Transcription
Life on the Infinite FarmRichard Evan Schwartz
To all my math teachers.
1Note to the reader2When I started writingthis book, I planned tomake it simple enough sothat anyone old enoughto read could enjoy it.3When I looked at thepages that mentionatoms and specialrelativity, I wondered ifI had gone too far.But then I went alittle crazy and putin all kinds of fancyideas that I foundinteresting.4I ended up making botha simple version andan advanced version,but I couldn't decidewhich one to show you.65This book has all thepages, simple andadvanced, but I putlittle black markersin the corners of allthe advanced ones.Then I had an ideaabout how to showyou both versions atthe same time.78If you want toread the simpleversion, thenskip the pageswith themarkers. Thebook will stillmake sense.If you arefeelingadventurousand want theadvancedversion,then justread it all.
Who are the animalson the infinite farm?
Gracie is one of the animals.She is aninfinite cow.3124
Gracielovesshoes.She hasoneon everyfoot .5678
and her feetgo on forever!1917151311911001214161820
Flambeau, aninfinite sheep,mulls overshearing seasonwhile Simon, an infinitedonkey, chats with Gerry,.
an infinite gopher.
Ezekiel is a goat withan infinite horn, buthe thinks of himselfas mostly a zebra.
Hammerwood is aninfinite crocodile. Let'szoom out to see him better.
Look at those teeth!They go on and on. And so do the bubbles.
Hammerwood loves bubble gum,even though it is hard for him tochew it allat once.
Penn also lives on the farm. Heis an infinite chicken. On the infinitefarm, chickens have teeth too.
Penn hasstaring contestswith his sisterVictoria, andalso with.
Bill, an owl with aninfinite eye.
Delores, an infinitesquid, lives in the farm's pond.
Her tentacles branch outendlessly, and she decorates themwith jewelry. She is crazy about ringsand bracelets. She likes to swim onSaturday nights with her friend.
Nelson, a sharkwhose head extendsforever in bothdirections.
The farm has lots ofother animals too.Some of them are sostrange I don't knowwhat to call them.Maybe YOU can thinkof names for them.
Do the animals have anyproblems on the farm?
Oh yes,lots ofproblems!3Gracie hasa problemwith shoes.Her friendsknow thatshe lovesshoes, sothey bringher newones aspresents.
She wants to wear the NEW shoes .But she doesn'twant to stopwearing any ofher OLD shoes.
She already has a shoeon every foot, so whatis she going to do?
Hammerwood hasproblems withhis teeth.He has so much funblowing bubbles .
.that he often forgetsto brush his teeth.To be honest, bad thingssometimes happen toHammerwood's teeth.It isn't pretty up close.It can get so bad that .
f hi sotsomteseethsol.heeth fodseeneHhdnancee tesothowrg'trchewingnew ones.af ix hehisnteeth?H ow c a
Delores has a problemwith her jewelry. Everyweek her cousin Binswims over and asks toborrow some of it.
Bin wants todecorate all histentacles for hisSaturday night dances.
but Delores wants to decorate all of HERtentacles for her Saturday nightswims with Nelson. Howcan they split the jewelryso that each hasenough forall of theirtentacles?
Here is how the animalssolve their problems.
Gracie prepares to put on hernew shoes by standing behindthem on an infinitely longsticky strip.
She gives each leftleg a little shake .then steps out ofeach left shoe .
and into the onein front of it.
Then she doesthe same thingwith all herright feet.
Now she wears the newshoes AND all of her oldshoes. There aren'tany shoes left overat the back end because.Gracie doesn't have a back end!
Gracie moves off thesticky tape and then.she is ready for another present.
Hammerwood snaps intoaction once he losesabout 9 out of 10 teeth.
The first thing hedoes is brush hisremaining teeth.
Then his friends fromthe infinite ant colony .put powerful rubberbands around his teeth.
For technical reasons, theyanchor the first two bands to a stick.At night, the rubberbands contract .
and slide his teethback into place.
The ants carry off the rubberbands and then his teeth areas good as new.
Delores agrees to lendBin some jewelry.Their tentacles branch out in thesame way, so on Saturday afternoon.they linethem uplike twoinfinite handspressingtogether.
Little fishcome andunclasp thejewelry.The fishmoveeachpieceonesteptowardsDelores' head.
Each pieceof jewelrymoves inward.and isreplaced byTWO piecesthatcomefrombehind it!
Something fancier happensto the piece of jewelry aroundDelores' neck. A fish unwinds itand moves it to her forehead.It becomes hertiara forthe evening.
The fishmoveonepieceof eachpairover toBin.andthenswimaway.
Except forthe tiara,Delores hasthe samepattern ofjewelry as inthe beginning.And now Bin has it too.Bin thanksDelores for beingso amenable andthen the cousinsswim off to theirSaturday night fun.
Any questions?
How doesDelores gether jewelryback fromBin?Easy. OnSundaymorningthey line upagain andreverse thewhole process.
Where doesHammerwoodget his gum?ALICE'SGUM SHOPCreditcardsaccepted.
