meanarises as the solution of a true optimization problem. Here is one such problem.

E OH Q D

G

F

B

C

S

θ

Figure 2. Newton’s frustum.

Newton’s frustum and the golden meanIn 1687, in his Philosophiæ Naturalis Principia Mathematica Newton stated, withoutdemonstration, the following geometric solution to the problem of the frustum of acone:

As if upon the circular base C E B H from the centre O , with the radius OC , andthe altitude O D [Figure 2], one would construct a frustum C BG F of a cone,which should meet with least resistance than any other frustum constructed withthe same base and altitude and going forwards towards D in the direction of itsaxis: bisect the altitude O D in Q, and produce O Q to S, so that QS may beequal to QC , and S will be the vertex of the cone whose frustum is sought.

[9, vol. 1, p. 333]

In the special case that OC = O D = r , that is, the base radius and altitude areequal, we call the resulting frustum the golden frustum. Note that for the golden frus-tum O Q = r/2 and QS = QC = √

5r/2. Thus,

O S = O Q + QS = 1 + √5

2r = φr,

where φ = (1 + √5)/2 is the golden ratio. So if θ is the angle between the generator

of the frustum and the plane of its lower base (see Figure 2), then we have a goldenfrustum precisely when

tan θ = φ.

146 © THE MATHEMATICAL ASSOCIATION OF AMERICA

Moreover, a golden frustum can be obtained from a so-called golden rectangle as fol-lows (see Figure 3): Rotate the right angled triangle of base r and height φr about itslongest side. Then cut the cone so constructed at a height r .

θ

φr

r

r

r*

Figure 3. Golden frustum and golden rectangle.

Note that if r∗ is the radius of the upper base of the golden frustum, then

φ = tan θ = φr − r

r∗ .

Therefore,

r∗ =(

1 − 1

φ

)r = r

φ2,

where we have used the second of the following identities

φ2 = φ + 1 and 1 − 1

φ− 1

φ2= 0, (1)

which follow from the fact that φ is the positive root of

x2 − x − 1 = 0.

Newton’s model of resistanceTo derive Newton’s frustum, we need a mathematical expression for the resistance ofa frustum moving in a rare medium with constant velocity.

Following Tikhomirov [11], we consider a frustum of a cone of altitude h, withlower base of radius r > 0 and upper base of radius x ∈ [0, r ] (see Figure 4). Weassume that each particle has mass m and that the density of the rare medium is aconstant ρ. We also assume that the particles hit the frustum from above with constantvelocity v and that the collisions are all perfectly elastic; that is to say:

1. If p1 and p2 denote the momenta of each particle, before and after the collision,and v = |v|, then |p1| = |p2| = mv.

2. The collisions follow the law of reflection, i.e., the angle of incidence equals theangle of reflection.


x

r

m

h

v�t

r – x

v

θ

Figure 4. Frustum of a cone hit by particles of a rare medium.

By Newton’s second and third laws, the resistance R(x) will be the vertical com-ponent of the momentum gained by the frustum from the total number of collisionsreceived per unit time.

Let P = p2 − p1 denote the change of momentum of a single particle that hits thefrustum (see Figure 5). If Rs denotes the vertical component of P for a particle hittingthe side of the frustum, then

Rs = |P| cos θ.

Since the collisions are perfectly elastic P is perpendicular to the side of the frustumand, using the sine law,

|P|sin(π − 2θ)

= mv

sin θ.

Hence |P| = 2mv cos θ and therefore

Rs = 2mv cos2 θ.

Note that if the particle hits the upper base of the frustum, then the vertical componentRu of the change of momentum is

Ru = |P| = |2mv| = 2mv.

h

r

x

Rs

P = p2 – p1

p2

p1

θ

θθ

Figure 5. Change of momentum P of a single particle hitting the frustum.


Thus, if Ns is the number of particles that hit the side of the frustum per unit time,and Nu is the corresponding quantity for the particles that hit the upper base, then

R(x) = Nu Ru + Ns Rs

= 2mvNu + 2mvNs cos2 θ. (2)

Observe now (see Figure 4) that the volume occupied by the particles that hit the upperbase of the frustum during the time �t is πx2v �t . Since the total mass of this volumeis m Nu�t ,

ρ = m Nu �t

πx2v �t,

and thus

Nu = πρx2v

m.

Similarly, the volume filled by the particles that hit the side of the frustum during thetime �t is equal to π(r 2 − x2)v �t . Since the mass of this volume is m Ns �t , then

ρ = m Ns �t

π(r 2 − x2)v �t

and therefore

Ns = πρ(r 2 − x2)v

m.

Substituting Nu and Ns in (2) we find that

R(x) = 2πρv2[x2 + (r 2 − x2) cos2 θ].Noting from Figure 4 that

cos θ = r − x√(r − x)2 + h2

,

we arrive at

R(x) = 2πρv2

[x2 + (r 2 − x2)

(r − x)2

(r − x)2 + h2

].

