The Magic Of Mathematics-slicer.pdf

•

i

i

Discovering

the

Spell

of

othematics

,z

lJ

M*L«*

author of

The

Mathematics

Calendars

The

_Joy

of

Mathematics

MAGIC

Discovering

the

Spell

of Mathematics

THEONI

PAPPAS

Copyright

© 1994

_by

Theoni

_Pappas.

All

_rights

reserved. No

_part

of this_{work may be}

_reproduced

or

copied

in_{any form}or

_by

_anymeanswithoutwritten

_permission

from Wide World

_{Publishing/Tetra.}

Portions of this book have

appeared

in

_previously

_published

works,butweretooessentialto notbe included.

Wide World

_{Publishing/Tetra}

P.O.Box476

SanCarlos, CA 94070

Printed intheUnited States of America.

Second

Printing,

October 1994.

Library

of

_Congress

_{Cataloging-in-Publication}

Data

Pappas,

Theoni,

The

_magic

of mathematics:

_discovering

the

_spell

of mathematics

_/

Theoni

_Pappas.

p. cm.

Includes

_{bibliographical}

references and index,

ISBN 0-933174-99-3

1.

_{Mathematics--Popular}

works. I.Title.

QA93 , P368 1994

510-dc20 94-11653

who have

created

and

are

_creating

CONTENTS

PREFACE

1

MATHEMATICS

IN

EVERYDAY

THINGS

3

MAGICAL MATHEMATICAL

WORLDS

33

MATH

EMATICS

&

ART

63

THE MAO

IC

OF

NUMBERS

97

MATHEMATICAL MAGIC

IN

NATURE

119

MATHEMATICAL MAGIC FROM

THE

PAST

143

MATHEMATICS

PLAYS ITS MUSIC

173

THE

REVOLUTION

OF COMPUTERS

189

MATHEMATICS

& THE

MYSTERIES

OF

LIFE

223

MATHEMATICS

AND ARCHITECTURE

243

THE SPELL

OF

LOGIC,

RECREATION &

GAMES

265

SOLUTIONS

311

BIBLIOGRAPHY

315

PREFACE

Youdon't have to solve

_problems

or

beamathematician todiscover the

magic

of mathematics. This bookIs a collection of Ideas — ideas withan

_underlying

mathematical theme. It is notatextbook. Do not

_expect

to become

_proficient

in a

_topic

or find an idea

exhausted. The

_Magic

_of

Mathematics delves into the world of

ideas,

explores

the

_spell

mathematicscastson ourlives,and

_helps

you discover mathematics where you least

expect

it.

Many

think of mathematics as a

_rigid

fixed curriculum.

_Nothing

could be further from the truth. The human mind

_continually

creates mathematical ideas and

_fascinating

new worlds

independent

ofourworld — and

presto

these ideas connect to our

world almost as Ifa

_magic

wand had been waved. The way in

which

_objects

fromonedimension can

_disappear

into another, a

new

_point

can

_always

be_{found between any}two

_points,

numbers

operate,

equations are solved,

graphs

produce pictures,

infinity

solves

_problems,

formulasare

_generated

— all

seemto _possess a

magical quality.

Mathematical ideasare

_figments

of the

_imagination.

Itsideasexist

in alien worlds and its

_objects

are

_produced

_by

sheer

_logic

and

creativity.

A

_perfect

_square or circle exists in a mathematical

world, while our world has

_only

_{representations}

of

_things

mathematical.

The

_topics

and

concepts

whicharementionedineach

_chapter

are

by

no meansconfinedtothatsection. On the

_contrary,

_examples

can

_easily

cross overthe

_arbitrary

boundaries of

_chapters.

Evenif itwere

_possible,

itwould be undesirableto restrictamathematical idea to a

_specific

area. Each

topic

is

_essentially

_{self-contained,} and canbe

_enjoyed

independently.

I

_hope

this book will bea

stepping

stoneinto mathematical worlds.

Print_{Gallery by}M.C.Escher.

EVERYDAY THINGS

THE MATHEMATICS OF FLYING

THE

MATHEMATICS

OF A

TELEPHONE CALL

PARABOLIC REFLECTORS

&

YOUR

HEADLIGHTS

COMPLEXITY AND

THE

PRESENT

MATHEMATICS & THE CAMERA

RECYCLING THE NUMBERS

BICYCLES,

POOL TABLES & ELLIPSES

THE

RECYCLING NUMBERS

LOOKOUT FOR TESSELLATIONS

STAMPING OUT MATHEMATICS

MOUSE'S

TALE

A

MATHEMATICAL

VISIT

THE

EQUATION

OF TIME

4 THE MAGIC OF MATHEMATICS

There isnobranch of

mathematics,

however

abstract,

which may not

someday

be

_applied

to

phenomena

of

the

real

world.

—

Nikolai

Lobachevsky

So

many

things

with which we come into contact in our

_daily

routines have a mathematical basis or connection. These _range

from

_taking

a

_plane

_flight

tothe

_shape

ofamanhole. Oftenwhen

one least

_expects,

one finds mathematics is involved. Here is a

THE

MATHEMATICS

OF FLYINC

The grace and ease of the

_flight

of birds have

_always

tantalized

human's desire to

_fly.

Ancient stories_{from many cultures}attest to

interest in various

_flying

creatures.

_{Viewing hang}

_gliders,

one

realizes that the

_flight

of Daedalus and Icaruswas

_probably

not

just

aGreek

_{myth. Today}

enormous sized alrcrafts lift themselves

and theircargointothe domain of the bird. Thehistorical

_steps

to

achieve

_flight,

as we now know It, has

literally

had Its _{up and}

downs.

_Throughout

_{the years, scientists. Inventors, artists,}

mathematiciansand other

professions

have been

_{Intrigued by}

the Idea of

_flying

and have

developed designs,

prototypes,

and

_experiments

ineffortstobeairborne.

Here Isacondensed outline of the

_history

of

_flying:

•Kiteswereinvented

_by

the Chinese

(400-300

B.C.).

•Leonardo daVinci

_{scientifically}

studied the

_flight

_of

birds and

6 THE MAGIC OF MATHEMATICS

• _{Italian mathematician Giovanni Borelli}

proved

that human

musclesweretooweakto

_support

_flight

(1680).

