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Python for Mathematical Visualization A Four-Dimensional Case Study David Dumas University of Illinois at Chicago May 20, 2017

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Goal Describe a mathematical visualization project that uses Python. Highlight characteristics of Python that make it suitable for this application.

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The PML Visualization Project Joint work with François Guéritaud http://dumas.io/PML

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Steps dumas.io/PML Enumerate the objects Calculate coordinates Render an image

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Topological enumeration dumas.io/PML START FINISH

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Topological enumeration dumas.io/PML START FINISH Obstacle

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Topological enumeration dumas.io/PML START FINISH

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Topological enumeration dumas.io/PML START FINISH

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Topological enumeration dumas.io/PML START FINISH

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Topological enumeration dumas.io/PML START FINISH

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Topological enumeration dumas.io/PML START FINISH

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Topological enumeration dumas.io/PML START FINISH

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Topological enumeration dumas.io/PML START FINISH Not a simple path

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Topological enumeration dumas.io/PML START FINISH

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Topological enumeration dumas.io/PML START FINISH Simple path space

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Topological enumeration dumas.io/PML Simple loops on the five-holed sphere Holes

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Topological enumeration dumas.io/PML Simple loops on the five-holed sphere START & FINISH

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Topological enumeration dumas.io/PML Simple loops on the five-holed sphere START & FINISH

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Topological enumeration dumas.io/PML Simple loops on the five-holed sphere

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Topological enumeration dumas.io/PML Simple loops on the five-holed sphere

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Topological enumeration dumas.io/PML Simple loops on the five-holed sphere

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Topological enumeration dumas.io/PML Simple loops on the five-holed sphere

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Topological enumeration dumas.io/PML Simple loops on the five-holed sphere

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Topological enumeration dumas.io/PML Simple loops on the five-holed sphere Holes

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Topological enumeration dumas.io/PML Simple loops on the five-holed sphere

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Topological enumeration dumas.io/PML Simple loops on the five-holed sphere

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Topological enumeration dumas.io/PML Simple loops on the five-holed sphere 2 3

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Topological enumeration dumas.io/PML Nonperipheral simple closed curves on the five-holed sphere 2 3

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Topological enumeration dumas.io/PML Nonperipheral simple closed curves on the five-holed sphere (NSCC) 2 3

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Topological enumeration dumas.io/PML Nonperipheral simple closed curves on the five-holed sphere (NSCC)

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Topological enumeration dumas.io/PML Nonperipheral simple closed curves on the five-holed sphere (NSCC)

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Topological enumeration dumas.io/PML Nonperipheral simple closed curves on the five-holed sphere (NSCC)

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Topological enumeration dumas.io/PML Nonperipheral simple closed curves on the five-holed sphere (NSCC)

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Topological enumeration dumas.io/PML Nonperipheral simple closed curves on the five-holed sphere (NSCC)

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Topological enumeration dumas.io/PML Nonperipheral simple closed curves on the five-holed sphere (NSCC)

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Topological enumeration dumas.io/PML Nonperipheral simple closed curves on the five-holed sphere (NSCC)

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Topological enumeration dumas.io/PML Nonperipheral simple closed curves on the five-holed sphere (NSCC)

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Topological enumeration dumas.io/PML Nonperipheral simple closed curves on the five-holed sphere (NSCC)

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Generators dumas.io/PML a

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Generators dumas.io/PML a b

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Generators dumas.io/PML a b c

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Generators dumas.io/PML a b c d

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Generators dumas.io/PML a b c d e

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Generators dumas.io/PML A B C D E

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Generators dumas.io/PML a b c d e

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Encoding curves in strings dumas.io/PML ab a b c d e

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Encoding curves in strings dumas.io/PML ab a b c d e

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Encoding curves in strings dumas.io/PML ab a b c d e

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Encoding curves in strings dumas.io/PML ab a b c d e

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Encoding curves in strings dumas.io/PML ab a b c d e

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Encoding curves in strings dumas.io/PML bc a b c d e

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Encoding curves in strings dumas.io/PML ac a b c d e

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Encoding curves in strings dumas.io/PML CA a b c d e

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Encoding curves in strings dumas.io/PML aC a b c d e

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Encoding curves in strings dumas.io/PML aC a b c d e

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Encoding curves in strings dumas.io/PML abDcd a b c d e

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Naïve generator dumas.io/PML All words in alphabet 'abcdeABCDE'. from itertools import product N = 5 # Word length for w in product('abcdeABCDE',repeat=N): print(''.join(w))

