Henry Segerman: Material Harmony in Mathematics

2024 Tus sau: Seth Attwood | [email protected]. Kawg hloov kho: 2023-12-16 16:06

Raws li cov lus dab neeg, Pythagoras yog thawj zaug pom tias ob txoj hlua sib npaug sib npaug tawm suab zoo nkauj yog tias lawv qhov ntev yog cuam tshuam nrog cov lej me me. Txij thaum ntawd los, tib neeg tau txaus siab los ntawm kev sib txuas tsis meej ntawm kev zoo nkauj thiab kev ua lej, cov ntaub ntawv sib haum xeeb ntawm cov ntaub ntawv, kev vibrations, symmetry - thiab kev paub zoo tshaj plaws ntawm cov lej thiab kev sib raug zoo.

Qhov kev sib txuas no yog ephemeral, tab sis tangible; nws tsis yog rau tsis muaj dab tsi uas cov kws ua yeeb yam tau siv cov cai ntawm geometry tau ntau xyoo thiab tau txais kev tshoov siab los ntawm kev cai lej. Henry Segerman pom tias nws nyuaj rau tso tseg cov tswv yim no: tom qab tag nrho, nws yog ib tug lej los ntawm txoj haujlwm thiab kev ua haujlwm.

Klein lub raj mis "Los ntawm lub hlwb gluing npoo ntawm ob lub Mobius strips," hais Henry Segerman, "koj tuaj yeem tau txais Klein lub raj mis, uas kuj muaj ib qho chaw. Ntawm no peb pom Klein lub raj mis ua los ntawm Mobius strips nrog ib puag ncig ntug.

Es tsis txhob, yuav ua li cas nws yuav saib nyob rau hauv peb-dimensional qhov chaw. Txij li thaum thawj "round" Mobius strips mus rau infinity, ces xws li lub raj mis Klein yuav mus infinity ob zaug thiab hla nws tus kheej, uas tuaj yeem pom hauv cov duab puab. " Ib daim qauv loj ntawm cov duab puab no ua rau lub Department of Mathematics thiab Statistics ntawm University of Melbourne.

Fractals

Henry hais tias "Kuv tau yug los rau hauv tsev neeg ntawm cov kws tshawb fawb, thiab kuv xav tias kuv qhov kev txaus siab rau txhua yam uas yuav tsum tau muaj kev xav txog kev xav tau muaj feem cuam tshuam nrog qhov no," Henry hais. Niaj hnub no nws twb kawm tiav ntawm Oxford kawm tiav thiab kev kawm kws kho mob ntawm Stanford Universities, thiab tuav txoj haujlwm ntawm Associate Professor ntawm University of Oklahoma.

Tab sis kev ua haujlwm zoo ntawm kev tshawb fawb tsuas yog ib sab ntawm nws tus kheej ntau yam: ntau dua 12 xyoo dhau los, tus lej pib tsim cov txheej xwm kos duab … hauv ntiaj teb virtual ntawm Lub Neej Thib Ob.

Qhov kev simulator peb-dimensional nrog cov ntsiab lus ntawm kev sib raug zoo network tau nrov heev, tso cai rau cov neeg siv tsis yog tsuas yog sib txuas lus, tab sis kuj muab lawv cov "avatars" virtual thiab thaj chaw rau kev lom zem, ua haujlwm, thiab lwm yam.

Lub npe: Henry Segerman

Yug xyoo 1979

Kev Kawm: Stanford University

Lub nroog: Stillwater, USA

Motto: "Ua ib lub tswv yim, tab sis qhia kom meej li sai tau."

Segerman tuaj ntawm no, muaj riam phom nrog cov qauv thiab cov lej, thiab npaj nws lub ntiaj teb virtual hauv kev ua lej, sau nws nrog cov duab tsis tau pom dua, spirals thiab tesseracts, plaub-dimensional hypercubes. "Qhov tshwm sim yog qhov projection ntawm plaub-dimensional hypercube nyob rau hauv peb-dimensional ntug ntawm Second Life - uas nws tus kheej yog ib tug projection ntawm peb-dimensional virtual ntiaj teb no mus rau ib tug ob-dimensional, tiaj tus screen," sau tus kws kos duab.