• _{FrenchmenJean Pilatre de Rozierand}

Marquis

d! Arlandes

made the

_first

hotairballoonascent

(1783).

• _{British inventor. Sir}

_{George Cayley,}

_designed

_the

_airfoil

(cross-section)

of

a

_wing,

built

_andflew

(1804)

thefirst

model

glider, andfounded

thescience

_{ofaerodynamics.}

•

Germany's

OttoLilienthal deviseda

_system

tomeasurethe

_lift

produced

by

experimental

wings

andmade

_thefirst

_successful

manned

_{gliderflights}

between 1891-1896.

• _{In 1903 OrviUe and Wilbur}

Wright

made the

_first

_engine

powered propeller

driven

_{airplane flights. They}

_experimented

with wind tunnels and

_{weighing systems}

tomeasure the

_lift

and

_drag

_ofdesigns.

_They

_perfected

their

flying techniques

and

machinestothe

_point

that

_by

1905their

_flights

had reached38

minutes in

_length

covering

adistance

_of

20 miles!

Here'showwe

_get

_off

the

_ground:

Inorderto

_fly,

therearevertical and horizontal forces thatmust

be balanced.

_Gravity

(the downward vertical force)

keeps

us

earthbound.To counteractthe

_pull

of

_gravity,

lift(avertical

upward

force)

mustbecreated. The

_shape

ofwingsand the

_design

of

airplanes

IsessentialIn

_creating

lift. The

_study

ofnature's

_design

of

_wings

and of birdsIn

_flight

holds the

_key.

Itseemsalmost

sacrilegious

to

_quantify

the

_elegance

of the

_flight

ofbirds,but without

the mathematicaland

_{physical analyses}

of the

components

of

flying,

today's

airplanes

wouldneverhave left the

_ground.

Onedoes not

_always

think ofairas asubstance,since it isinvisible. Yet air is amedium, as water. The

wing

ofan

_airplane,

as wellas the

airplane

itself, divides or slices the air as it _passes

_through

it. Swiss mathematician Daniel Bernoulli

(1700-1782)

discovered thatasthe

_speed

of_gasorfluidincreases its pressure decreases.

Bernoulli's

_principle1

_explains

how the

_shape

ofa

_wing

creates

the

_speed

ofairand

_thereby

decreases theair_{pressure of the}air

passing

overit. Since the bottom of the

_wing

does not have this

curve, the

speed

of the air

_passing

under the

wing

is slowerand thusits air_pressureis

_higher.

The

_high

airpressure beneath the

wing

moves or

_pushes

toward the low pressure above the wing,

and thus lifts the

_plane

into the air. The

weight

(the

pull

of

gravity)

isthe vertical force thatcounteractsthelift of the

_plane.

Thewing'sshape

makesthe distance

overthe_toplonger,

whichmeansair musttravelover

thetopfaster, makingthe pressureonthe_top

ofthe_winglower than underthe

wing.Thegreater

pressure below the wingpushesthe

wingup.

When the_wingis

ata_steeper

_angle,

thedistanceover

the_topiseven

longer, thereby

increasingthe

liftingforce.

Drag

and thrust are the horizontal forces which enter the

_flying

picture.

Thrust

_pushes

the

_plane

forward while

_{drag pushes}

it

backwards. Abird creates thrust

by flapping

its

wings,

while a

plane

reliesonits

_propellers

or

_jets.

Fora

_plane

tomaintainalevel and

_straight

_flight

all the forces

acting

on it must

_equalize

one

another, i.e. bezero. The liftand

_gravity

must bezero, while the

thrust and

_drag

mustbalance.

_During

take off the thrustmustbe

greater

than the

drag,

butin

_{flight they}

mustbe

_equal,

otherwise the

_{plane's speed}

would be

_continually

increasing.

8 THE MACIC OF MATHEMATICS

When the

speed

ofairoverthe

top

of the

_wing

IsIncreased, the lift

will also Increase.

_By

_Increasing the

_wing's

_angle

to the

approaching

air, called the

angle

of

attack, the

_speed

overthe

_top

of the

_wing

canbefurtherIncreased. If this

_angle

Increases to

approximately

15or more

_degrees,

the liftcan

_stop

_abruptly

and the birdor

_plane

_begins

tofall Insteadof

_rising.

When this takes

_place

it is called the

_{angle of}

stall. The

_angle

of stall makes theairform vortexesonthe

_top

of the

_wing.

These vibrate the

_{wing causing}

the

lifttoweakenand the force of

_gravity

tooverpower the lift force.

Not

_having

been endowed with the

_flying

_equipment

of birds,

humans have utilizedmathematical and

_{physical principles}

tolift

themselves and other

_things

off the

_ground.

_Engineering

_designs

and features2 have been

_{continually adapted}

to

_Improve

an

aircraft's

_performance.

lLaws governingtheflow ofairfor

_{airplanes apply}

to_{many otheraspects}

Inour lives, suchas _{skyscrapers, suspension}

_bridges,

certain _computer

diskdrives,water_{and gas pumps, and turbines.}

^The

_flaps

andslotsare_changes

_adapted

tothe_wingwhichenhance lift. The

flap

Isa_hinged sectionthat when_engaged

_changes

thecurvatureof the_wing and addsto the liftforce. Slots are_openings In the_wingthat

Every

time _you

_pick-up

the

telephone

receiver to

_place

a call,

send a

_fax,

or modem Information

THE

MATH EMATI

C

OFATELEPHONI

CALI

—

youare

_entering

a

_phenomenally

complicated

andenormousnetwork. Thecommunication netthat encompasses the

globe

is

_amazing.

It is difficult to

_Imagine

how

many calls are fielded and directed each

_day

over this network.

How does a

_system

which is

_{"broken-up" by}

varied

_systems

of

different countries and bodies of water

_operate?

Howdoesa

_single

phone

call find its _waytosomeone in_your

_city,

state oranother

country?

In the

_early

_{years of the}

_telephone,

one

_picked-up

thereceiverand

cranked the

phone

to

_get

an

_operator.

Alocal

_operator

came on

the line from the local switch board and said "number

_please",

and _{from there connected you with the}

_party

_youwere

_trying

to

10 THEMAGICOFMATHEMATICS

methods usedto convertand direct calls. Mathematics

_Involving

sophisticated types

of linear

_programming,

_coupled

with

_binary

systems

andcodes,makesenseoutofa

_potentially

_precarious

situation.

