the 11 most beautiful equations

8/10/2019 The 11 Most Beautiful Equations

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The 11 Most Beautiful Mathematical EquationsBy Clara Moskowitz, LiveScience Senior Writer LiveScience!com Wed, Jan 30, 2013

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This equation of se!ial relati"it# des!ri$es time dilation% &a href'(htt)**www%shuttersto!+%!om*(-Shuttersto!+.&*a/iew Photo

This equation of se!ial relati"it#

athemati!al equations arent ust useful 4 man# are quite $eautiful% 5nd man# s!ientists admit the# are often fond of arti!ular formulas not ust for their fun!tion, $ut for their form, and the simle, oeti! truths the# !ontain%

While !ertain famous equations, su!h as 5l$ert Einsteins E ' m!62, ho7 most of the u$li! 7lor#, man# less familiar formulas ha"e their !hamions amon7 s!ientists% 8i"eS!ien!e as+ed h#si!ists, astronomers and mathemati!ians for their fa"orite equations9 heres what we found)

:eneral relati"it#

The equation a$o"e was formulated $# Einstein as art of his 7round$rea+in7 7eneral theor# of relati"it# in 1;1t is still ama=in7 to me that one su!h mathemati!al equation !an des!ri$e what sa!e?time is all a$out,( said Sa!e Teles!oe S!ien!e >nstitute astroh#si!ist ario 8i"io, who nominated the equation as his fa"orite% (5ll of Einsteins true 7enius is em$odied in this equation%( @Einstein Aui=) Test Bour nowled7e of the :eniusD

(The ri7ht?hand side of this equation des!ri$es the ener7# !ontents of our uni"erse-in!ludin7 the dar+ ener7# that roels the !urrent !osmi! a!!eleration.,( 8i"io elained% (The left?hand side des!ri$es the 7eometr# of sa!e?time% The equalit# reFe!ts the fa!t that in Einsteins 7eneral relati"it#, mass and ener7# determine the 7eometr#, and !on!omitantl# the !ur"ature, whi!h is a manifestation of what we !all 7ra"it#%( @G Weird Ha!ts 5$out :ra"it#D

(>ts a "er# ele7ant equation,( said #le Cranmer, a h#si!ist at Iew Bor+ ni"ersit#,addin7 that the equation re"eals the relationshi $etween sa!e?time and matter and ener7#% (This equation tells #ou how the# are related 4 how the resen!e of the sun wars sa!e?time so that the Earth mo"es around it in or$it, et!% >t also tells #ou h

ow the uni"erse e"ol"ed sin!e the Ki7 Kan7 and redi!ts that there should $e $la!+holes%(

Standard model

5nother of h#si!s rei7nin7 theories, the standard model des!ri$es the !olle!tion offundamental arti!les !urrentl# thou7ht to ma+e u our uni"erse%

The theor# !an $e en!asulated in a main equation !alled the standard model 8a7ra


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n7ian -named after the 1Lth?!entur# Hren!h mathemati!ian and astronomer Joseh8ouis 8a7ran7e., whi!h was !hosen $# theoreti!al h#si!ist 8an!e Mion of the S85C Iational 5!!elerator 8a$orator# in California as his fa"orite formula%

(>t has su!!essfull# des!ri$ed all elementar# arti!les and for!es that we"e o$ser"ed in the la$orator# to date 4 e!et 7ra"it#,( Mion told 8i"eS!ien!e% (That in!ludes,

of !ourse, the re!entl# dis!o"ered Ni77s-li+e. $oson, hi in the formula% >t is full# self?!onsistent with quantum me!hani!s and se!ial relati"it#%(

The standard model theor# has not #et, howe"er, $een united with 7eneral relati"it#,whi!h is wh# it !annot des!ri$e 7ra"it#% @>nfo7rahi!) The Standard odel ElainedD

Cal!ulus

While the Orst two equations des!ri$e arti!ular ase!ts of our uni"erse, another fa"orite equation !an $e alied to all manner of situations% The fundamental theoremof !al!ulus forms the $a!+$one of the mathemati!al method +nown as !al!ulus, and lin+s its two main ideas, the !on!et of the inte7ral and the !on!et of the deri"ati"e%