Hilbert's nkhaus: ib txoj kab txuas ntxiv ua rau qhov chaw ntawm lub voos xwmfab, tsis txhob cuam tshuam lossis cuam tshuam nrog nws tus kheej.

Hilbert curves yog cov qauv fractal, thiab yog tias koj zoom hauv, koj tuaj yeem pom tias qhov chaw ntawm qhov nkhaus no ua raws li cov duab ntawm tag nrho. "Kuv tau pom lawv ntau txhiab zaus hauv cov duab kos duab thiab cov qauv hauv computer, tab sis thaum kuv thawj zaug coj cov duab puab 3D hauv kuv txhais tes, kuv pom tam sim ntawd nws kuj yog caij nplooj ntoos hlav," Segerman hais. "Lub cev embodiment ntawm cov tswv yim lej yog ib txwm xav tsis thoob nrog qee yam."

Txawm li cas los xij, nws nyiam ua haujlwm nrog cov khoom sculptures ntau dua. Segerman hais tias "Muaj ntau cov ntaub ntawv nthuav dav thoob plaws peb txhua lub sijhawm," - Hmoov zoo, lub ntiaj teb tiag tiag muaj bandwidth loj heev, uas tseem tsis tau muaj nyob hauv Web.

Muab ib tug neeg ua tiav, ib daim ntawv tseem ceeb - thiab nws yuav tam sim ntawd perceive nws nyob rau hauv tag nrho nws complexity, tsis tos rau loading. Yog li txij li xyoo 2009, Segerman tau tsim me ntsis ntau tshaj li ib puas sculptures, thiab txhua tus ntawm lawv yog ib qho kev pom thiab, raws li qhov ua tau, lub cev muaj pes tsawg leeg ntawm kev paub txog lej thiab cov cai.

Polyhedra

Qhov kev hloov pauv ntawm Segerman qhov kev sim ua yeeb yam nrog 3D luam ntawv yog qhov txawv txav rov ua dua qhov hloov pauv ntawm cov tswv yim lej. Ntawm nws qhov kev sim thawj zaug yog cov khoom siv classical Platonic, ib txheej ntawm tsib daim duab sib luag, folded hauv cov duab peb sab, pentagons thiab squares. Lawv tau ua raws li ib nrab ntawm cov polyhedra - 13 Archimedean cov khib nyiab, uas nws lub ntsej muag yog tsim los ntawm cov tsis sib xws tsis tu ncua polygons.

Stanford Rabbit 3D qauv tsim nyob rau hauv 1994. Ua los ntawm ze li ntawm 70,000 daim duab peb sab, nws ua haujlwm yooj yim thiab nrov ntawm kev ua haujlwm ntawm software algorithms. Piv txwv li, ntawm tus luav, koj tuaj yeem sim qhov ua tau zoo ntawm cov ntaub ntawv compression lossis nto smoothing rau cov duab hauv computer.

Yog li ntawd, rau cov kws kho mob tshwj xeeb, daim ntawv no yog tib yam li cov lus "Nyob ib co ntxiv ntawm cov mos Fabkis yob" rau cov neeg uas nyiam ua si nrog computer fonts. Stanford Bunny sculpture yog tib tus qauv, qhov saum npoo ntawm uas yog paving nrog cov tsiaj ntawv ntawm lo lus luav.

Twb tau cov ntaub ntawv yooj yim no, tau tsiv los ntawm cov duab kos duab ob sab thiab lub ntiaj teb zoo tshaj plaws ntawm kev xav mus rau peb qhov kev muaj tiag, ua rau muaj kev qhuas sab hauv rau lawv cov laconic thiab kev zoo nkauj zoo meej. Kev sib raug zoo ntawm kev ua lej zoo nkauj thiab kev zoo nkauj ntawm kev pom lossis suab ntawm kev kos duab zoo li tsis yooj yim rau kuv.