40

3

^l^. 16'

_J976

/*U*X~W*A~*-How_{does your}voicetravel? Yourvoice

_produces

sounds whichare

converted Inthereceivertoelectrical

_{signals. Today}

these

electrical

_impulses

can

be carried and convertedIna

variety

of ways.

They

may be

changed

tolaser

light signals

whicharethen

carried

_along

fiber

_optics

cables1,

they

maybe

convertedto radio

_signals

and transmitted overradioor microwavelinks from towerto toweracross a

country,

or

_they

mayremainas electrical

_signals

along

the

_phone

lines. Most of

thecalls

connectedinthe USAaredone

_by

anautomatic

u^ft

N

***** kX**4-u4 u-fcwv^U^ -fri,tf^.c- 4L~-~i Tt-t. e^4r*y**f **7

4tt~ZE^ce.i

?k7\

lf~«Zio-t<.

—

£•-*-+.

fan*. —

J

_***~r- £* —

4,

Pages from the notebook ofAlexander Graham Bell in

which he writes

_of

hisfirst

telephone

messageon his

inventiontalkingtohis assistant,Mr.Watson. "Ithen shouted

intoMthe_following sentence:'Mr. Watson—Come here—I

want tosee_you.'Tb my

delight

hecameand declared that he hadheardandunderstoodwhat J said."

switching

system.

Presently

theelectronic

_{switching system}

isthe

fastest. Its

_system

has a _program which contains the needed

information for all

_aspects

of

_telephone

operationswhile

_keeping

track of which

_telephones

are

_being

used and which

_paths

are

available. Callscanbetransmitted

_by

electriccurrents at different

frequencies

orconvertedto

_{digital signals.}

Either method enables

multiple

conversations to be transmitted

_along

the same wires.

Themost modern

_systems

convertcalls Into

_{digital signals}

which

are thenencoded witha

_binary

_{number sequence. The Individual}

callscanthus travel

_{"simultaneously" along}

thelinesina

_specified

order until

_they

aredecoded for theirdestinations.

Whenacallis

_placed,

the

_system

chooses the best

_path

for the call

and sends a chain ofcommands to

_complete

the

_circuitry.

The

entire_{process takes} afraction ofasecond.

_Ideally

itwould takea

direct route to the other

_party—

thatwould be desirable from the

view

_point

of theeconomicsofdistance andtime. Butif thedirect lineisat

_{capacity servicing}

othercalls, thenew

callmustbesent

along

thebest of the alternative routes. Here iswhere linear

programming2

comesInto the

_picture.

Visualize the

telephone

routing

problem

as a

_complex

geometric

solid with millions of facets. Eachvertex

_represents

a

possible

solution. The

_challenge

istofindthebest solution without

12 THE MAGICOFMATHEMATICS

Danzig developed

the

simplex

method to find the solution to

complex

linear

_programming

_problems.

The

_simplex

method. In

essence, runs

_along

the

_edges

of the solid,

checking

one corner

afteranother,while

always

heading

for the best solution. As

_long

as the number of

_{possibilities}

is no more than 15,000to 20,000,

this method _manages to find the solution

_efficiently.

In 1984,

mathematician Narendra Karmarkar discovered a method that

drastically

cuts down the time needed to _{solve very cumbersome}

linear

_programming

_problems,

such as the best routes for

telephone

calls over

_long

distances. The Karmarkar

_algorithm

takes ashort-cut

_by

_going

_through

the middle of the solid. After

selecting

an

_arbitrary

interior

point,

the

_algorithm

warps the

entire structure sothatit

_reshapes

the

_problem

which

_brings

the

chosen

point

exactly

Into the center. The next

_step

is to find a

new

_point

In the direction of the best _{solution and to warp the} structure

_again,

and

_bring

thenew

_point

Into the center. Unless the

_warping

is done, the direction that appears to

_give

the best

improvement each time is an illusion. These

_repeated

transformations are based on

_concepts

of

_projective

_geometry

and lead

_rapidly

tothe best solution.

Today,

the old

_telephone

salutation "number

_please"

takes on a

double

_meaning.

The once

_simple

_{process of}

_picking

_{up your}

telephone

receiverand

_placing

a_call, nowsetsInto motionavast and

_complicated

network that reliesonmathematics.

^Depending

on the _type of lines used, the number of"simultaneous"

conversationscan_{range from 96}to over 13000. Fiber_{optic systems}can

carry even more information than the traditional

_{copper/aluminum}

cables.

^Linear

programming

techniques

areused tosolvea_varietyof

_problems.

Usually

the

_problems

entail many conditions and variables. A

_simple

case _{may be} an

_agricultural

_problem:

A

_farmer

wants to decide howto most _effectively use _his/her land to maximize _production and _profit Conditions and variables would involve such things as

_considering

different crops, how much land each croprequires, how much

_yield

each

produces

per acre, and how much revenue each _brings when sold. To

solve sucha

_problem,

onewriteslinear

_Inequalities

and/orequationsfor each condition and looksata_{2-dimensional}

_graph

ofa

_polygonal

_region

PARABOLK

When you flick the switch of your

R E F L E

CTO

R

S & YO UI

headlights

from

_bright

to dim,

_HEADLIOHT!

mathematicsisat work. Tobe

specific, the

_principles

ofa

_parabola

do

the trick. Thereflectorsbehind

the

headlights

are

_parabolic

In

_shape.

Infact,

they

are

paraboloids

(3-dimensional

_parabolas

formed

_by

rotating

a

parabola1

aboutits axisof

symmetry).

The

_bright

beam

iscreated

_by

a

_light

source

located atthe focal

_point

of the

_parabolic

re

flectors. Thus, the

_light

_{rays travel}out

_parallel

tothe

_parabola's

axisof

_symmetry.

When the

_lights

aredimmed, the

_light

source

changes

location. It isno

_longer

atthefocus, andas aresultthe

light

rays do not travel

_parallel

tothe axis. The low beams now

point

down and up. Those

pointing

upare _shielded, sothat

_only

thedownward lowbeamsarereflectedashorter distance than the

high

beams.

