life on the infinite farm - brown universityres/farm/rgb.pdfdelores has a problem with her jewelry....
TRANSCRIPT
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Life
on
the
Infin
ite F
arm
Rich
ard
Eva
n S
chw
artz
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To all my math teachers.
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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.
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If you want to read the simple version, then skip the pages with the markers. Thebook will still make sense.
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ho are the animalson the infinite farm?
W
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Gracie is one of the animals. She is an infinite cow.
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3 4
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65 7 8
Gracieloves shoes.She has one on every foot ...
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and her feetgo on forever!
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1817
1112
1314
1516
1920
910
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while Simon, an infinite donkey, chats with Gerry,...
Flambeau, aninfinite sheep,mulls overshearing season
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an infinite gopher.
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Ezekiel is a goat withan infinite horn, but he thinks of himselfas mostly a zebra.
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Hammerwood is an infinite crocodile. Let'szoom out to see him better.
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Look at those teeth!
They go on and on. And so do the bubbles.
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Hammerwood loves bubble gum,even though it is hard for him tochew it allat once.
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Penn also lives on the farm. Heis an infinite chicken. On the infinitefarm, chickens have teeth too.
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Penn hasstaring contestswith his sisterVictoria, and also with...
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Bill, an owl with aninfinite eye.
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Delores, an infinitesquid, lives in the farm's pond.
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Her tentacles branch outendlessly, and she decorates themwith jewelry. She is crazy about ringsand bracelets. She likes to swim onSaturday nights with her friend...
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Nelson, a shark whose head extends forever in both directions.
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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.
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o the animals have anyproblems on the farm?
D
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3
Gracie hasa problem
with shoes.Her friends know that she loves shoes, so
they bringher newones as
presents.
Oh yes, lots of problems!
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She wants to wear the NEW shoes ...
But she doesn't want to stopwearing any of her OLD shoes.
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She already has a shoe on every foot, so what
is she going to do?
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Hammerwood has problems with his teeth.
He has so much funblowing bubbles ...
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...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 ...
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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?
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Delores has a problemwith her jewelry. Every
week her cousin Bin swims over and asks to
borrow some of it.
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Bin wants todecorate all his
tentacles for hisSaturday night dances...
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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?
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ere is how the animalssolve their problems...
H
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Gracie prepares to put on hernew shoes by standing behindthem on an infinitely longsticky strip.
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She gives each left leg a little shake ...
then steps out of each left shoe ...
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and into the onein front of it.
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Then she doesthe same thingwith all herright feet.
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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...
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Gracie moves off thesticky tape and then...
she is ready for another present.
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Hammerwood snaps intoaction once he loses
about 9 out of 10 teeth.
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The first thing he does is brush his remaining teeth.
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Then his friends from the infinite ant colony ...
put powerful rubberbands around his teeth.
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For technical reasons, they anchor the first two bands to a stick.
At night, the rubber bands contract ...
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and slide his teeth back into place.
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The ants carry off the rubberbands and then his teeth are as good as new.
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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.
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Little fish come andunclasp the jewelry.
The fishmove each piece one step towards Delores' head.
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and isreplaced byTWO piecesthat comefrom behind it!
Each piece of jewelrymoves inward...
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It becomes hertiara for the evening.
Something fancier happensto the piece of jewelry aroundDelores' neck. A fish unwinds itand moves it to her forehead.
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The fishmoveone pieceof eachpairover toBin...
and then
swim away.
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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.
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ny questions? A
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Easy. OnSunday
morningthey line up
again andreverse the
whole process.
How does Delores gether jewelry back from Bin?
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Where does Hammerwoodget his gum?
ALICE'S GUM SHOP
Creditcards
accepted.
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Simple. He points his mouth up aninfinite branchof a gum treeand takes whathe wants.
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Can Gracie swing all theway around, like thehand on a clock?
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No. She would get stuck onthe trees.
But otherwise she is quite
flexible.
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??
How do theanimals get around each other?
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Gracie walks underneath
Penn and theyhardly notice each other.
That is a very interestingquestion. For one thing,
the animals oftenhave very different sizes.
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Also, the animals aresurprisingly acrobatic. Gracie's cousin Boopis
can roll right over her.
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Some of the animals have special tricks. Nelson's head can swing
open like a door in some places...
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and Delores is built like a flexible puzzle.She can pull herself apartto let anotheranimal through.
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1 2 48
1632
641282565121024204840968192163843276865536
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How do the infiniteanimals fit in the pond?
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?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.
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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.
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Bill's eye is also one of these enclosed infinite spaces. Heloves watching Flambeau andthe other sheep. He can seetheir whole crater perfectly.
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Craters could have their own craters...
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And so on.
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Can the infinitefarm fit on Earth?
?
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?
The infinite farm doesn't close up like Earth, so how could it fit on Earth? It seems impossible.
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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.
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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.
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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.
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How doesday changeto night onthe farm?
Yeah, howdoes thesun getaround
the farm?
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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...
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or you couldimagine thatthe suns and moons dancearound the farm in a complicated, multi-dimensionalpattern...
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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
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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?
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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...
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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
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GEOMETRYmakestherules.
Well, I guess there's time for one
last question.
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Can we VISITthe infinite farm?
SCHOOL BUS
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It depends on what you mean by "visit".
You won't find the infinite farm on Earth...
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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
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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