Life

on

the

Infin

ite F

arm

Rich

ard

Eva

n S

chw

artz

2

To all my math teachers.

3

When I looked at the pages that mentionatoms and special

relativity, I wondered ifI had gone too far.

3

But then I went alittle crazy and putin all kinds of fancyideas that I found

interesting.

2

When I started writing this book, I planned to

make it simple enough so that anyone old enough to read could enjoy it.

1

4

I ended up making botha simple version and

an advanced version, but I couldn't decide

which one to show you.

Note to the reader

Then I had an idea about how to show

you both versions at the same time.

5

If you arefeelingadventurousand want theadvancedversion, then just read it all.

8

This book has all thepages, simple and

advanced, but I putlittle black markersin the corners of allthe advanced ones.

6

If you want to read the simple version, then skip the pages with the markers. Thebook will still make sense.

7

4

ho are the animalson the infinite farm?

W

5

Gracie is one of the animals. She is an infinite cow.

21

3 4

6

65 7 8

Gracieloves shoes.She has one on every foot ...

7

and her feetgo on forever!

10

1817

1112

1314

1516

1920

910

8

while Simon, an infinite donkey, chats with Gerry,...

Flambeau, aninfinite sheep,mulls overshearing season

9

an infinite gopher.

10

Ezekiel is a goat withan infinite horn, but he thinks of himselfas mostly a zebra.

11

Hammerwood is an infinite crocodile. Let'szoom out to see him better.

12

Look at those teeth!

They go on and on. And so do the bubbles.

13

Hammerwood loves bubble gum,even though it is hard for him tochew it allat once.

14

Penn also lives on the farm. Heis an infinite chicken. On the infinitefarm, chickens have teeth too.

15

Penn hasstaring contestswith his sisterVictoria, and also with...

16

Bill, an owl with aninfinite eye.

17

Delores, an infinitesquid, lives in the farm's pond.

18

Her tentacles branch outendlessly, and she decorates themwith jewelry. She is crazy about ringsand bracelets. She likes to swim onSaturday nights with her friend...

19

Nelson, a shark whose head extends forever in both directions.

20

The farm has lots of other animals too. Some of them are so strange I don't know what to call them.Maybe YOU can think of names for them.

21

o the animals have anyproblems on the farm?

D

22

3

Gracie hasa problem

with shoes.Her friends know that she loves shoes, so

they bringher newones as

presents.

Oh yes, lots of problems!

23

She wants to wear the NEW shoes ...

But she doesn't want to stopwearing any of her OLD shoes.

24

She already has a shoe on every foot, so what

is she going to do?

25

Hammerwood has problems with his teeth.

He has so much funblowing bubbles ...

26

...that he often forgets to brush his teeth.

To be honest, bad thingssometimes happen to Hammerwood's teeth.

It isn't pretty up close.It can get so bad that ...

27

he loses most of his teeth.

He needs those teeth for chewing

and he can't grow new ones.

How can he fix his teeth?

28

Delores has a problemwith her jewelry. Every

week her cousin Bin swims over and asks to

borrow some of it.

29

Bin wants todecorate all his

tentacles for hisSaturday night dances...

30

but Delores wants to decorate all of HER tentacles for her Saturday night

swims with Nelson. How can they split the jewelry

so that each has enough forall of theirtentacles?

31

ere is how the animalssolve their problems...

H

32

Gracie prepares to put on hernew shoes by standing behindthem on an infinitely longsticky strip.

33

She gives each left leg a little shake ...

then steps out of each left shoe ...

34

and into the onein front of it.

35

Then she doesthe same thingwith all herright feet.

36

Gracie doesn't have a back end!

Now she wears the newshoes AND all of her old shoes. There aren't any shoes left overat the back end because...

37

Gracie moves off thesticky tape and then...

she is ready for another present.

38

Hammerwood snaps intoaction once he loses

about 9 out of 10 teeth.

39

The first thing he does is brush his remaining teeth.

40

Then his friends from the infinite ant colony ...

put powerful rubberbands around his teeth.

41

For technical reasons, they anchor the first two bands to a stick.

At night, the rubber bands contract ...

42

and slide his teeth back into place.

43

The ants carry off the rubberbands and then his teeth are as good as new.

44

Delores agrees to lendBin some jewelry.

Their tentacles branch out in thesame way, so on Saturday afternoon...

they line them up like two

infinite handspressingtogether.

45

Little fish come andunclasp the jewelry.

The fishmove each piece one step towards Delores' head.

46

and isreplaced byTWO piecesthat comefrom behind it!

Each piece of jewelrymoves inward...

47

It becomes hertiara for the evening.

Something fancier happensto the piece of jewelry aroundDelores' neck. A fish unwinds itand moves it to her forehead.

48

The fishmoveone pieceof eachpairover toBin...

and then

swim away.

49

Bin thanksDelores for being so amenable and then the cousins swim off to their

Saturday night fun.

And now Bin has it too.

Except forthe tiara,

Delores hasthe samepattern ofjewelry as inthe beginning.

50

ny questions? A

51

Easy. OnSunday

morningthey line up

again andreverse the

whole process.

How does Delores gether jewelry back from Bin?

52

Where does Hammerwoodget his gum?

ALICE'S GUM SHOP

Creditcards

accepted.

53

Simple. He points his mouth up aninfinite branchof a gum treeand takes whathe wants.

