{"id":9087,"date":"2023-03-14T07:09:28","date_gmt":"2023-03-14T11:09:28","guid":{"rendered":"https:\/\/blogs.mathworks.com\/community\/?p=9087"},"modified":"2023-03-14T16:10:05","modified_gmt":"2023-03-14T20:10:05","slug":"celebrate-pi-day-with-a-pi-patch","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/community\/2023\/03\/14\/celebrate-pi-day-with-a-pi-patch\/","title":{"rendered":"Celebrate pi day with a pi patch"},"content":{"rendered":"
\n
Today's guest article is by <\/span>Adam Danz<\/span><\/a> whose name you might recognize from the MATLAB Central community. Adam is a developer on the MATLAB Graphics and Charting team. About 20 years ago Adam memorized pi to a measly 200 decimal places which is 1\/50 of the digits he'll visualize today using MATLAB.<\/span><\/div>\n

It's March 14th. Let's get irrational.<\/h2>\n
Today is pi day, the most popular irrational number at any party, transcendental to all number systems, with an infinite number of decimal places.<\/div>\n
\n

But how many decimal places are needed? <\/span> NASA's Jet Propulsion Laboratory uses 15 decimal places<\/a> of pi for interplanetary navigation. With 15 decimals of pi, the earth's circumference can be calculated from its 12756 km (7926 mi) diameter with an error the size of a molecule. The figure below shows the increase in precision (decrease in error) of calculating the earth's circumference as more digits of pi are included in the calculation c=\u03c0d<\/span>.<\/p>\n