Tom qab tag nrho, ntau tus neeg yog acutely paub txog ib daim ntawv ntawm no kev zoo nkauj, kiag li tsis to taub lwm yam. Cov tswv yim lej tuaj yeem muab txhais ua cov ntawv pom lossis suab, tab sis tsis yog txhua yam, thiab tsis yog ze li yooj yim li nws zoo li,”Segerman ntxiv.

Tsis ntev, ntau thiab ntau cov ntaub ntawv ua raws li cov duab classical, txog rau cov uas Archimedes lossis Pythagoras tsis tuaj yeem xav txog - niaj hnub polyhedra uas sau Lobachevsky qhov chaw hyperbolic yam tsis muaj ib ntus.

Cov duab zoo li no nrog cov npe zoo kawg li "tetrahedral honeycomb ntawm kev txiav txim 6" lossis "hexagonal mosaic honeycomb" tsis tuaj yeem xav txog yam tsis muaj duab pom ntawm tes. Lossis - ib qho ntawm cov duab puab los ntawm Segerman, uas sawv cev rau lawv nyob rau hauv peb qhov chaw peb-seem Euclidean.

Platonic solids: ib lub tetrahedron, octahedron thiab icosahedron folded nyob rau hauv tsis tu ncua daim duab peb sab, raws li zoo raws li ib lub voos xwmfab thiab ib tug icosahedron muaj xws li squares raws li pentagons.

Plato nws tus kheej cuam tshuam rau lawv nrog plaub lub ntsiab lus: "ntxhiab" octahedral hais, hauv nws lub tswv yim, folded cua, "kua" icosahedrons - dej, "dense" cubes - lub ntiaj teb, thiab ntse thiab "thorny" tretrahedrons - hluav taws. Lub caij thib tsib, dodecahedron, raug suav hais tias los ntawm tus kws tshawb fawb los ua ib feem ntawm lub ntiaj teb ntawm cov tswv yim.

Tus kws kos duab ua haujlwm pib nrog tus qauv 3D, uas nws tsim hauv cov pob kws Rhinoceros. Los ntawm thiab loj, qhov no yog li cas nws xaus: kev tsim cov duab puab nws tus kheej, luam cov qauv ntawm lub tshuab luam ntawv 3D, Henry tsuas yog xaj los ntawm Shapeways, lub zej zog online loj ntawm 3D luam ntawv nyiam, thiab tau txais cov khoom tiav ua los ntawm cov yas lossis hlau-bronze-raws li hlau matrix composites. "Nws yooj yim heev," nws hais. "Koj tsuas yog upload tus qauv rau ntawm qhov chaw, nyem qhov Ntxiv rau lub laub khawm, tso ib qho kev txiav txim, thiab ob peb lub lis piam nws yuav xa tuaj rau koj los ntawm kev xa ntawv."

Daim duab Yim Ntxiv Xav txog kev khi ib lub pob hauv ib qho khoom thiab tom qab ntawd tshem nws; cov kab noj hniav seem yog hu ua qhov sib ntxiv ntawm cov node. Cov qauv no qhia txog qhov sib ntxiv ntawm ib qho ntawm cov pob qij txha yooj yim tshaj plaws, daim duab yim.

kev zoo nkauj

Thaum kawg, evolution ntawm Segerman tus lej sculptures coj peb mus rau hauv lub complex thiab mesmerizing teb ntawm topology. Cov ceg ntawm kev ua lej no kawm txog cov khoom thiab deformations ntawm qhov chaw tiaj tus thiab qhov chaw ntawm qhov sib txawv, thiab lawv cov yam ntxwv dav dav yog qhov tseem ceeb rau nws dua li classical geometry.