(>n simle words, @itD sa#s that the net !han7e of a smooth and !ontinuous quantit#,su!h as a distan!e tra"elled, o"er a 7i"en time inter"al -i%e% the dieren!e in the "alues of the quantit# at the end oints of the time inter"al. is equal to the inte7ral of the rate of !han7e of that quantit#, i%e% the inte7ral of the "elo!it#,( said el+ana Kra+alo"a?Tre"ithi!+, !hair of the math deartment at Hordham ni"ersit#, who !hose this equation as her fa"orite% (The fundamental theorem of !al!ulus -HTC. allows us todetermine the net !han7e o"er an inter"al $ased on the rate of !han7e o"er the entire inter"al%(

The seeds of !al!ulus $e7an in an!ient times, $ut mu!h of it was ut to7ether in the1Qth !entur# $# >saa! Iewton, who used !al!ulus to des!ri$e the motions of the lanets around the sun%

P#tha7orean theorem

5n (oldie $ut 7oodie( equation is the famous P#tha7orean theorem, whi!h e"er# $e7innin7 7eometr# student learns%

This formula des!ri$es how, for an# ri7ht?an7led trian7le, the square of the len7th ofthe h#otenuse, !, -the lon7est side of a ri7ht trian7le. equals the sum of the squares of the len7ths of the other two sides -a and $.% Thus, a62 $62 ' !62

(The "er# Orst mathemati!al fa!t that ama=ed me was P#tha7orean theorem,( saidmathemati!ian Maina Taimina of Cornell ni"ersit#% (> was a !hild then and it seemed to me so ama=in7 that it wor+s in 7eometr# and it wor+s with num$ers( @< Seriousl# ind?Ko77lin7 ath Ha!tsD

1 ' 0%;;;;;;;;;%

This simle equation, whi!h states that the quantit# 0%;;;, followed $# an inOnite strin7 of nines, is equi"alent to one, is the fa"orite of mathemati!ian Ste"en Stro7at= of Cornell ni"ersit#%

(> lo"e how simle it is 4 e"er#one understands what it sa#s 4 #et how ro"o!ati"eit is,( Stro7at= said% (an# eole dont $elie"e it !ould $e true% >ts also $eautifull#


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$alan!ed% The left side reresents the $e7innin7 of mathemati!s9 the ri7ht side reresents the m#steries of inOnit#%(

Se!ial relati"it#

Einstein ma+es the list a7ain with his formulas for se!ial relati"it#, whi!h des!ri$es

how time and sa!e arent a$solute !on!ets, $ut rather are relati"e deendin7 on the seed of the o$ser"er% The equation a$o"e shows how time dilates, or slows down,the faster a erson is mo"in7 in an# dire!tion%

(The oint is its reall# "er# simle,( said Kill urra#, a arti!le h#si!ist at the CERI la$orator# in :ene"a% (There is nothin7 there an 5?le"el student !annot do, no !omle deri"ati"es and tra!e al7e$ras% Kut what it em$odies is a whole new wa# of loo+in7 at the world, a whole attitude to realit# and our relationshi to it% Suddenl#, the ri7id un!han7in7 !osmos is swet awa# and rela!ed with a ersonal world, related to what #ou o$ser"e% Bou mo"e from $ein7 outside the uni"erse, loo+in7 down, to one of the !omonents inside it% Kut the !on!ets and the maths !an $e 7rased $# an#one that wants to%(

urra# said he referred the se!ial relati"it# equations to the more !omli!ated formulas in Einsteins later theor#% (> !ould ne"er follow the maths of 7eneral relati"it#,( he said%

Eulers equation

This simle formula en!asulates somethin7 ure a$out the nature of sheres)

(>t sa#s that if #ou !ut the surfa!e of a shere u into fa!es, ed7es and "erti!es, andlet H $e the num$er of fa!es, E the num$er of ed7es and / the num$er of "erti!es, #ou will alwa#s 7et / E H ' 2,( said Colin 5dams, a mathemati!ian at Williams Colle7e in assa!husetts%

(So, for eamle, ta+e a tetrahedron, !onsistin7 of four trian7les, si ed7es and four"erti!es,( 5dams elained% (>f #ou $lew hard into a tetrahedron with Fei$le fa!es,#ou !ould round it o into a shere, so in that sense, a shere !an $e !ut into four fa!es, si ed7es and four "erti!es% 5nd we see that / E H ' 2% Same holds for a #ramid with O"e fa!es 4 four trian7ular, and one square 4 ei7ht ed7es and O"e "erti!es,( and an# other !om$ination of fa!es, ed7es and "erti!es%

(5 "er# !ool fa!t The !om$inatori!s of the "erti!es, ed7es and fa!es is !aturin7 somethin7 "er# fundamental a$out the shae of a shere,( 5dams said%