Derivation of the golden frustumTo solve Newton’s problem for the frustum of a cone in the special case h = r , wemust minimize

S(x) = x2 + (r 2 − x2)(r − x)2

(r − x)2 + r 2

for x ∈ [0, r ].


Although this problem can be solved using standard calculus techniques or the in-equality method of Tikhomirov [11], we will solve it here using only the fact that φ

satisfies the identities (1). Simplifying the formula for S(x) we have

S(x) = x2 + (r 2 − x2)(r − x)2 + r 2 − r 2

(r − x)2 + r 2

= x2 + (r 2 − x2)

(1 − r 2

(r − x)2 + r 2

)

= r 2 − r 2 − x2

(r − x)2 + r 2r 2.

Next we note that

1

φ− r 2 − x2

(r − x)2 + r 2= (1 + φ)x2 − 2xr + (1 − φ)r 2 + r 2

φ((r − x)2 + r 2)

= φ2(1 + φ)x2 − 2xφ2r + φ2(1 − φ)r 2 + r 2φ2

φ3((r − x)2 + r 2)

= φ4x2 − 2xφ2r + (1 − φ2)r 2 + r 2φ2

φ3((r − x)2 + r 2)

= (φ2x − r)2

φ3((r − x)2 + r 2),

where we have used the first identity in (1) to go from the second to the third line.Combining the last identity with the simplified formula of S(x), and using the secondidentity in (1) we obtain

S(x) =(

1 − 1

φ

)r 2 + r 2

φ3

(φ2x − r)2

(r − x)2 + r 2

= r 2

φ2+ r 2

φ3

(φ2x − r)2

(r − x)2 + r 2.

Hence, the minimum value μ of S(x) for x ∈ (−∞, ∞) is attained when

r∗ = r

φ2,

and is given by

μ = r 2

φ2.

Since φ > 1, we have that 0 < r ∗ < r and, therefore, μ is the minimum value ofS(x) for x ∈ [0, r ]. Furthermore, if θ is the angle between the generator of the goldenfrustum (see Figure 3) and the plane of its lower base, then using r∗ = r/φ2 and thesecond identity in (1) we find that

tan θ = r

r − r ∗ = 1

1 − (1/φ2)= φ.


Thus, using only some algebraic properties of the golden mean, we have provideda simple and transparent derivation of the golden frustum, and recovered all the factsthat we derived above from Newton’s geometric construction.

Misconceptions, POEM’s and Newton’s frustumLike many others, we have, in the past, been beguiled by the many enthusiastic writers(for example [5], [6], [10]) who praise prodigally the idea that the golden mean isan indicator of optimal beauty. We even thought that if the shape of the Apollo 11Command Module Columbia was a golden frustum, then our response to that nineyear old boy was entirely correct. We were so pleased about this conception of thegolden frustum that we even expressed some views along these lines in [2].

However, as pointed out in this JOURNAL by Markowsky [8] and later Falbo [3],most aesthetic ideas about the golden ratio are misconceptions. Falbo, in particular,points out the strong similarity between the algebraic and geometric properties of thegolden mean φ, and the positive root φp of

x2 − px − 1 = 0, (3)

where p is a fixed positive number. Falbo calls

φp = p + √p2 + 4

2

the pth order extreme mean, or POEM. For p = 1 we obtain the golden mean. Aremarkable subclass of the POEM’s is obtained when p = n is a positive integer. Thissubclass is referred to by Fowler [4] as the class of NOEM’s (for nth Order ExtremeMeans) and by Huylebrouck [7] as the family of metallic ratios. In addition to thegolden mean, important members of this family are the silver mean σAg = 1 + √

2 =φ2 and the bronze mean σBr = (3 + √

13)/2 = φ3. In his proof of the irrationality of

ζ(2) = 1 + 1

22+ 1

32+ 1

42+ · · · = π2

6,

Huylebrouck [7] needs to solve an optimization problem whose solution happens to begiven in terms of φ. Furthermore, Huylebrouck gives a parallel proof of the irrational-ity of Apery’s number ζ(3) in which σAg plays the role of φ.

Following Newton’s directions once again (see Figure 2), we note that if OC = rand O D = h = pOC = pr , where p is a real positive number, then O Q = pr/2 and

QS = QC =√

p2 + 4

2r.

Therefore

O S = O Q + QS = p + √p2 + 4

2r = φpr.

Thus, Newton’s frustum is precisely the one for which the angle θ , between thegenerator of the frustum and the plane of its lower base, satisfies

tan θ = φp.