54

Can Gracie swing all theway around, like thehand on a clock?

55

No. She would get stuck onthe trees.

But otherwise she is quite

flexible.

56

??

How do theanimals get around each other?

57

Gracie walks underneath

Penn and theyhardly notice each other.

That is a very interestingquestion. For one thing,

the animals oftenhave very different sizes.

58

Also, the animals aresurprisingly acrobatic. Gracie's cousin Boopis

can roll right over her.

59

Some of the animals have special tricks. Nelson's head can swing

open like a door in some places...

60

and Delores is built like a flexible puzzle.She can pull herself apartto let anotheranimal through.

61

62

1 2 48

1632

641282565121024204840968192163843276865536

63

How do the infiniteanimals fit in the pond?

64

?Like this! The space insidethe pond is warped in a waythat makes it infinite from thethe inside but finite from theoutside. To us outsiders, Delores' spots appear to shrink as they approach the edge, but they are really all the same size. The same goes for Nelson's teeth and eyes.

65

GF L

B

Flambeau and his herdlive in a crater thatis like the pond,except thatthey walk on thebottom.

From the outside,

Flambeau'sfriends Bolyai,

Gauss, and Lobachevskyseem to be different sizes, but

they are really all the same size.

66

Bill's eye is also one of these enclosed infinite spaces. Heloves watching Flambeau andthe other sheep. He can seetheir whole crater perfectly.

67

Craters could have their own craters...

68

And so on.

69

Can the infinitefarm fit on Earth?

?

70

?

The infinite farm doesn't close up like Earth, so how could it fit on Earth? It seems impossible.

71

But, think back to how the pondand those craters take up a finite

amount of space from the outsider's point of view.

But think back to how the pond andthose craters only take up a finiteamount of space from the outside.

72

Oh, I see. It isn't such a stretch to imagine that the farm itself is one of those craters,and that the crater is on Earth.

73

Yes,precisely,but there isa catch. OnEarth thephysicallawswouldpreventone ofthosecratersfrom existing. So, the infinite farmcan't REALLYfit on Earth.

*

*A crater like this seems to requirematter to exist on all scales, whichis incompatible with the atomic natureof our world. Even setting this aside,the crater might generate a gravitational field that would crushour entire universe.

74

How doesday changeto night onthe farm?

Yeah, howdoes thesun getaround

the farm?

75

You could imagine that there is an infinite grid

of suns and moons hovering over the farm...

and that these celestial lights change their

color and brightnessto suit the time of day...

76

or you couldimagine thatthe suns and moons dancearound the farm in a complicated, multi-dimensionalpattern...

77

I like to think thatlight appears

on the farm withoutexplanation or cause,

like a new idea.

So, if there are nofurther

questions...

or you could imagine thatthere are no suns and moons,

by an infiniteflock ofglowingbirds.

andinsteadthat light isbroughteachday

78

How didGracie get on

the farm?

How canEzekiel's

neck supporthis infinite horn?

Would the farmhave infiniteatoms and/or

infinite biomolecules?

Since most of his teeth wouldhave to move faster than the

speed of light, isn't Hammerwood's teeth-fixingprocess incompatible withthe constraints of special

relativity?

79

PHYSICSCHEMISTRYBIOLOGY

OK, OK, slow down! Let me tryto answer all these questions at

the same time. In our universe thesciences are relentless enforcers.

They seem to look into every situation, no matter how large or

small, and make sure that it unfolds according to the law...

80

m2c cc

UAAG

On the infinite farm,the sciences are more like negligent babysitters who check in from timeto time but mostlyjust play around.

Meanwhile, out on the farm,...

UAG

81

GEOMETRYmakestherules.

Well, I guess there's time for one

last question.

82

Can we VISITthe infinite farm?

SCHOOL BUS

83

It depends on what you mean by "visit".

You won't find the infinite farm on Earth...

84

THE

CAT IN

THE HAT

COMES

BACK

GEOMETRY AND THE IMAGINATION

Hyperbolic Geometry

RE

AL

LY

BIG

NU

MB

ER

S

Hilbert H

otel

Proceedings ofthe Flat Earth soc.

Non

Am

enable

GroupsZENO

Dot

s and

You

METRIC

SPACES

OF

NON

POSITIVE

CURVATURE

but if you read betweenthe lines of geometrybooks...

you might find it there.

GA

LL

ER

Y O

F

TH

E IN

FIN

ITE

THE GEOMETRYAND TOPOLOGYOF 3-MANIFOLDS

GE

OM

ET

RIC

GR

OU

P

TH

EO

RY

1978

85

About the author: I am the Chancellor's Professor of Mathematics at Brown University. In my spare time I like to draw, write computer

programs, listen to music, walk on the beach, cycle, or go to the gym.

About the illustrations: I drew the pictures for thisbook using Inkscape, a great drawing program.

Support: Even though this book is not part of my math research, the research support I got duringthe time I wrote it gave me extra freedom to work

on it. I thank the National Science Foundation, the Simons Foundation, and the Isaac Newton

Institute for their generous support.

For theirenthusiasmand helpfulsuggestions

I thankTaeer Bar-YamBrienne Brown

Diana DavisCornelia Drutu

Alexandra DrutuSteven FrankelEko Hironaka

Lilith SchwartzLucina Schwartz

Ren Yi