Euler8a7ran7e equations and Ioethers theorem

(These are rett# a$stra!t, $ut ama=in7l# owerful,( IBs Cranmer said% (The !oolthin7 is that this wa# of thin+in7 a$out h#si!s has sur"i"ed some maor re"olutions

in h#si!s, li+e quantum me!hani!s, relati"it#, et!%(

Nere, 8 stands for the 8a7ran7ian, whi!h is a measure of ener7# in a h#si!al s#stem, su!h as srin7s, or le"ers or fundamental arti!les% (Sol"in7 this equation tells #ou how the s#stem will e"ol"e with time,( Cranmer said%

5 sino of the 8a7ran7ian equation is !alled Ioethers theorem, after the 20th !entur# :erman mathemati!ian Emm# Ioether% (This theorem is reall# fundamental to h#si!s and the role of s#mmetr#,( Cranmer said% (>nformall#, the theorem is that if #o


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ur s#stem has a s#mmetr#, then there is a !orresondin7 !onser"ation law% Hor eamle, the idea that the fundamental laws of h#si!s are the same toda# as tomorrow -time s#mmetr#. imlies that ener7# is !onser"ed% The idea that the laws of h#si!s are the same here as the# are in outer sa!e imlies that momentum is !onser"ed% S#mmetr# is erhas the dri"in7 !on!et in fundamental h#si!s, rimaril# due to @IoethersD !ontri$ution%(

The Callan?S#man=i+ equation

(The Callan?S#man=i+ equation is a "ital Orst?rin!iles equation from 1;Q0, essential for des!ri$in7 how nai"e ee!tations will fail in a quantum world,( said theoreti!al h#si!ist att Strassler of Rut7ers ni"ersit#%

The equation has numerous ali!ations, in!ludin7 allowin7 h#si!ists to estimate the mass and si=e of the roton and neutron, whi!h ma+e u the nu!lei of atoms%

Kasi! h#si!s tells us that the 7ra"itational for!e, and the ele!tri!al for!e, $etween two o$e!ts is roortional to the in"erse of the distan!e $etween them squared% na simle le"el, the same is true for the stron7 nu!lear for!e that $inds rotons and neutrons to7ether to form the nu!lei of atoms, and that $inds quar+s to7ether to formrotons and neutrons% Nowe"er, tin# quantum Fu!tuations !an sli7htl# alter a for!esdeenden!e on distan!e, whi!h has dramati! !onsequen!es for the stron7 nu!lear for!e%

(>t re"ents this for!e from de!reasin7 at lon7 distan!es, and !auses it to tra quar+s and to !om$ine them to form the rotons and neutrons of our world,( Strassler said% (What the Callan?S#man=i+ equation does is relate this dramati! and dii!ult?to?!al!ulate ee!t, imortant when @the distan!eD is rou7hl# the si=e of a roton, to more su$tle $ut easier?to?!al!ulate ee!ts that !an $e measured when @the distan!eD ismu!h smaller than a roton%(

The minimal surfa!e equation

(The minimal surfa!e equation somehow en!odes the $eautiful soa Olms that formon wire $oundaries when #ou di them in soa# water,( said mathemati!ian Hran+ or7an of Williams Colle7e% (The fa!t that the equation is nonlinear, in"ol"in7 owers and rodu!ts of deri"ati"es, is the !oded mathemati!al hint for the surrisin7 $eha"ior of soa Olms% This is in !ontrast with more familiar linear artial dierential equations, su!h as the heat equation, the wa"e equation, and the S!hrUdin7er equationof quantum h#si!s%(

The Euler line

:len Whitne#, founder of the useum of ath in Iew Bor+, !hose another 7eometri!al theorem, this one ha"in7 to do with the Euler line, named after 1Lth?!entur# Swiss mathemati!ian and h#si!ist 8eonhard Euler%

(Start with an# trian7le,( Whitne# elained% (Mraw the smallest !ir!le that !ontainsthe trian7le and Ond its !enter% Hind the !enter of mass of the trian7le 4 the oint where the trian7le, if !ut out of a ie!e of aer, would $alan!e on a in% Mraw the three altitudes of the trian7le -the lines from ea!h !orner erendi!ular to the oositeside., and Ond the oint where the# all meet% The theorem is that all three of the oints #ou ust found alwa#s lie on a sin7le strai7ht line, !alled the Euler line of the trian7le%(


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Whitne# said the theorem en!asulates the $eaut# and ower of mathemati!s, whi!h often re"eals surrisin7 atterns in simle, familiar shaes%

the 11 most beautiful equations

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