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Naïve generator dumas.io/PML Simple words in alphabet 'abcdeABCDE'. from itertools import product N = 5 # Word length for w in product('abcdeABCDE',repeat=N): if is_simple_curve(w): print(''.join(w))

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Naïve generator dumas.io/PML Simple words in alphabet 'abcdeABCDE'. from itertools import product N = 5 # Word length for w in product('abcdeABCDE',repeat=N): if is_simple_curve(w): print(''.join(w)) # TODO: Implement is_simple_curve()

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Exponential haystack dumas.io/PML aa, ab, ac, ad, ae, aB, aC, aD, aE, ba, bb, bc, bd, be, bA, bC, bD, bE, ca, cb, cc, cd, ce, cA, cB, cD, cE, da, db, dc, dd, de, dA, dB, dC, dE, ea, eb, ec, ed, ee, eA, eB, eC, eD, Ab, Ac, Ad, Ae, AA, AB, AC, AD, AE, Ba, Bc, Bd, Be, BA, BB, BC, BD, BE, Ca, Cb, Cd, Ce, CA, CB, CC, CD, CE, Da, Db, Dc, De, DA, DB, DC, DD, DE, Ea, Eb, Ec, Ed, EA, EB, EC, ED, EE

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Exponential haystack dumas.io/PML aaa, aab, aac, aad, aae, aaB, aaC, aaD, aaE, aba, abb, abc, abd, abe, abA, abC, abD, abE, aca, acb, acc, acd, ace, acA, acB, acD, acE, ada, adb, adc, add, ade, adA, adB, adC, adE, aea, aeb, aec, aed, aee, aeA, aeB, aeC, aeD, aBa, aBc, aBd, aBe, aBA, aBB, aBC, aBD, aBE, aCa, aCb, aCd, aCe, aCA, aCB, aCC, aCD, aCE, aDa, aDb, aDc, aDe, aDA, aDB, aDC, aDD, aDE, aEa, aEb, aEc, aEd, aEA, aEB, aEC, aED, aEE, baa, bab, bac, bad, bae, baB, baC, baD, baE, bba, bbb, bbc, bbd, bbe, bbA, bbC, bbD, bbE, bca, bcb, bcc, bcd, bce, bcA, bcB, bcD, bcE, bda, bdb, bdc, bdd, bde, bdA, bdB, bdC, bdE, bea, beb, bec, bed, bee, beA, beB, beC, beD, bAb, bAc, bAd, bAe, bAA, bAB, bAC, bAD, bAE, bCa, bCb, bCd, bCe, bCA, bCB, bCC, bCD, bCE, bDa, bDb, bDc, bDe, bDA, bDB, bDC, bDD, bDE, bEa, bEb, bEc, bEd, bEA, bEB, bEC, bED, bEE, caa, cab, cac, cad, cae, caB, caC, caD, caE, cba, cbb, cbc, cbd, cbe, cbA, cbC, cbD, cbE, cca, ccb, ccc, ccd, cce, ccA, ccB, ccD, ccE, cda, cdb, cdc, cdd, cde, cdA, cdB, cdC, cdE, cea, ceb, cec, ced, cee, ceA, ceB, ceC, ceD, cAb, cAc, cAd, cAe, cAA, cAB, cAC, cAD, cAE, cBa, cBc, cBd, cBe, cBA, cBB, cBC, cBD, cBE, cDa, cDb, cDc, cDe, cDA, cDB, cDC, cDD, cDE, cEa, cEb, cEc, cEd, cEA, cEB, cEC, cED, cEE, daa, dab, dac, dad, dae, daB, daC, daD, daE, dba, dbb, dbc, dbd, dbe, dbA, dbC, dbD, dbE, dca, dcb, dcc, dcd, dce, dcA, dcB, dcD, dcE, dda, ddb, ddc, ddd, dde, ddA, ddB, ddC, ddE, dea, deb, dec, ded, dee, deA, deB, deC, deD, dAb, dAc, dAd, dAe, dAA, dAB, dAC, dAD, dAE, dBa, dBc, dBd, dBe, dBA, dBB, dBC, dBD, dBE, dCa, dCb, dCd, dCe, dCA, dCB, dCC, dCD, dCE, dEa, dEb, dEc, dEd, dEA, dEB, dEC, dED, dEE, eaa, eab, eac, ead, eae, eaB, eaC, eaD, eaE, eba, ebb, ebc, ebd, ebe, ebA, ebC, ebD, ebE, eca, ecb, ecc, ecd, ece, ecA, ecB, ecD, ecE, eda, edb, edc, edd, ede, edA, edB, edC, edE, eea, eeb, eec, eed, eee, eeA, eeB, eeC, eeD, eAb, eAc, eAd, eAe, eAA, eAB, eAC, eAD, eAE, eBa, eBc, eBd, eBe, eBA, eBB, eBC, eBD, eBE, eCa, eCb, eCd, eCe, eCA, eCB, eCC, eCD, eCE, eDa, eDb, eDc, eDe, eDA, eDB, eDC, eDD, eDE, Aba, Abb, Abc, Abd, Abe, AbA, AbC, AbD, AbE, Aca, Acb, Acc, Acd, Ace, AcA, AcB, AcD, AcE, Ada, Adb, Adc, Add, Ade, AdA, AdB, AdC, AdE, Aea, Aeb, Aec, Aed, Aee, AeA, AeB, AeC, AeD, AAb, AAc, AAd, AAe, AAA, AAB, AAC, AAD, AAE, ABa, ABc, ABd, ABe, ABA, ABB, ABC, ABD, ABE, ACa, ACb, ACd, ACe, ACA, ACB, ACC, ACD, ACE, ADa, ADb, ADc, ADe, ADA, ADB, ADC, ADD, ADE, AEa, AEb, AEc, AEd, AEA, AEB, AEC, AED, AEE, Baa, Bab, Bac, Bad, Bae, BaB, BaC, BaD, BaE, Bca, Bcb, Bcc, Bcd, Bce, BcA, BcB, BcD, BcE, Bda, Bdb, Bdc, Bdd, Bde, BdA, BdB, BdC, BdE, Bea, Beb, Bec, Bed, Bee, BeA, BeB, BeC, BeD, BAb, BAc, BAd, BAe, BAA, BAB, BAC, BAD, BAE, BBa, BBc, BBd, BBe, BBA, BBB, BBC, BBD, BBE, BCa, BCb, BCd, BCe, BCA, BCB, BCC, BCD, BCE, BDa, BDb, BDc, BDe, BDA, BDB, BDC, BDD, BDE, BEa, BEb, BEc, BEd, BEA, BEB, BEC, BED, BEE, Caa, Cab, Cac, Cad, Cae, CaB, CaC, CaD, CaE, Cba, Cbb, Cbc, Cbd, Cbe, CbA, CbC, CbD, CbE, Cda, Cdb, Cdc, Cdd, Cde, CdA, CdB, CdC, CdE, Cea, Ceb, Cec, Ced, Cee, CeA, CeB, CeC, CeD, CAb, CAc, CAd, CAe, CAA, CAB, CAC, CAD, CAE, CBa, CBc, CBd, CBe, CBA, CBB, CBC, CBD, CBE, CCa, CCb, CCd, CCe, CCA, CCB, CCC, CCD, CCE, CDa, CDb, CDc, CDe, CDA, CDB, CDC, CDD, CDE, CEa, CEb, CEc, CEd, CEA, CEB, CEC, CED, CEE, Daa, Dab, Dac, Dad, Dae, DaB, DaC, DaD, DaE, Dba, Dbb, Dbc, Dbd, Dbe, DbA, DbC, DbD, DbE, Dca, Dcb, Dcc, Dcd, Dce, DcA, DcB, DcD, DcE, Dea, Deb, Dec, Ded, Dee, DeA, DeB, DeC, DeD, DAb, DAc, DAd, DAe, DAA, DAB, DAC, DAD, DAE, DBa, DBc, DBd, DBe, DBA, DBB, DBC, DBD, DBE, DCa, DCb, DCd, DCe, DCA, DCB, DCC, DCD, DCE, DDa, DDb, DDc, DDe, DDA, DDB, DDC, DDD, DDE, DEa, DEb, DEc, DEd, DEA, DEB, DEC, DED, DEE, Eaa, Eab, Eac, Ead, Eae, EaB, EaC, EaD, EaE, Eba, Ebb, Ebc, Ebd, Ebe, EbA, EbC, EbD, EbE, Eca, Ecb, Ecc, Ecd, Ece, EcA, EcB, EcD, EcE, Eda, Edb, Edc, Edd, Ede, EdA, EdB, EdC, EdE, EAb, EAc, EAd, EAe, EAA, EAB, EAC, EAD, EAE, EBa, EBc, EBd, EBe, EBA, EBB, EBC, EBD, EBE, ECa, ECb, ECd, ECe, ECA, ECB, ECC, ECD, ECE, EDa, EDb, EDc, EDe, EDA, EDB, EDC, EDD, EDE, EEa, EEb, EEc, EEd, EEA, EEB, EEC, EED, EEE

