The Mathematics Enthusiast Volume 18 Number 1 Numbers 1 & 2 Article 3 1-2021 Unit shapes and a wealth of calculus problems Robert G. Donnelly Alexander F. Thome Follow this and additional works at: https://scholarworks.umt.edu/tme Let us know how access to this document benefits ou.y Recommended Citation Donnelly, Robert G. and Thome, Alexander F. (2021) "Unit shapes and a wealth of calculus problems," The Mathematics Enthusiast: Vol. 18 : No. 1 , Article 3. Available at: https://scholarworks.umt.edu/tme/vol18/iss1/3 This Article is brought to you for free and open access by ScholarWorks at University of Montana. It has been accepted for inclusion in The Mathematics Enthusiast by an authorized editor of ScholarWorks at University of Montana. For more information, please contact [email protected]. Donnelly & Thome, p. 10 Unit shapes and a wealth of calculus problems Robert G. Donnelly and Alexander F. Thome Department of Mathematics & Statistics, Murray State University ABSTRACT: For a given family of similar shapes, what we call a \unit shape" strongly analogizes the role of the unit circle within the family of all circles. Within many such families of similar shapes, we present what we believe is naturally and intrinsically unital about their unit shapes. We present a number of calculus problems related to extremal questions about collections of unit shapes, and we recapitulate some isoperimetric problems in terms of unit shapes. We close by presenting some problems (some of which are open) and by proffering perhaps a new perspective on the π vs. τ debate. Keywords: area, perimeter, semiperimeter, simple closed curve, Blob Pythagorean Theorem, rigid motion, congruence, scaling, similarity, Isoperimetric Inequality The Mathematics Enthusiast, ISSN 1551-3440, vol. 18, nos. 1-2, pp. 10{20 2021 c The Authors & Dept. of Mathematical Sciences { The University of Montana TME, vol. 18, nos. 1-2, p. 11 1 Introduction A striking and frequently pondered fact commonly encountered in first-semester Calculus is that d πr2 = π · 2r; dr an equality which can be interpreted as saying that, for circles, the derivative of area is perimeter (i.e. circumference) when each of these quantities is understood with respect to circle radius r. This familiar identity is often interpreted/explained using differentials, as one can see, for example, in the internet discussions [3] and [8]. Here is how this line of reasoning generally goes: Express circle area A and perimeter P as functions of radius: A(r) = π r2 and P(r) = π · 2r. Then a small change dr in the radius r results in an exact change in area r + (r + dr) ∆A = π(r + dr)2 − πr2 = π · 2 dr = π · 2rdr + π(dr)2; 2 which is the area of an annulus of thickness dr along the circle's boundary. If dr is small, then this thin piece of area is about the same as the area of a rectangle of length π · 2r and height dr, i.e. the contribution of the term π(dr)2 is relatively negligible. So ∆A ≈ dA = π · 2rdr = P(r)dr as dr ! 0. One could view this geometric reasoning as an interpretation/explanation of the differential relationship dA A0(r) = = P(r) (1.1) dr for circles. Perhaps one reason that fact (1.1) catches our eye is that it does not immediately square with other similar scenarios. For example, if we express circle area and perimeter in terms of diameter d, then π 2 dA 0 π A(d) = 4 d , P(d) = πd, but dd = A (d) = 2 d 6= πd = P(d). Or, if we express square area A and 2 dA 0 perimeter P in terms of side length s, then A(s) = s , P(s) = 4s, but ds = A (s) = 2s 6= 4s = P(s). Among the many papers that take more general viewpoints on fact (1) | see for example [6], [10] | our favorite is from a College Mathematics Journal note from 1997 by Jingcheng Tong [9]. Let us briefly summarize the perspective of [9] for planar figures using notions that we more fully develop in 1 x2 and x3 below. Let C be a shape in the plane with perimeter P(C), semiperimeter S(C) = 2 P(C), and area A(C). Suppose λ is a positive real number. Apply the planar scaling function that stretches or compresses the plane by a factor of λ. Let Cλ be the image of shape C under this transformation. In x2 we will more carefully consider the following intuitively plausible facts: P(Cλ) = λP(C), S(Cλ) = λS(C), 2 and A(Cλ) = λ A(C). Next, set υ := S(C)=A(C). If