The

_parabola

is an ancientcurve that was

discovered

_by

Menaechmus (circa 375-325 B.C.)

while hewas

_trying

to

_duplicate

the cube. Over

the centuries, new uses and discoveries

involving

the

_parabola

have been made. For

example,

itwas Galileo

_(1564-1642)

who showed that a

_trajectile's

_path

was

_{parabolic. Today}

one can _go into a

hardwarestoreandfinda

_highly

_energyefficient

_parabolic

electric

heater whichuses

_only

1000wattsbut

_produces

thesamenumber

of BTU thermalunitsas aheater that

_operates

on 1500watts.

1

Parabolaisthe setof all_pointsina

_plane

whichare

_equidistant

froma

14 THE MAGIC OF MATHEMATICS

COMPLEXITY &

THE

PRESENT

"The hours

_from

seven 'til

_nearly

midnight

are

_normally

_quiet

ones on

the

_bridge.

...

Beginning

almost

exactly

at seven _o'clock, ...it

_just

looked as

_{if everybody}

in

Manhattan who ownedamotorcarhaddecidedtodriveouton

_Long

Island that

evening."

Asthis

_excerpt

from

TheLaw

_by

RobertM,

CoatesIllustrates,

sometimes

_things

just

seemtotake

place

withno

apparent

reason. Nor Isthere % a

_warning

thata

particular

event Is abouttotake

place.

Wehave all

_experienced

such events and

usually

attributed themto "coincidence", sincethere were no

apparent

indicatorsto

predict

otherwise

Complexity

Isan

_emerging

science_{which may hold} answers orat

least

_explanations

tosuch

_questions

as:

Howisitthat

•_theuniverse

_emerged

_{outof the void?}

•_{cellsknowwhich}

organsand

_parts

to become andwhen?

•on

_January

17, 1994 Los

_Angeles

suffered an

earthquake

of

_{unexpected magnitude}

anddestruction?

• _{theSoviet Union's}

long

reign

overits satellite countries

collapsed

insuchashorttime?

•

Yugoslavia

was thrown

_suddenly

into severe internal

wars? • _a

species

that has not

_changed

_{for millions of years}

suddenly

experiences

amutation? •_forno

apparent

reason the stock market_surges

_upward

or

_plunges

downward?

The listisendless. The

_underlying

common factor of these events

isthat each

_represents

a_very

_{complex system.}

A

_system

_governed

by

an enormous number and

_diversity

of factors, which are

delicately

balanced,

tittering

between

stability

and chaos. The

factors whichacton sucha

_system

are ever

_growing

and

changing. Consequently,

a

_{complex system}

is

_always

In astate of

potential chaos i.e. at the

edge of

chaos. There seems tobe a

continual

_tug

of war between order and chaos.

Spontaneous

self-organizing

dynamics

areanessential

_part

ofa

_{complex system.}

It isthemeans

_by

which the

_{system regains}

_{equilibrium by}

changing

and

_adapting

Itself to

_constantly

_changing

_factors/

circumstances.Those

_studying

thisnewsciencedrawon ahost of

mathematical and scientificideas, suchas chaos

_theory,

fractals,

probability,

artificial

_{intelligence, fuzzy}

_logic,

etc. Thesescientists

and mathematicians feel that

_today's

mathematics,

_along

with

other tools and

_high

tech innovations, are

_capable

of

_creating

a

complexity

framework thatcan

_{impact major aspects}

ofour

_global

16 THE MAGIC OFMATHEMATICS

MATHEMATICS

&

THE

CAMERA

Ever wonder about the

_f-stop

number ofacamera?Where did It

get

Its name? How Is It

determined? "f stands for the mathematical term

_factor.

The

brightness

of the

_photographic

_image

on film

_depends

on the

aperture

and focal

_length

of the lens.

Photographers

usewhatIs knownasthe f-number

system

to relate focal

length

and

aperture.

The

f-stop

Is calculated

_by

measuring

the diameter of the

aperture

and

dividing

It into the focal

length

of the lens. For

example,

f4= 80mm

lens/20mm aperture.

fl6=80mm

lens/5mm aperture.

Weseethelens

_opening

issmaller

(the aperture

decreases) asthe

f-stop

number increases.

_Working

with

_f-stop

numbers and

shutter

speeds,

you can

_manually

decide how much of the

Here mathematicalunitsand

symbols

wereusedto

_get

the

_point

acrossabout

_recycling

_paper!

RECYCLINC

THE

NUMBER!

•Aton

of

_virgin

_paper*aton

of

_recycled

_paper

•A ton

of

_recycled

_paperuses4102

kwh less

_energy.

•A ton

of

_recycled

_paperuses7000

_gallons

less

water to

produce.

•Aton

of

_recycled

paper

produces

60

_pounds

less air

pollution.

•A ton

of

_recycled

_paper

_produces

3

cubic

_yards

less solid

waste.

•A ton

of

_recycled

_paperuses

less

tax

_money

for landfill.

•A ton

of

_recycled

_paperuses17

fewer

_logged

trees.

—the

numbers

behind

_recycling

and landfill—

•_37%

_{of all landfill is}

comprised

of

_paper.

•

Only

29%

of

_{all newspapers}

_produced

are

_{recycled by}

the

consumer.

•_{165 million}

_cubic

_yards

_{of landfill}

_are

_needed

_forour

paper

wastes per year.

•97%

of the

_virgin

forests of the continental

USA

have been

cutdown in

the

past

200 years.

IWasOnceA Tree...Newsletter, Spring 1990, AlonzoPrinting,HaywardCA.

18 THE MAGICOFMATHEMATICS

BICYCLES,

POOL

TABLES

&

ELLIPSES

The

_ellipse,

_along

with other conic section curves,was studied

_by

the

Greeksas

_early

asthe 3rd

_century

If this ballis hit

_through

the

location of the

_focus,

marked

with an _X, it will bounce off

the cushion and go to the

other focus where the

_pocket

is located.

B.C.. Most of

usassociate the

_ellipse

with

an

_angled

circleorthe orbital

_path

of a

_planet,

but

elliptical

shapes

and

properties

also lend themselvesto

An_ellipsehastwo_foci,and thesum_ofthe distances

fromthe

foci

to anypointoftheellipse

always equals

the

length

ofits_majoraxis.le.

_\PF1\

+

_\PF2\.=

\AB\.

contemporary

nonscientific

_{applications.}

Who would have

imagined

thatan

_ellipse

would find itselfin the

_design

of

_bicycle

gears and

_pool

tables?