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Exponential haystack dumas.io/PML

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Twisting dumas.io/PML bc a b c d e cd

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Twisting dumas.io/PML bc a b c d e cd

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Twisting dumas.io/PML bc a b c d e ? cd

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Twisting dumas.io/PML bc a b c d e ? cd

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Twisting dumas.io/PML bc a b c d e ? cd

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Twisting dumas.io/PML bc a b c d e bd Tcd

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Twisting dumas.io/PML bc a b c d e bd Tcd bDcd 2

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Twisting is search & replace dumas.io/PML Tcd is equivalent to substituting: a → a b → b c → d d → Dcd e → e

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Twisting is search & replace dumas.io/PML Tab Tbc Tcd Tde Tea a → b a a a Aea b → Bab c b b b c → c Cbc d c c d → d d Dcd e d e → e e e Ede a

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Twists generate everything dumas.io/PML Fact: Any nonperipheral simple closed curve can be transformed to any other one by applying twists Tab , Tbc , Tcd , Tde , Tea and their inverses.

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Twist-based generator dumas.io/PML seed curves

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Twist-based generator dumas.io/PML twists seed curves

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Twist-based generator dumas.io/PML twists frontier

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Twist-based generator dumas.io/PML twists frontier

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Twist-based generator dumas.io/PML twists frontier

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Twist-based generator dumas.io/PML curves

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Twist-based generator dumas.io/PML

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Twist-based generator dumas.io/PML

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Twist-based generator dumas.io/PML

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Twist-based generator dumas.io/PML

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Twist-based generator dumas.io/PML

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Twist-based generator dumas.io/PML

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Twist-based generator dumas.io/PML curves

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Twist-based generator dumas.io/PML Python’s set type stores collections of distinct elements and supports efficient boolean operations. depth = 5 twists = { Tab, Tab_inv, Tbc, Tbc_inv } # etc curves = set() frontier = {'ab', 'bc', 'cd', 'de', 'ea'} for _ in range(depth): latest = \ { T(x) for x in frontier for T in twists } frontier = latest.difference(curves) curves.update(frontier)

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Coordinates dumas.io/PML How to visualize the entire set of NSCCs? Assign coordinates.

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Coordinates dumas.io/PML How to visualize the entire set of NSCCs? Assign coordinates. In 1986, W. P. Thurston described a way to associate a 4-dimensional vector to each NSCC on the five-holed sphere.

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Coordinates dumas.io/PML How to visualize the entire set of NSCCs? Assign coordinates. In 1986, W. P. Thurston described a way to associate a 4-dimensional vector to each NSCC on the five-holed sphere. The resulting cloud of points densely fills a 3-dimensional hypersurface in 4-space. This hypersurface is PML.

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Length & dlength dumas.io/PML Curve x Tiling matrix (x) length L(x)

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Length & dlength dumas.io/PML Curve x Tiling matrix (x) length L(x) 6.467 . . .

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Length & dlength dumas.io/PML Curve x Tiling matrix (x) length L(x) 6.467 . . . 1 1 1 1

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Length & dlength dumas.io/PML Curve x Tiling matrix (x) length L(x) 6.467 . . . 2 2 2 2

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Computing matrices dumas.io/PML Numpy’s matrix algebra + recursive splitting to compute ρ(x): import numpy as np def representation(gen_matrices): def _rho(x): if len(x) == 0: # identity matrix return np.eye(2) elif len(x) == 1: # generator matrix return gen_matrices[x] else: N = len(x) return np.dot(_rho(x[:N//2]),_rho(x[N//2:])) return _rho rho = representation( {'a': np.array( [[0,1],[1,1]] )} ) # etc print(rho('aaaaa')) # -> [[3 5], [5 8]]

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Length and dlength dumas.io/PML eps = 0.0001 rho0 = representation( {'a': np.array( [[0,1],[1,1]] )} ) rho1 = representation( {'a': np.array( [[0,1],[1+eps,1]] )} ) L0 = 2.0*np.arccosh(0.5*np.trace(rho0('aaaaa'))) L1 = 2.0*np.arccosh(0.5*np.trace(rho1('aaaaa'))) dL = (L1 - L0) / eps print(dL) Four calculations like this give the four coordinates of the vector dL associated to a curve.