It turns out that now, all the results we have obtained for the golden frustum haveparallel results (and parallel proofs) for Newton’s frustum, with φ replaced by φp and(1) by the identities

φ2p = pφp + 1 and 1 − p

φp− 1

φ2p

= 0,

which follow from the fact that φp satisfies (3). In particular,

The frustum of a cone, with base radius r and height h = pr , subject to least re-sistance when moving in a rare medium is the one for which the angle θ betweenthe generator of the frustum and the plane of its lower base satisfies

tan θ = φp,

or, equivalently, the one for which the radius of the upper base is

r ∗ = r

φ2p

.

Is the Command Module a Newton frustum?We have not been able to obtain exact measurements of this piece of spacecraft, so wecannot resolve this question. We leave it to an enterprising reader to decide whetherthe Apollo Command Module is a Newton frustum (or even a golden frustum).

ConclusionWe have demonstrated a close relationship between Falbo’s POEM’s and Newton’saerodynamical problem for the frustum of a cone. In fact, we solved this minimizationproblem using only two simple algebraic properties of the POEM’s. We also expressedthe significant quantities of the minimizing frustum in terms of these ratios. This wayof viewing Newton’s aerodynamical problem for the frustum of a cone shows that thePOEM’s not only enjoy the algebraic and geometric properties established by Falbo,but also a remarkable minimizing physical property.

Concerning φ, on the one hand, we have another example of a connection betweenthe golden ratio and a standard optimization problem. On the other hand, sadly, wehave another situation in which an interesting property of the golden mean is matchedby those of the POEM’s.

Acknowledgment. We appreciate very much the time, energy, and patience expended by theeditor to help us improve the presentation of this paper. The first author was partially supportedby El Consejo Nacional de Ciencia y Tecnologıa in Mexico, Grant 89639.

Summary. There are numerous situations in the literature in which the golden mean has beennaively interpreted as an indicator of optimal beauty; but only a few in which this mean arisesas the solution of a true optimization problem. In this article we present one such problem. Wedemonstrate a close relationship between the golden mean and a special case of Newton’s aero-dynamical problem for the frustum of a cone. Then, we exhibit a parallel relationship betweenthe general case of this problem and a family of means called POEM’s (for pth Order ExtremeMeans). This shows that the POEM’s not only share with the golden mean the algebraic and


geometric properties established by Falbo, but also a remarkable minimizing physical prop-erty. Sadly, we have another situation in which an interesting property of the golden mean ismatched by those of the POEM’s.

References

1. Command module at the National Air and Space Museum, from Wikipedia, the free encyclopedia; availableat http://en.wikipedia.org/wiki/File:NASA_Apollo_11_command_module.jpg.

2. J. Cruz-Sampedro, L. E. Rivera-Fernandez and M. Tetlalmatzi-Montiel, Formas aerodinamicas y la razonaurea, Miscelanea Matematica 47 (2008) 11–22 (in Spanish).

3. C. Falbo, The golden ratio—a contrary viewpoint, College Math. J. 36 (2005) 123–134.4. D. H. Fowler, A generalization of the golden section, Fibonacci Quart. 20 (1982) 146–158.5. M. C. Ghyka, The Geometry of Art and Life, Dover, New York, 1977.6. H. E. Huntley, The Divine Proportion, Dover, New York, 1970.7. D. Huylebrouck, Similarities in irrationality proofs for π , ln 2, ζ(2), and ζ(3), Amer. Math. Monthly 108

(2001) 222–231. doi:10.2307/26953838. G. Markowsky, Misconceptions about the golden ratio, College Math. J. 23 (1992) 2–19. doi:10.2307/

26861939. I. Newton, Principia, vol. I, The Motion of Bodies, trans. A. Motte, University of California Press, Los

Angeles, CA, 1962.10. D. Pedoe, Geometry and the Visual Arts, Dover, New York, 1983.11. V. M. Tikhomirov, Stories About Maxima and Minima, Mathematical World, vol. 1, American Mathematical

Society–Mathematical Association of America, Providence, RI, 1990.

The Sweetness of Primes

The Professor loved prime numbers more than anything in the world. I’d beenvaguely aware of their existence, but it never occurred to me that they couldbe the object of someone’s deepest affection. He was tender and attentive andrespectful; by turns he would caress them or prostrate himself before them; henever strayed far from his prime numbers. Whether at his desk or at the din-ner table, when he talked about numbers, primes were most likely to make anappearance. At first, it was hard to see their appeal. They seemed so stubborn,resisting division by any number but one and themselves. Still, as we were sweptup in the Professor’s enthusiasm, we gradually came to understand his devotion,and the primes began to seem more real, as though we could reach out and touchthem. I’m sure they meant something different to each of us, but as soon as theProfessor would mention prime numbers, we would look at each other with con-spiratorial smiles. Just as the thought of a caramel can cause your mouth to water,the mere mention of prime numbers made us anxious to know more about theirsecrets.

—Y. Ogawa, The Housekeeper and the Professor(reviewed on page 170)


poem’s and newton’s aerodynamic frustum - imelda trejo · poem’s and newton’s aerodynamic...

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