_Today

some

_bicycles

have been

manufacturedwith afront

_elliptical

_gear and circular rear_gears.

The

_drawing,

onthe_{previous page,} illustrates how this

_design

can

utilize thedownward thrust of

leg

power anda

_{quick upward}

return.

_{Elliptipools, elliptical shaped pool}

tables,

are

_designed

to

utilize the reflection

_property

of the

ellipse's

twofoci.Asillustrated

onthe_{previous page,} the

_elliptipool

hasone

_pocket

locatedatone

of the two focus

points

of the

_ellipse.

Aball hit sothat it passes

through

the

_ellipse's

_non-pocket

focus will bounce offthe side of

the table andtravel the reflected

path

overtothe

pocket

(the

other

20 THE MAGIC OF MATHEMATICS

LOOKOUT FOR

TESSELLATIONS

u „

This Escher-like transformation

_by

MarkSlmonsonIllustrates theuse

of tessellationsas aform of visual

communication. This

_graphic

appeared

In The Utne Reader and on the cover of Transactions, a

Metropolitan

Transportation

Communication

_publication.

Reprinted

courtesy

of Mark Slmonson.

_{Bluesky Graphics,}

STAMPING

OU1

MATHEMATICS

One

_usually

doesn't

expect

to encountermathematical Ideasona

triptothe

_post

office, buthereare a

few of the _stamps that have been

printed

with mathematical themes. _{These and many other Ideas} have

_appeared

on such

_popular

Items as

_posters,

television, T-shirts,

post-Its,

mugs,

bumper

stickers, andstickers.

7*^**^V

US10FORMULAS MAT£MATKAS ODE CAMBIABOHIAFU DE LA TIERS* The_Pythagoreantheorem—Nicaragua

The

_Pythagorean

theorem —Greece

Bolyai—Rumania

Gauss—-Germany

Mathematical Formulas —Israel

22 THE MAGIC OFMATHEMATICS

THE

MOUSE'S TALE

'Fury

said to a _{mouse, That} he met in the

house,

' Letus both go toInw. / will proseouto you.— Come,I'11 takeno denial; "Wemust hnvca Saidtho mousoto tha our, 'Sucha trial, dflursll', With111) Jjrjror Judge, wuulil Uo WMllna-uururdKtll. 1I'II ll« lil<JI«, IMIb* CMnolaf Mrnrri U»»l»fe trial: For roally this morning 1 'vo nothing todo.'

'Contrariwise,'

continued

Twee-dledee.

"if itwasso,it

_might

be;

and if itwereso,it would be: but asit isn

't,

it ain 't, That's

_logic.'

—Lewis

Carrol

Alice inWonderland

Charles

_Lutwidge

_Dodgson

(1832-1898)

was a mathematician, lecturer of mathematics, creator of

_puzzles

and _games, renowned Victorian

children's

_photographer

and author of

mathematics

books1,

children's

stories,

_poetry

and essays on social

issues. When

_writing

children's tales his

_pseudonym

wasLewis Carroll. It

seems that

_Dodgson

didnot want to

be associated or connected with his

Lewis Carroll

_identity.

In fact, he

wouldreturnmail addressedtoLewis

Carroll. When the Bodleian

_Library

cross-referenced

_Dodgson

and

Carroll,

Dodgson

took

_exception

to the connection.

Dodgson

had a

_passion

for word

games. In 1991, two

_teenagers

from New

_Jersey

discovereda

_four-way

_pun

(both

visual and verbalin_nature)inthe poem,A Caucus-Raceand

a

_Long

Tale from

_Chapter

TwoinAliceinWonderland. Aliceistold

thetale

_by

a mouseina_poem

_shaped

likea

_long

tail. Inaddition

tothis tale and taillink, students

_Gary

Graham and

_Jeffrey

Maiden_{discovered that when the poem}waswritten in stanzaform the

'wttftktw.te

hmxi

_^tfy^Hiiln

*$ipf$4

'tfkljJmf,

shir.

^2-mtoptipi,

Vti

fa

jury*

• '""

'

_*kk

wtipte

tm$4?'«tiA

condemn

_$mt£&&

(the

mouse's

_body)

anda

_long

third line

(the

mouse'stall).

Lastly,

they

found thata

tall-rhyme

Isa

_poetic

structure defined

_by

a

_pair

of

_rhyming

lines followed

_by

another line of different

_length.

Do

you thinkLewisCarroll

planned

all this

intentionally?

1_{Euclid and His ModemRivals,}An_Elementary TreatiseonDeterminants, AliceinWonderland, TheHuntingoftheSnark,Phantasmagoriaand Other Poems,ThroughtheLookingGlass areofa few of_Dodgsonworks.

24 THEMAGIC OFMATHEMATICS

A

MATHEMATICAL

VISIT

Not

_quite

surewhatto

_expect,

I_rang

the doorbell A voiceaskedmeto

please

push

the

_firstfive

terms

_of

the Fibonacci sequence.

Fortunately,

Ihad donesome

research

_after

_my

_magazine

_assigned

methe

_story

onthe home

_of

the renownedmathematician,Selath.

I

_pushed

1,1,2,3,5and the door

_slowly

_opened.

As I

_passed

through

the

_doorway,

Iwasstruck

by

the

catenary

stone

shaped

archway

independently suspended

atthe entrance.

_After

a

minute,Selath entered

_saying,

_"May

I

_offer

_you

_something

_after

your

long

drive?'

"TdreaRy

appreciate

a

glass of

cold water,"I

replied.

"Pleasecome withme,"he said,

leading

the way. As I

_followed,

Icouldn't

help

noticingthe manyuniqueand unusual

_objects.

Inthe

kitchen,we cametoa

_peculiar

_{table with many}

_legs.

Selath

_pulled

an

_equally

unusualbottle

_from

the

_{refrigerator.}

I musthave hada

quizzical

expression, for

Selath

began,

"While you drtnk yourwater,

we

_might

aswell start thetourhereinthe kitchen."As_{you noticed}

tangramtables

_for

_dining

becauseitsseven_componentscanbe

rearranged

intoas_many

_shapes

asthe

_tangram

_puzzle.

