Squigonometry: The Study of Imperfect Circles

292 Pages·2022·6.692 MB·English

by  Robert D. Poodiack , William E. Wood

SUMS Readings
Robert D. Poodiack
William E. Wood
Squigonometry:
The Study
of Imperfect
Circles
Springer Undergraduate Mathematics Series
SUMS Readings
Advisory Editors
Mark A. J. Chaplain, St Andrews, UK
Angus Macintyre, Edinburgh, UK
Simon Scott, London, UK
Nicole Snashall, Leicester, UK
Endre Süli, Oxford, UK
Michael R. Tehranchi, Cambridge, UK
John F. Toland, Bath, UK
SUMS Readings is a collection of books that provides students with opportunities
to deepen understanding and broaden horizons. Aimed mainly at undergraduates,
the series is intended for books that do not fit the classical textbook format, from
leisurely-yet-rigorous introductions to topics of wide interest, to presentations of
specialised topics that are not commonly taught. Its books may be read in parallel
with undergraduate studies, as supplementary reading for specific courses,
background reading for undergraduate projects, or out of sheer intellectual
curiosity. The emphasis of the series is on novelty, accessibility and clarity of
exposition,aswellasself-studywitheasy-to-followexamplesandsolvedexercises.
Robert D. Poodiack (cid:129) William E. Wood
Squigonometry: The Study
of Imperfect Circles
123
RobertD.Poodiack William E. Wood
Department ofMathematics Department ofMathematics
Norwich University University of Northern Iowa
Northfield,VT, USA Cedar Falls,IA, USA
ISSN 1615-2085 ISSN 2197-4144 (electronic)
SpringerUndergraduate MathematicsSeries
ISSN 2730-5813 ISSN 2730-5821 (electronic)
SUMSReadings
ISBN978-3-031-13782-2 ISBN978-3-031-13783-9 (eBook)
https://doi.org/10.1007/978-3-031-13783-9
MathematicsSubjectClassification: 33-01,51Nxx,46B10,33B10,14Hxx
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Preface
Our main goals in writing Squigonometry: The Study of Imperfect Circles were to
compile results in generalized trigonometry that had been known for years, to
introduce new facts and relationships that we had only recently discovered, and
to create a book that exposes students to the rewards available from taking ideas
fromtheirearlyexperiencesinmathematicsandpushingonthem.Inparticular,we
wantedtowriteabookthatcouldenticeundergraduatestudentstoexploretheback
alleyways of a familiar corner of mathematics.
Squigonometry brings a modern perspective to some old mathematics, offering
many opportunities for students to explore a variety of mathematical niches and
maybe discover something new, or see something old in a new way.
Generalization and abstraction have always been part of the fiber of mathe-
matics. To take something we have known to be true for a long time, a canonical
equation, for example, and disassemble it, thoroughly examine it, and then
reassemble it with some different parts is often the key to discovering structure.
Doingthiscanoftenactasagatewaytoanewmathematicalworld.Thisbookisan
introduction to generalizations of concepts that students learn early in their math-
ematical careers, namely,
1. Intheplane,thedistancebetweentwopointsðx1;y1Þandðx2;y2Þisgivenbythe
Pythagorean theorem:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
d ¼ ðx (cid:2)x Þ2þðy (cid:2)y Þ2:
2 1 2 1
2. Theunitcircle,thesetofallpointsðx;yÞthatareadistanceof1unitfromð0;0Þ,
has equation x2þy2 ¼1.
3. The points on the unit circle can be given as ðx;yÞ¼ðcosðtÞ;sinðtÞÞ where t is
the measure of the angle subtended along the arc of the circle from ð1;0Þ to
ðx;yÞ. (This gives the relationship cosðtÞ2þsinðtÞ2 ¼1.)
As students continue through calculus, they learn that dcosðtÞ¼(cid:2)sinðtÞ and
dt
dsinðtÞ¼cosðtÞandthattheparametertrepresentsnotonlyananglemeasure,but
dt
isalsotwicetheareaofthesectoroftheunitcirclesweptoutbytheangle,aswell
vii
viii Preface
asthelengthofthearcoftheunitcirclesubtendedbytheangle.Studentsalsolearn
the importance and ubiquity of p in these contexts.