Simple. He pointshis mouth up aninfinite branchof a gum treeand takes whathe wants.
Can Gracie swing all theway around, like thehand on a clock?
No. She would get stuck onthe trees.But otherwiseshe is quiteflexible.
?How do theanimals getaround eachother?
That is a very interestingquestion. For one thing,the animals oftenhave very different sizes.Gracie walksunderneathPenn and theyhardly notice each other.
Also, the animals aresurprisingly acrobatic.Gracie's cousin Boopiscan roll right over her.
Some of the animals have specialtricks. Nelson's head can swingopen like a door in some places.
and Delores is builtlike a flexible puzzle.She can pullherself apartto let anotheranimal through.
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How do the infiniteanimals fit in the pond?
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 toshrink as they approachthe edge, but they arereally all the same size.The same goes forNelson's teethand eyes.?
LFlambeau and his herdlive in a crater thatis like the pond,except thatthey walkon thebottom.GFFrom theoutside,BFlambeau'sfriends Bolyai,Gauss, and Lobachevskyseem to be different sizes, butthey are really all the same size.
Bill's eye is also one of theseenclosed infinite spaces. Heloves watching Flambeau andthe other sheep. He can seetheir whole crater perfectly.
Craters couldhave theirowncraters.
And so on.
?Can theinfinitefarm fit onEarth?
?The infinite farm doesn'tclose up like Earth, sohow could it fit on Earth?It seems impossible.
But, think back to how the pondand thoseupa finiteBut thinkback tocratershow thetakepondandthose cratersonly takeup a finiteamountof spacefrom theamount of spacefrom thepointoutside.outsider'sof view.
Oh, I see. It isn't such a stretchto imagine that the farm itselfis one of those craters,and that thecrater is onEarth.
Yes,precisely,but there isa catch. OnEarth thephysicallaws*wouldpreventone ofthosecratersfromexisting.So, theinfinite farmcan't REALLYfit on Earth.crater like this seems to require* Amatterto exist on all scales, whichis incompatible with the atomic natureof our world. Even setting this aside,the crater might generate agravitational field that would crushour entire universe.
How doesday changeto night onthe farm?Yeah, howdoes thesun getaroundthe farm?
You could imagine thatthere is an infinite gridof suns and moonshovering over the farm.and that these celestiallights change theircolor and brightnessto suit the time of day.
or you couldimagine thatthe suns andmoons dancearound thefarm in acomplicated,multi-dimensionalpattern.
or you could imagine thatthere are no suns and moons,andinsteadthat light isbroughteachdayby an infiniteflock ofglowingbirds.I like to think thatlight appearson the farm withoutexplanation or cause,like a new idea.So, if there are nofurtherquestions.
How didGracieget onthe farm?How canEzekiel'sneck supporthis infinite horn?Would the farmhave infiniteatoms and/orinfinitebiomolecules?Since most of his teeth wouldhave to move faster than thespeed of light, isn'tHammerwood's teeth-fixingprocess incompatible withthe constraints of specialrelativity?
BIOLOGYCHEMISTRYPHYSICSOK, OK, slow down! Let me tryto answer all these questions atthe same time. In our universe thesciences are relentless enforcers.They seem to look into everysituation, no matter how large orsmall, and make sure that itunfolds according to the law.
UAGccc UAA GOn the infinite farm,the sciencesare more likenegligent babysitterswho check in from timeto time but mostlyjust play around.m2Meanwhile,out onthe farm,.
GEOMETRYmakestherules.Well, I guessthere's time for onelast question.
Can we VISITthe infinite farm?SCHOOL BUS
It depends on whatyou mean by "visit".You won't find theinfinite farm on Earth.
Proceedings ofthe Flat Earth soc.Hilbert HotelGEOMETRICGROUPTHEORYGEOMETRY ANDTHE IMAGINATIONREALLY BIG NUMBERSNonAmenableGroupsZENOTHECAT INTHE HATCOMESBACKTHE GEOMETRYAND TOPOLOGYOF 3-MANIFOLDS1978METRICDots and YouHyperbolic GeometrySPACESOFNONPOSITIVECURVATUREGALLERY OFTHE INFINITEbut if you read betweenthe lines of geometrybooks.you mightfind it there.
About the author: I am the Chancellor's Professorof Mathematics at Brown University. In myspare time I like to draw, write computerprograms, listen to music, walk on thebeach, 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 mymath research, the research support I got duringthe time I wrote it gave me extra freedom to workon it. I thank the National Science Foundation,the Simons Foundation, and the Isaac NewtonInstitute for their generous support.For theirenthusiasmand helpfulsuggestionsI thankTaeer Bar-YamBrienne BrownDiana DavisCornelia DrutuAlexandra DrutuSteven FrankelEko HironakaLilith SchwartzLucina SchwartzRen Yi
This book has all the pages, simple and advanced, but I put little black markers in the corners of all the advanced ones. 6 If you want to read the simple . chew it all at once. 14 Penn also lives on the farm. He is an infinite chicken. On the infinite farm, chickens have teeth too. 15 Penn has