Hereinthe

kitchenitsmadeintoa_square

_shape

_today,

whileTve

_arranged

the

oneinthe

_living

roomintoa

_triangle,

since Iam

_expecting

two

_guests

for

dinner. Thewater container iswhats knownas aKlein

bottle—itsinside and outsideare one.

_If

you lookatthe

_{floor you'll}

notice

_only

two

shapes

of

tilesareused." "Yes,"I

_replied,

"

but the

_design

doesn't

seemto

_repeat

_anywhere."

"Very perceptive."

Selath seemed

_pleased

with my response. "ThesearePenrose

tiles.Thesetwo

_shapes

can

cover a

_plane

ina

_{non-repeating}

fashion."

"Please continue,"I

_urged.

"

Tm most anxious toseeall the

mathematical

_parts

_of

your

home."

"Well,

actually

almost all

_of

my

houseismathematical.

Anywhere

yousee

_wallpaper

Tve

_{designed special}

tessellation

_patternsfor

wallsalaEscher. Lets

_proceed

tothe

_Op

room.

_Every

item inhereisan

_optical

illusion. In

_fact,

_reality

inthis

roomisanillusion. Furniture,

fvdures, photos, everything!

For

example,

the couchismade

_from

modulo cubesinblackand white

fabric

stackedto_givethe

feeling of

an

_oscillating

illusion. The

sculpture

inthe middlewas

_designed

to_{show convergence and}

26 THEMACIC OF MATHEMATICS

This

_lamp's

base,

viewedfrom

this

location,makes the

_impossible

tribar."

"Fascinating!

Icould

_spend

hours

discovering

things

inthisroom,"I

replied

enthusiastically.

"Sincewe are on a

_tight

schedule,lets

movetothenextroom," Selath saidashe

led the way.

Weenteredadarkenedroom.

"Watch yourstephere. Come

this waytothe

parabolic

screen,"Selathdirected.

As I

_peered

intothe disca

movingscene

_appeared.

"Is

thisavideo camera?"Iasked.

"Oh,

no,"Selathlaughed,

"Icall it_myantiquesurveillance

system.

The lens above the hole

captures

light

inthe

day

timeand rotatesto

_project

scenes_{outside myhome,}much thesame_waya camerawould. It iscalleda cameraobscura. Ihavea

_special

lens

for night viewing."

Iwas

_{busily taking}

notes,

realizing

Iwould havemuch additional

research todo

_before

_writing

_{my article.}

Glancing

aroundInoted,

_{'Yourfluorescent}

clockseemstobe

_off."

My

watch read 5:30 pm while his read21:30.

eight

becauseTm

_working

on

_eight

hour

_cycles

this week. So 24:00

hows would be 30:00hours,8:00 would be10:00,andsoon."

Selath

_explained.

"Whatever works best

_for

_you,"

I

_replied,

abit

_confused.

"Now,let's go to the masterbedroom."

And

_off

wewent,

_passing

allsorts

_ofshapes

and

_objects

rdnever seeninahome

_before.

28 THE MAGIC OF MATHEMATICS

movable

_{geodesic skylights. They}

are

_designed

to

_optimize

theuse

of

solar

_energy."

"Marvelous,but whereisthe bed?'Iasked.

"Just

_push

the buttononthis wooden

cube,

and you willsee abed

unfold

withahead board andtwoend tables."

"Whata

_great

_waytomakeabed,"I

_replied.

"Thereare_manymore

_things

tosee,buttime isshort. Lets gointhe

bathroomsoyoucan seethemirrorsoverthe basin. Comethis way.

Nowlean

_forward."

To_my

_surprise

Isawan

_infinite

number

_of

images

ofmyself

repeated.

Themirrorswere

_reflecting

backand

_forth

intoone

another ad

infinitum.

"Now turnaround andnoticethismirror. Whats

different

about it?"

Selathasked.

"My part

ison_{the wrong}side,"I

_replied.

'Tothe_contrarythis

mirror1

lets yousee

_yourself

as_youare

_really

seen

_by

_others,"Selath

_explained.

Just then thedoorbell rang. The dinner

guests

hadarrived.

_"Why

don't you

stay

todinner?' Selath asked. 'Youhaven'tseenthe

living

room

_yet,

andFmsure

_you'll

_{enjoy meeting}

_my

_guests."

Itwashardtoconceal

_by

enthusiasm."Butyour tableis set

_for

three," Iblurted.

"No

_problem.

With the

_tangram

tableIcan

_just

_rearrangea

_few

parts

and we'llhavea

_rectangle.

1Made

fromtwo mirrors

_placed

at

_{right angles}

toeach other. The

right-angled

mirrorsarethen

_positioned

sothat

_they

_{will reflect your} reflection.

THE

EQUATION

OFTIME

If you have ever used a

sundial, you may have

noticed that the time

registered

on the sundial

differed

_slightly

from that

on_{your watch.} This

differenceIs tied Into the

_length

of

daylight

during

the year. In the 15th

_century,

Johannes

Kepler

formulated

three laws that

_governed

planetary

motion.

_Kepler

described how the Earth travels

around theSun Inan

_elliptical

orbit, and also

_explained

that the line

_{segment joining}

theSunand

the Earthsweeps out

equal

areas

(sectors)

In

_equal

Intervals of time

_along

its orbit. The Sun Is

located at one of the foci of the

ellipse

thereby

making

each

sector's area

_equal

for a fixed

timeInterval and the arc

_lengths

of thesectors

_unequal.

Thus the

Earth's orbit

speed

varies

_along

its

_path.

This accounts for the

variations in the

_lengths

of

, „ , , , , ,,„. , ,, ,A 10th century pocket sundial

daylight

during

differenttimesof _^^

msix^^ listedoneach

year. Sundials

rely

on

_daylight,

side-Astlckteplacedin.the holeof

30 THEMAGIC OF MATHEMATICS

and

_daylight

_depends

on the time _{of the year and}

_geographic

location.Onthe otherhand,thetimeintervals ofourotherclocks

are consistent. The difference between asundial's time and an

ordinary

clockis referred toas the

equation

of

time. Thealmanac

lists the

_equation

_of

time

chart,

which indicates how manyminutes

fastorslow the sundialisfrom the

_regular

clocks. For

_example,

the chart may look like theonebelow.

Equation

of Time Chart

(The

negative

and

_positive

numbers indicate the minutes the sundial

is

slower or faster than an

_ordinary

clock.