we let Dλ := Cυλ, then λ = A(Dλ)=S(Dλ). Set 2 2 P(λ) := P(Dλ) = λ(υP(C)), S(λ) := S(Dλ) = λ(υS(C)), and A(λ) := A(Dλ) = λ (υ A(C)). Then A0(λ) = 2λ(υ2A(C)) = 2λ(υS(C)). So, dA A0(λ) = = 2S(λ) = P(λ): (1.2) dλ That is, when we use the ratio of area{to{semiperimeter as the indexing parameter for a family of similar shapes, then perimeter is the derivative of area with respect to this parameter. Here we adopt the nomenclature of [6] and refer to this parameter as the Tong inradius for such a family of similar shapes. In this paper, we use the perspective of [9] to develop the apparently new concept of a unit shape within any given family of similar shapes. We will see that the relationship of this unit shape to its other similar shapes strongly analogizes the relationship of the unit circle to all circles. As examples, we will characterize unit right triangles, unit triangles in general, unit rectangles, unit rhombi, unit parallelograms, and unit ellipses. Further, we consider extremal problems relating to unit shapes, such as: Which unit triangle encloses the minimum area? We demonstrate that for many such questions, there are equivalent isoperimetric problems. We propose what we believe is naturally unital about each of the unit shapes considered here and remark on how such notions might contribute to the π vs. τ debate. Throughout we propose many related calculus problems, some of which are open. We formulate all these ideas using the language of first-, second-, and third-semester Calculus & Analytic Geometry together with some elementary linear algebra. Donnelly & Thome, p. 12 2 Shapes, families of shapes, and similarity We begin by setting up some preliminary geometric notions framed in the language of elementary calculus and linear algebra. Our work will mostly take place in the Euclidean plane, which is here denoted R2. A curve is some continuous function f : I −! R2 from an interval I on the real line to the Euclidean plane with coordinate functions, perhaps designated as xf and yf , such that f(t) = (xf (t); yf (t)). When such a curve is continuously differentiable on [a; b] (taking a right-derivative at a and a left-derivative at 0 0 b q 0 2 0 2 b) such that the derivatives xf (t) and yf (t) do not simultaneously vanish, then a xf (t) + yf (t) dt is some positive real number; we call f nicely rectifiable. The curve f is piecewise´ nice if it is continuous on [a; b] and there is some partition a = a0 < a1 < ··· < an−1 < an = b of the interval such that the restriction of f to each subinterval [ai−1; ai] (i.e. each \piece" of f) is nicely rectifiable. For us, a shape C will be a piecewise nice simple closed curve in the Euclidean plane, notated as C :[a; b] −! R2 with C t 7! (xC(t); yC(t)) and C(a) = C(b). Most often we identify the shape (or curve) C with its image C([a; b]) in the plane. A rigid motion M : R2 −! R2 of the Euclidean plane R2 is the composition of a rotation R, (possibly) followed by a reflection S, and followed by a translation T . Two shapes C and D are congruent, written C =∼ D, if D = M(C) for some rigid motion M. A rotation R of the plane through an angle θ can be expressed via matrix multiplication: x R cos θ − sin θ x 7−! ; y sin θ cos θ y where a rotation through θ = 0 corresponds to the identity transformation of the plane. A reflection S a of the plane across a line through the origin in the direction of some unit vector can be expressed b thusly: 2 2 x S a − b 2ab x 7−! : y 2ab b2 − a2 y h A translation T of the plane by a vector is given by k x T x h x + h 7−! + = : y y k y + k For clarity, we sometimes write T = T(h;k). The rigid motion M may or may not include a reflection S; we use a binary exponent M = 1 to indicate the inclusion of S when we write M as a composite function M and M = 0 to indicate the exclusion of S. So, M = T(h;k) ◦ S ◦ R. We leave it as an exercise for the reader to confirm that if M and N are rigid motions of the plane, then so is M −1 ◦ N. 2 2 Let λ be a positive real number. The planar scaling function Lλ : R −! R is given by Lλ(x; y) := (λx, λy). (The notation \L," which abbreviates the word \length-modifying," is our mnemonic for M remembering that Lλ is scaling function.) Observe that for a rigid motion M = T(h;k) ◦ S ◦ R, 0 0 M we have Lλ ◦M = M ◦Lλ, where M = T(λh,λk) ◦S ◦R.