.<\/p>\n<\/div>\n

\"\"<\/div>\n
The value of pi in MATLAB<\/span>, like most other technical computing environments, loses precision after the 15th decimal place due to limitations of floating point arithmetic<\/a>. To get more than 15 decimal places of pi we can set variable precision that converts symbolic expressions to floating-point numbers, provided by the Symbolic Math Toolbox<\/a>.<\/div>\n
Since we're being irrational, let's get 10,000 decimal places of pi.<\/div>\n
\n
\n
% Requires the Symbolic Math toolbox<\/span><\/span><\/div>\n<\/div>\n
\n
symExpression = sym(pi); % Symbolic expression; e.g. sym(pi), exp(sym(1)), sym(1\/23)<\/span><\/span><\/div>\n<\/div>\n
\n
nDecimalPlaces = 10000; % Number of decimal places<\/span><\/span><\/div>\n<\/div>\n
\n
<\/div>\n<\/div>\n
\n
decimalChar = char(vpa(mod(abs(symExpression),1),nDecimalPlaces+3))<\/span><\/div>\n
\n
\n
\n
decimalChar = <\/span>'0.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506016842739452267467678895252138522549954666727823986456596116354886230577456498035593634568174324112515076069479451096596094025228879710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821682998948722658804857564014270477555132379641451523746234364542858444795265867821051141354735739523113427166102135969536231442952484937187110145765403590279934403742007310578539062198387447808478489683321445713868751943506430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675142691239748940907186494231961567945208095146550225231603881930142093762137855956638937787083039069792077346722182562599661501421503068038447734549202605414665925201497442850732518666002132434088190710486331734649651453905796268561005508106658796998163574736384052571459102897064140110971206280439039759515677157700420337869936007230558763176359421873125147120532928191826186125867321579198414848829164470609575270695722091756711672291098169091528017350671274858322287183520935396572512108357915136988209144421006751033467110314126711136990865851639831501970165151168517143765761835155650884909989859982387345528331635507647918535893226185489632132933089857064204675259070915481416549859461637180270981994309924488957571282890592323326097299712084433573265489382391193259746366730583604142813883032038249037589852437441702913276561809377344403070746921120191302033038019762110110044929321516084244485963766983895228684783123552658213144957685726243344189303968642624341077322697802807318915441101044682325271620105265227211166039666557309254711055785376346682065310989652691862056476931257058635662018558100729360659876486117910453348850346113657686753249441668039626579787718556084552965412665408530614344431858676975145661406800700237877659134401712749470420562230538994561314071127000407854733269939081454664645880797270826683063432858785698305235808933065757406795457163775254202114955761581400250126228594130216471550979259230990796547376125517656751357517829666454779174501129961489030463994713296210734043751895735961458901938971311179042978285647503203198691514028708085990480109412147221317947647772622414254854540332157185306142288137585043063321751829798662237172159160771669254748738986654949450114654062843366393790039769265672146385306736096571209180763832716641627488880078692560290228472104031721186082041900042296617119637792133757511495950156604963186294726547364252308177036751590673502350728354056704038674351362222477158915049530984448933309634087807693259939780541934144737744184263129860809988868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55256375679'<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
To use each decimal digit independently, convert the character array to a numeric vector.<\/div>\n
\n
\n
decvec = decimalChar(3:end-3) - '0'<\/span><\/span><\/div>\n
\n
\n
\n
\n
decvec = <\/span>1\u00d710000<\/span><\/div>\n<\/div>\n
\n
1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0<\/div>\n
<\/div>\n
<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Create a pi patch<\/span><\/h2>\n
10,000 decimals of pi are efficiently visualized as a single Patch object<\/span>. Each digit is assigned an angular coordinate evenly distributed within 10 wedges, one wedge for each value 0 to 9. Lines segments connect adjacent digits starting with 1-to-4, 4-to-1, 1-to-5, 5-to-9, 9-to-2 and so on, moving slightly counterclockwise within each wedge. Each line segment is colored according to its length.<\/div>\n
\n
\n
% Assign each digit an angular coordinate based on its value 0:9<\/span><\/span><\/div>\n<\/div>\n
\n
nDecimals = numel(decvec);<\/span><\/div>\n<\/div>\n
\n
theta = ((0:36:324)+(0:36\/nDecimals:36)')*pi\/180;<\/span><\/div>\n<\/div>\n
\n
ang = theta(sub2ind(size(theta),1:nDecimals,decvec+1));<\/span><\/div>\n<\/div>\n
\n
<\/div>\n<\/div>\n
\n
% Compute the length of each line segment; used to set color<\/span><\/span><\/div>\n<\/div>\n
\n
[x,y] = pol2cart(ang,1);<\/span><\/div>\n<\/div>\n
\n
[~,~,d] = uniquetol(hypot(diff(x),diff(y)));<\/span><\/div>\n<\/div>\n
\n
alpha = min(0.85,max(0.006, 1000\/nDecimals));<\/span><\/div>\n<\/div>\n
\n
<\/div>\n<\/div>\n
\n
% Plot line segments using the edge property of a single Patch object<\/span><\/span><\/div>\n<\/div>\n
\n
fig = figure(Color='k'<\/span>);<\/span><\/div>\n<\/div>\n
\n
fig.Position(3:4) = 600;<\/span><\/div>\n<\/div>\n
\n
movegui(fig)<\/span><\/div>\n<\/div>\n
\n
ax = axes(fig);<\/span><\/div>\n<\/div>\n
\n
<\/div>\n<\/div>\n
\n
p = patch(ax,XData=x,YData=y,FaceColor='none'<\/span>,EdgeAlpha=alpha);<\/span><\/div>\n<\/div>\n
\n
set(p,'FaceVertexCData'<\/span>, [d;nan], 'EdgeColor'<\/span>,'Flat'<\/span>)<\/span><\/div>\n<\/div>\n
\n
colormap(ax,jet(10))<\/span><\/div>\n<\/div>\n
\n
axis equal off<\/span><\/span><\/div>\n<\/div>\n
\n
<\/div>\n<\/div>\n
\n
% Add pi character<\/span><\/span><\/div>\n<\/div>\n
\n
text(0,0.05,'\\pi'<\/span>, ...<\/span><\/span><\/div>\n<\/div>\n
\n
HorizontalAlignment='Center'<\/span>, ...<\/span><\/span><\/div>\n<\/div>\n
\n
FontUnits='normalized'<\/span>, ...<\/span><\/span><\/div>\n<\/div>\n
\n
FontSize = 0.2, ...<\/span><\/span><\/div>\n<\/div>\n
\n
Color = 'k'<\/span>);<\/span><\/div>\n
\n
\n
<\/div>\n
<\/div>\n
Run this section to add angular labels for each wedge.<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
% wedge lines<\/span><\/span><\/div>\n
th = linspace(0,2*pi,1000);<\/span><\/div>\n
gaps = 0 : pi\/5 : 2*pi;<\/span><\/div>\n
isgap = any(abs(th-gaps') < pi\/120);<\/span><\/div>\n
th(isgap) = nan;<\/span><\/div>\n
radius = 1.08;<\/span><\/div>\n
[segx,segy] = pol2cart(th,radius);<\/span><\/div>\n
hold on<\/span><\/span><\/div>\n
plot(segx,segy,'-'<\/span>,LineWidth=1,Color=[.8 .8 .8])<\/span><\/div>\n
<\/div>\n
% wedge labels<\/span><\/span><\/div>\n
midAng = gaps(1:end-1) + pi\/10;<\/span><\/div>\n
tradius = radius + .08; <\/span><\/div>\n
[tx,ty] = pol2cart(midAng,tradius);<\/span><\/div>\n
<\/div>\n
text(tx, ty, string(0:9), ...<\/span>,<\/span><\/span><\/div>\n
FontUnits='normalized'<\/span>,...<\/span><\/span><\/div>\n
FontSize=0.05, ...<\/span><\/span><\/div>\n
Color=[.8 .8 .8], ...<\/span><\/span><\/div>\n
HorizontalAlignment='center'<\/span>,...<\/span><\/span><\/div>\n
VerticalAlignment='middle'<\/span>);<\/span><\/div>\n<\/div>\n
To animate the visualization, each line segment was plotted individually and a frame was captured every 200 iterations using getframe<\/span>, written to a gif file using imwrite<\/span> with a delay of 0.2 sec.<\/div>\n
\"AnimatedPolarPi_50dFrames.gif\"<\/div>\n

Visualizing patterns in decimals<\/h2>\n
We can use this visualization to investigate patterns in several approximations of pi by replacing the symExpression<\/span> variable with sym(22\/7)<\/span>, sym(333\/106)<\/span>, sym(355\/113)<\/span>, and sym(103993)\/sym(33102)<\/span>.<\/div>\n
\"piApproximations.png\"<\/div>\n

More fun<\/h2>\n