Ntawm no, lub voos xwmfab tuaj yeem hloov tau yooj yim rau hauv lub pob, zoo li plasticine, thiab lub khob nrog lub kov tuaj yeem dov rau hauv lub khob noom cookie yam tsis muaj qhov tseem ceeb hauv lawv - piv txwv zoo uas muaj nyob rau hauv Segerman's elegant Topological tso dag.

Lub tesseract yog ib lub voos xwmfab plaub-dimensional: ib yam li ib lub xwmfab tuaj yeem tau los ntawm kev hloov cov ntu ntu rau nws ntawm qhov deb sib npaug ntawm nws qhov ntev, lub voos xwmfab tuaj yeem tau los ntawm kev luam ib lub xwmfab hauv peb qhov ntev, thiab los ntawm kev txav ib lub voos xwmfab. nyob rau hauv plaub, peb yuav "kos" ib tesseract, los yog hypercube. Nws yuav muaj 16 vertices thiab 24 lub ntsej muag, qhov projections uas rau hauv peb qhov chaw peb-dimensional zoo li me me li peb-dimensional lub voos xwmfab.

"Hauv kev ua lej, qhov kev xav zoo nkauj yog qhov tseem ceeb heev, cov lej nyiam "zoo nkauj" theorems, - tus kws kos duab sib cav. - Nws yog qhov nyuaj los txiav txim siab seb qhov kev zoo nkauj no muaj nyob hauv, li, qhov tseeb, hauv lwm qhov xwm txheej. Tab sis kuv xav hais tias qhov kev zoo nkauj ntawm lub theorem yog nyob rau hauv nws simplicity, uas tso cai rau koj kom nkag siab txog ib yam dab tsi, pom ib co yooj yim kev twb kev txuas uas yav tas los seemed incredibly complex.

Nyob rau hauv lub plawv ntawm kev zoo nkauj lej tuaj yeem ua kom dawb huv, zoo minimalism - thiab xav tsis thoob hais tias "Aha!" ". Qhov kev zoo nkauj sib sib zog nqus ntawm kev ua lej tuaj yeem ua rau tsis txaus ntseeg raws li qhov dej khov nyob mus ib txhis ntawm Snow Queen's palace. Txawm li cas los xij, tag nrho cov kev sib haum xeeb txias no cuam tshuam txog qhov kev coj noj coj ua sab hauv thiab tsis tu ncua ntawm Lub Ntiaj Teb uas peb nyob. Kev ua lej tsuas yog ib hom lus uas haum rau lub ntiaj teb no elegant thiab complex.

Paradoxically, nws muaj cov ntaub ntawv lub cev thiab cov ntawv thov rau yuav luag txhua nqe lus hauv cov lus ntawm cov qauv lej thiab kev sib raug zoo. Txawm tias feem ntau paub daws teeb meem thiab "artificial" kev tsim kho yuav sai lossis tom qab pom daim ntawv thov hauv ntiaj teb tiag.

Ib qho kev tso dag topological: los ntawm qee qhov kev xav, qhov chaw ntawm lub voj voog thiab lub khob cij yog "zoo ib yam", los yog, qhov tseeb dua, lawv yog homeomorphic, vim lawv muaj peev xwm hloov mus rau ib leeg yam tsis muaj kev tawg thiab nplaum, vim yog maj mam deformation.

Euclidean geometry tau los ua ib qho kev xav ntawm lub ntiaj teb nyob ruaj ruaj, qhov sib txawv ntawm kev xam pom tau los ua ke rau Newtonian physics. Qhov zoo kawg Riemannian metric, raws li nws tau muab tawm, yog qhov tsim nyog los piav txog Einstein lub ntiaj teb tsis ruaj khov, thiab ntau qhov chaw hyperbolic tau pom daim ntawv thov hauv txoj hlua txoj kev xav.

Nyob rau hauv no coj txawv txawv ntawm kev paub daws teeb meem xam zauv thiab cov zauv rau lub hauv paus ntawm peb kev muaj tiag, tej zaum, lus dag lub secrets ntawm kev zoo nkauj uas peb yuav tsum xav tias qab tag nrho cov txias xam ntawm cov lej.