_Naturally

the

table

does not

take into consideration

_daylight

difference

within time

_zones.)

DATE Jan Feb Mar

April

May

VARIATION 1 -3 15 -9 1 -13 15 -14 1 -3 15 -9 1 -4 15 0 1 +3 15 +4

// thesundialshows 11:50on

May

15, its time should be

WHY

ARE

MANHOLES

ROUND?

Why

Isthe

_shape

ofamanhole circular?

Why

nota_square,

_{rectangular, hexagonal,}

or

_elliptical

shape?

Is itbecauseacircle's

_shape

Ismore

_pleasing?

ThereIsamathematicalreason.

MetamorphosisW_byM.C. Esclwr.

MATHEMATICAL

WORLDS

HOW

MATHEMATICAL

WORLDS

ARE FORMED

GEOMETRIC

WORLDS

NUMBER WORLDS

THE WORLDS

OF

DIMENSIONS

THE WORLDS

OF

INFINITIES

FRACTAL WORLDS

MATHEMATICAL WORLDS

IN

LITERATURE

34 THE MAGIC OF MATHEMATICS

Howcanit be that

mathematics,

a

_product

ofhuman

thought independent

of

experience,

isso

admirably

adapted

to the

_objects

of

reality.

—Albert Einstein

Mathematics

islinked and used

_by

so_many

_things

inourworld,

yet

delves in its own worlds—worlds so

_strange,

so

_perfect,

so

totally

alien to

_things

of our world. A

_complete

mathematical

worldcanexiston the

_{pin point}

ofaneedle orinthe infinitesetof

numbers. Onefinds such worlds

_composed

of

_{points, equations,}

curves, knots, fractals,

and so on. Until one

understands how

mathematicalworlds and

_systems

areformed,someofits

worlds mayseem

contradictory.

For

_example,

one

might

ask howaninfinite

worldcanexist

_only

on a

tiny

line

_segment,

or a

world be created

_using

%

only

three

_points/This

chapter

seeks to

_explore

the

_magic

ofsomeof these

mathematical worldsand _

—£ delve into their domains.

XT

Asdiscussedlater, the_countingnumbers

Why,

sometimes I've believed

as_manyassix

_{impossible things}

before

breakfast.

—Lewis

Carroll

HOW

MATHEMATICAL

WORLDS

ARE

FORMED

Little didEuclid knowin300 B.C.whenhe

_began

to

_organize

geometric ideasinto a mathematical

_system

that hewas

_developing

the firstmathematicalworld. Mathematical worlds and their

elements abound — here _we find the world of arithmetic with its

elements thenumbers,worlds of

_algebra

withvariables, the world

of Euclidean

_geometry

with squares and

triangles,

topology

with

such

_objects

as the Mobius

_strip

and networks, fractals with

objects

that

_{continually change}

— all _are

Independent

worlds

_yet

are interrelated with one another. All form the universe of

mathematics. A universethatcanexistwithout

_anything

fromour

universe,

yet

a universe that describes and

_{explains things}

all

aroundus.

Every

mathematicalworld exists in a mathematical

_system.

The

system

setsthe

_ground

rulesfor the existenceof the

_objects

inits

world. It

_explains

how its

_objects

are

formed,

how

_they

_generate

new

_objects,

andhow

_they

are

_governed.

Amathematical

_system

is

_composed

of basicelements,whicharecalled

_undefined

terms.

These terms can be described, so that one has a

_feeling

of what

they

mean,but

_technically

_they

cannotbe defined.

_Why?

Because

ittakestermstoform

definitions,

and you haveto

_begin

withsome

terms. For these

_beginning

words there are no other terms that existwhichcanbe usedtodefine them.

Thebest_waytounderstand sucha

_system

is tolookatone.Here's

how a finite mini mathematical world

_might

take form. Assume

thisminiworld's undefinedtermsare

_points

and lines. Inaddition

to undefined terms, a mathematical

_system

also has axioms,

36 TH E MAC IC O F MATH EMATICS

we

_accept

as

_being

truewithout

_{proof. Definitions}

are newterms

we

_{describe/define using}

undefined terms or

_previously

defined terms.TheoremsareIdeas whichmust_{be proven}

_by

_using

_existing

axioms, definitionsortheorems.

What

_type

ofdefinitions, theoremsandaxiomscan ourminiworld

have?Herearesomethat

_might

evolve—

Undefined

terms:Pointsand lines.

Definition

1:A set

_of

_points

iscolltnear

_if

a linecontains the set.

Definition

2:A set

_of

_points

isnoncolltnear

if

aline cannot

containtheset.

Axiom 1: Our miniworld contains

_only

3 distinct

_points,

which do notlieon aline.

Axiom 2:

_Any

twodistinct

_points

makea

line.

Theorem 1:

_Only

three distinct linescan

exists In this

world.

proof:

Axiom1 statesthatthere

are3distinctpointsinthis

world._UsingAxiom2we

knowthateverypair of

these_pointsdeterminesa

line. Hencethreelinesare

formed by

the threepoints

ofthis world.

This

example

Illustrates howamathematical world

_might

evolve.

As new Ideas come to mind, one adds more undefined terms,

axioms,definitions,and theorems and

_{thereby expands}

the world.

The

_following

_{sections introduce you}tosomemathematical worlds

GEOMETRIC

...

The universe stands

WO RLDS

continually

open toour_gaze,

but it cannot

be

understood

unless

one

first learns to

comprehend

the

_language

and

_interpret

the characters in

which it

is

written. It

is

written

in

the

_language

of

mathematics,

and its characters

_{are...geometric figures,}

without which it is

humanly impossible

to

understand

a

single

word of

it; without these,

oneis

_wandering

about in a

dark

_{labyrinth.—Galileo}

Mathematics has many

types

of

geometries.

These include Euclidean and

_analytic

geometries

andahost of non-Euclidean

geometries.

Herewe find

_hyperbolic,

elliptic,

projective,

topological,

fractal

geometries.

Each

geometry

formsa mathematical

_system

with itsown

undefinedterms,

axioms, theorems anddefinitions.