The sine and cosine functions are so durable and so flexible that we can define
them in several different ways and obtain the same functions—the unit circle
approachanddifferentialequationrelationshipaboveyieldthesametwofunctions,
as does defining sine and cosine functions via integrals.
An entry into generalized trigonometry comes from the search for variants of
sine and cosine functions that can parameterize the closed curve jxjpþjyjp ¼1.
Onceweselectavariant,weneedtoanswersomequestions.Whatdoesthiscurve
looklike?Doalltheconnectionsbetweenarea,angle,andarclengthhold?Canwe
measuredistanceinthesamewayasbefore?Aretherewaysofapproximatingthese
functions?
The philosophy of the book is to follow these ideas in small steps and build up
consequencesofourchoicesineachchapter.Whilethechangeofa2toagenericp
in an equation may seem only cosmetic, the ripples from dropping that stone go a
longway,envelopingnotonlythetrigonometricfunctionsbutalsoourconceptions
of distance, and the hyperbolic, exponential, and logarithmic functions.
Audience and Structure
Squigonometry is suitable for an undergraduate reading course in mathematics as
well as a source for senior capstone research. Students should know the basic
trigonometric functions as well as both differential and integral single-variable
calculus, initial-value problems, and results on infinite series. Familiarity with
multivariable and vector calculus will allow additional insights into Chapters 13
and 18, and some basic tools from linear algebra would also assist in Chapter 18.
One notion that ties much of the book together is that of parameterization, the
ideathatwecansetx¼fðtÞandy¼gðtÞforsomevaluesoftandtheorderedpair
ðx;yÞ¼ðfðtÞ;gðtÞÞ describes a curve in the plane. A primer (or refresher) on
parametric equations can be found in Appendix C.
The book is meant to function as both an introduction to generalized
trigonometry and an invitation into further research. The chapters are short by
design and focus on one aspect of squigonometry. Fundamental results about our
new functions such as relational equations, derivatives, integrals, and series
approximations are all developed. Variations on p are introduced through the lens
ofso-calledspecialfunctions.Fromthere,readers canexploreoffshoottopicssuch
ashyperbolic,exponential,andlogarithmicfunctions;metricsandnorms;andmore
advancedtopicssuchaslemniscaticandellipticfunctions,andothergeneralizations
of the trigonometric functions that are different from the ones focused on here and
that have cropped up in recent years. There are historical reasons for the creation
of these generalized functions and these are sprinkled throughout the exposition.
Students should also come away with the idea that there is no last word on
squigonometric functions.
Preface ix
TheintroductioninChapter1givesabriefrecapofthewaysinwhichwedefine
the usual trigonometric functions as well as the relations among them. We rein-
troducefamiliarformulaswewillgeneralizelaterinthebook,andinChapter2,we
showhowreplacingtheperfectionofthecirclewithanimperfectshape—onewith
far less symmetry than the circle—can lead to issues.
Chapter3introducesustothep-trigonometric(orsquigonometric)functions,the
generalized trigonometric functions of Grammel and Shelupsky and the notion
of the p-circle, the locus of points related by the equation jxjpþjyjp ¼1. We also
define generalizations of the tangent, cotangent, secant, and cosecant.
In Chapter 4, we explore the idea of defining distance differently than we are
accustomedtobyintroducingthep-metric.Inparticular,wewillbeabletolookat
theunitp-circleasthesetofallpointsthatareoneunitawayfromtheorigininthe
p-metric.
We return in Chapter 5 to the integrals that led mathematicians to this study in
the first place, and look at the inverses of the squigonometric functions. We also
develop a generalized version of right triangle trigonometry using the p-metric.
Up until this point in the book, the relationships and formulas between the
various squigonometric functions are established through purely formal manipu-
lation. In Chapter 6, we introduce a generalization of pfor each p by appealing to
so-called special functions. We then establish domains and ranges for our gener-
alized trigonometric functions and their inverses.