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Length and dlength dumas.io/PML eps = 0.0001 rho0 = representation( {'a': np.array( [[0,1],[1,1]] )} ) rho1 = representation( {'a': np.array( [[0,1],[1+eps,1]] )} ) L0 = 2.0*np.arccosh(0.5*np.trace(rho0('aaaaa'))) L1 = 2.0*np.arccosh(0.5*np.trace(rho1('aaaaa'))) dL = (L1 - L0) / eps print(dL) Four calculations like this give the four coordinates of the vector dL associated to a curve. Next problem: 4D space is difficult to visualize!

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Stereographic projection dumas.io/PML “Open up” the surface in 4-space and flatten to 3-space.

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Stereographic projection dumas.io/PML “Open up” the surface in 4-space and flatten to 3-space. (p1 , p2 , p3 , p4 ) −→ ( p1 1 − p4 , p2 1 − p4 , p3 1 − p4 )

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Stereographic projection dumas.io/PML “Open up” the surface in 4-space and flatten to 3-space. (p1 , p2 , p3 , p4 ) −→ ( p1 1 − p4 , p2 1 − p4 , p3 1 − p4 )

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Stereographic projection dumas.io/PML “Open up” the surface in 4-space and flatten to 3-space. (p1 , p2 , p3 , p4 ) −→ ( p1 1 − p4 , p2 1 − p4 , p3 1 − p4 )

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Stereographic projection dumas.io/PML “Open up” the surface in 4-space and flatten to 3-space. (p1 , p2 , p3 , p4 ) −→ ( p1 1 − p4 , p2 1 − p4 , p3 1 − p4 )

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Stereographic projection dumas.io/PML “Open up” the surface in 4-space and flatten to 3-space. (p1 , p2 , p3 , p4 ) −→ ( p1 1 − p4 , p2 1 − p4 , p3 1 − p4 )

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Rendering dumas.io/PML Projecting each dL to a 3-vector (x, y, z), we obtain data like word L x y z aDBdbd 19.195 0.630 0.729 0.695 aDbcBd 16.790 0.596 0.470 1.133 bcbCBd 14.664 0.177 0.544 1.242 with > 106 rows. To make short curves more prominent, but all curves visible, we draw a sphere centered at (x, y, z) of radius 1/L for each curve.

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Rendering dumas.io/PML Projecting each dL to a 3-vector (x, y, z), we obtain data like word L x y z aDBdbd 19.195 0.630 0.729 0.695 aDbcBd 16.790 0.596 0.470 1.133 bcbCBd 14.664 0.177 0.544 1.242 with > 106 rows. To make short curves more prominent, but all curves visible, we draw a sphere centered at (x, y, z) of radius 1/L for each curve. Easy way to draw millions of spheres if real-time rendering is not essential?

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Rendering dumas.io/PML Party like it’s 1992

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Rendering dumas.io/PML Party like it’s 1992 with ray-tracing!

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POV-Ray dumas.io/PML POV-Ray is an open-source ray tracer that uses a scene description language (SDL) with syntax reminiscent of C. We generate SDL sphere primitives from the dL data using: outfile.write('sphere { <%f, %f, %f>, %f }\n' % (x,y,z,1.0/L)) → Appending some camera and lighting setup boilerplate gives a full scene file for rendering with POV-Ray.

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dumas.io/PML Fractal dust pmls05-001

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dumas.io/PML Rotating the pole (wide angle) pmls05-010

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dumas.io/PML Rotating the pole (close-up) pmls05-020

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dumas.io/PML Clifford flow pmls05-030

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Rings dumas.io/PML

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Rings dumas.io/PML

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dumas.io/PML pmls05-071

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dumas.io/PML Rotating the pole pmls05-041

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dumas.io/PML Rotating the pole II pmls05-061

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Twists dumas.io/PML

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Twists dumas.io/PML

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Twists dumas.io/PML

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Twists dumas.io/PML

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Twists dumas.io/PML

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Twists dumas.io/PML

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dumas.io/PML pmls05-081

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Twists dumas.io/PML

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Rings poster dumas.io/PML

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David Dumas david@dumas.io http://dumas.io github.com/daviddumas