Although

these

geometric

worlds may

usethesame names

This is an abstract

_{design of}

Henri Poincare's

(1854-1912)hyperbolicworld.Hereacircleisthe

boundary of this world. The sizes _ofthe inhabitants _change in relation to their distance

from the center. As _{they approach} the center

they

grow, andas

_they

move_awayfromthe

center_they shrink. Thus _they will neverreach the

boundary, and_forallpurposes, their worldis

38 THEMAC\COF MATHEMATICS

for their elements or

_properties,

_{their elements possess different}

characteristics. For

_example.

In Euclidean

_geometry

lines are

straight

andtwodistinct linescaneither intersectInone

_point,

be

parallel,

orbeskew. ButlinesIn

_{elliptic geometry}

arenot

_straight

lines but

_great

circles ofa

_sphere,

and therefore_anytwoofIts

distinctlines

_always

Intersectin two

_points.

Consider the word

_parallel.

In

Euclidean

_geometry

parallel

lines

are

_{always equidistant}

andnever

Intersect. Notsoin

_elliptic

or

hyperbolic

geometry.

Why?

Because every

great

circleofa

sphere

Intersectsanother.Thus,

elliptic

geometry

hasno

_parallel

lines. In

_hyperbolic

_geometry

_parallel

linesneverIntersect,but

_they

donot

resembleEuclidean lines.

_Hyperbolic

parallel

lines

_continually

comecloserand

closer

_together,

_yet

neverIntersect.

_They

are called

_asymptotic.

Euclidean,

hyperbolic,

and

_elliptic

geometries create three

_dramatically

different worlds with lines and

points,

etc., but whose

_properties

are universes

_apart.

Each of

these worldsIs amathematical

_system

unto Itself, and each has

applications

inouruniverse.

Theabove

diagram

showstwogreat circles,

line1 and2_intersecting

at_pointsA&B.

L^-

_¦}

In

_hyperbolic

_geometry, linesMand Nareboth_parallelto

line L and passthrough pointP. Mand Nare_asymptotic

NUMBER

WORLD!

Numbers can be considered the

first elements of mathematics.

Their

_{early symbols}

were

_probably

marks drawn in the earth to

indicate a number of

_things.

But ever since mathematicians

entered the scene the

_simple

world of

_counting

numbers has

never been the same.

_Many

_people

are familiar with

_integers,

Stone_Agenumberpatternsfoundtn LaPdeta,Spain.

fractions and decimals, and use these for their

_daily

computations.

But number worlds also include the rational and

irrational numbers, the

_complex

numbers, the

_never-ending

non-repeating

decimals, transcendental numbers, transfinite

numbers, and many many subsets of numbers thatarelinked

_by

specific properties,

suchas

_perfect

_{numbers whose proper factors}

total the number, or

_polygonal

numbers whose

_shapes

are

40 THEMACIC OF MATHEMATICS

Interesting

todelveIntothe

_{interrelationship}

ofnumbers, surmise how

_{they developed,}

and

_explore

theirvarious

_properties.

The

_counting

numbers date back to

_prehistoric

times. Consider the

_simple

marks of the Stone

_Age

number

patterns

fromLaPileta

Cave In southern

_Spain,

whichwas Inhabited over25,000 years

ago until the Bronze

_{Age (1500 B.C.).}

The numbern was known

over three thousand _{years ago,} when It was used in the

calculations ofacircle'sareaand

circumference,

and later shown

to be irrational and transcendental. Ancient civilizations were

aware that fractional

_quantities

existed. The

_Egyptians

used the

glyph

formouth,

O

,towritetheirfractions.

For

_example,

^^ was

_{1/3, ^^}

was

_1/10.

Irrational numbers were known

_by

the ancientmathematicians,

who devised

_fascinating

methods for

_{approximating}

their values.

Infact, theGreeks

_developed

the ladder method to

_approxlmatle

the

_|/2

while the

_Babylonians

used another method.

0=

₁₌

10=

_H=

100=

101

=

two

two two two two two

0 12 3 4 5

Hexagramsand their

_binary

_eqiduaLents.

Over the centuries different civilizations

_{developed symbols}

and

counting systems

fornumbers,andInthe 20th

_century

the

_binary

numbers and base twohave been

_put

towork with the

_computer

revolution. Gottfried Wilhelm Leibniz

(1646-1716)

first wrote

about the

_{binary system}

In his paper De

Progressione

Dyadica

(1679).

He

_corresponded

with Pere Joachim Bouvet, a Jesuit

missionary

inChina. Itwas

_through

Bouvetthat Leibniz learned that the I

_{Chlng hexagrams}

were connected to his

_binary

broken line and 1for the unbrokenline,the

_hexagrams

illustrated the

_binary

numbers. Centuries _prior to _this, the

_Babylonians

developed

and

_Improved

_{upon the Sumerian}

_sexagimal

_system

to

develop

a base 60 number

_system.

But this section on number

worldsIsnotabout number

_systems

butabout

_types

of numbers.

Let's take a

_glimpse

atthe first

_type

of numbers —the

counting

numbers.Inthe world of

_counting

numberswefind the undefined

termsare the numbers 1,2, 3 — with suchaxioms

asthe

orderin which two

_counting

numbersareadded doesnot

_affect

the

sum (a+b=b+a, called thecommutative

_property

_for

addition); the

orderin whichtwo

counting

numbersare

_{multiplied together}

does

not

_affect

the

_product

_(axb=bxa,

called the commutative

_property

_for

multiplication).

— andsuch theorems

asAnevennumber

_plus

an

evennumberis alsoan evennumber.And,Thesum

_of

anytwoodd

numbers is

_always

an evennumber. But the world of

_counting

numbers werenot

_enough

to solve all the

_problems

thatwereto

evolveoverthe years. Can_you

_Imagine

_tackling

a

_problem

whose

solutionwasthe valuexfor the

_equation

x+5=3 andnot

_knowing

about

_negative

numbers? What would havebeen some reactions

— the

problem

Is defective, there is no answer. Arabtexts

Introduced

_negative

numbersIn

_Europe,

but most mathematicians of

the the 16th and 17th centurieswere not

_willing

to

_accept

these

numbers. Nicholas

_Chuquet

(15th

century)

and Michael Stidel

(16th

century)

referred to

_negative

numbers as absurd.

_Although

Jerome Cardan _{(1501-1576) gave}

_negative

numbers as solutions

to

_equations,

he considered them as

_Impossible

answers. Even

Blaise Pascal said "Ihave known those who couldnotunderstand