In Chapters 7 and 8, we note that in generalizing the trigonometric functions in
this way, we may preserve knowledge of the area of the p-circle, but we lose
information such as the length of an arc and the angle subtended. (The case p¼2
has been the “go to” for years because it preserves all of that information.) In
particular, in Chapter 8, we develop a squigonometry inspired by the generaliza-
tions of p in [2, 30].
Chapter9introducesnotonlytheintegralformulaslistedbyGrammel[42],but
alsosometweaksofknownintegrationtechniquesfortrigonometricfunctions(such
as inverse trigonometric substitution) that will allow us to evaluate some integrals
that previously eluded mathematicians. We also look at three applications of this
kind of integration in Chapter 10, two of which are via mathematician Piet Hein:
finding the volume of the central square in Stockholm, and determining the exact
volumeofHein’sso-called“superegg.”Thethirdinvolvesthetrisectionofaregion
between two curves.
Chapter 11 introduces series representations for the squigonometric functions
andfortheinversesinesandtangents,aswellasextensionsofourfunctionstothe
complex plane. We then use these series in Chapter 12 to find reasonable series
approximations for the various p’s, as well as examine continued fraction repre-
sentation of the p’s, and how to find exact sums of related series.
Polarcoordinatesareoftenintroducedtostudentsinprecalculusorcalculusand
in Chapter 13, we introduce squircular coordinates. Just as polar coordinates
generalize to cylindrical and spherical coordinates, squircular coordinates
x Preface
generalize to p-cylindrical and p-spherical coordinates. This will allow us to
examine other three-dimensional objects such as superellipsoids.
Most of the previous research on generalized trigonometric functions also
includedgeneralizationsofthehyperbolicfunctions,andwegothereinChapter14.
We establish relationships between the generalized trigonometric and hyperbolic
functions via complex numbers and a special function called the Gudermannian.
We also determine the radii of convergence for the series representations of the
generalized transcendental functions discovered thus far. In an example of a
seemingly small change having perhaps unforeseen circumstances, this leads to
generalizations of the exponential and logarithmic functions in Chapter 15.
Lemniscates—curves with polar equation r2 ¼cosð2hÞ—provided the initial
impetus to generalize trigonometric functions and in Chapter 16, we dive into the
relationship between these lemniscatic functions, the related elliptic integrals, and
oursquigonometricfunctionsinthep¼3andp¼4cases.Inparticular,welookat
additionanddouble-angleformulasforDixonellipticfunctions.Wefurtherexplore
these relationships in Chapter 17 as well as how so-called elliptic functions—
generalizationsoftrigonometricfunctionsthatparameterizecertainellipses—fitin.
We had earlier used the p-metric to ease the discovery of relationships between
squigonometricfunctionsandtheirinverses.InChapter18,weintroducetherelated
idea of the p-norm (a metric measures distances between two things; a norm
measuresthesizeofasinglething).Normsandthenotionofconvexityallowustie
upsomelooseendsinourunderstandingofthep-metric,mostnotablythetriangle
inequality. With these tools in hand, we formally define the dual norm in
Chapter 19 with a focus on geometric understanding of the special relationship
between conjugate p-metrics (when 1 þ 1¼1).
p q
In particular, we look at the relationship between area and arclength, between
curvesandtheirso-calledpedals,andhowto“invert”thecurveswehaveexamined
thus far. We also get a geometric look at some classical theorems of analysis,
including Hölder’s inequality.
We wrap things up in Chapter 20 by briefly looking at a few other generaliza-
tions of trigonometric functions and some additional directions for students to
explore these functions.
The chapters are generally brief. Each chapter begins with a short introduction
discussing some of the history behind that chapter’s focus and then presents the
main results with proofs. Many of the proofs in the book are computational in
nature and use known results from calculus and the theory of special functions to
discover properties of our squigonometric functions.
Someofthesepropertiesaredevelopedfurtherintheexercisesectionsattheend
of each chapter. The exercises vary in difficulty from one-or two-line proofs to
extensiveapplications ofthematerial, butthe material necessaryto complete them
canbefoundinthetext.Theyaremeanttoofferpracticewithnewfoundskillsand
reinforceconnections.Theappendixincludessolutionsandhintstomanyexercises.