Miscellaneous math trivia that I don't have a good place for:

• Bifurcation diagrams

  	High-resolution bifurcation diagram of the logistic map xn+1 = rxn(1 - xn) as r goes from -2 to 4.
  	Detail of just the chaotic region from 3.56... to 4, with colors chosen to highlight some of the details.
  	Real parts of logistic map orbit points for complex r. Note that along the real line this corresponds to the standard 2-dimensional bifurcation diagrams shown above. Chaotic filaments other than the ones along the real line have been removed from this diagram for clarity. (The moire patterns seen on the surfaces are artifacts of the algorithm I used to generate this diagram. They shouldn't really be there.)

• Feigenbaum point of the logistic map

  The accumulation point of the initial cascade of period-doubling bifurcations (the Fiegenbaum point, where chaos begins) of the logistic map

  x_n+1 = rx_n(1 - x_n)

  is (probably)

  r_∞ ≈ 3.569945671870944901842005151386498936763836911514832378107975529921.

  This number was obtained by applying Wynn's epsilon algorithm (Mathematica's SequenceLimit ) to the sequence of values s_k less than r_∞ for which there is a superstable orbit of period 2^k. Here are the values s_k (obtained using Mathematica's InterpolateRoot), the ratios of their successive differences (which approach Feigenbaum's constant δ (high-precision)), and the result of applying Wynn's epsilon algorithm to the s_i up through s_k:

  k	Period	sk
  		sk - 1 - sk - 2 sk - sk - 1
  		Wynn
  0	20	2.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
  1	21	3.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
  2	22	3.498561699327701519998945381944539267886879036544426836572823638523385321243421028730810326484984401209228956102701
  		4.70894301354050331317654831774940588489937713693388586328734328463287309984427635487983913742171879
  3	23	3.55464086276882486536608185194849179182720001411430077699189722300180457537926270994088921878146246402948086932391222906846021441992712
  		4.68077099801069538163085065277607751050409025167936478280760178578837705547213746550186040787231064
  		3.569
  4	24	3.56666737985626851397263115745536809193795406600135721602859741834854866938184350179211105329000710120346175491605712223282276348165160021902958772
  		4.6629596111141025840102402120246233524704006984860871020445705151655373170682130326099444289824855360093424248869
  		3.569
  5	25	3.56924353163711033780824951091274555817662944104831530104686151721655393769709204113086384370114449867692654043724486963960226278754616641049760282070250
  		4.66840392591840023782173976316803489717188841829836912293353401456363929773457592311468955011787600127066223679434521317898031430930
  		3.5700
  6	26	3.56979529374994462051535252960697797567745917676502626322596394460629421029814460918934592566961921162608103
  		4.66895374096762278100225701336664038524138179324338912835235483037000301817746290913103770686124445041453
  		3.5700
  7	27	3.5699134654223485148409735196680118263186321889073686299704621656350089306235880668031885032433091981974660354505219530885409
  		4.6691571813288434823178489841538326584176522980723438128033504859700041692254275967309699629197851892701
  		3.56994567
  8	28	3.56993877423330548779344606756298692636115014624332439706973965078701729230296041671253280151443518691556964883775279625269419235851794128056
  		4.6691910024850961508886384643570191225472608959960444998081722301207869363703024135734006004971745610745
  		3.56994567
  9	29	3.569944194608064933243633343872173336470096797608740397228669656040271760698888744089005123993117100995416965550620687671732855083896080347106660002339808
  		4.6691994705477257746860463581920147713321370005858606726741742636515594370912340821819072485969470190058479777864999335
  		3.569945671872
  10	210	3.569945355486468580892802382308164248816385027775574063964940488091265203231781598749446905833720799577353723850730946175017984859174803141205234858763289
  		4.66920113460104223541026773993253799322632233219761236264738934971772101534764670513311477165323818535233889569135631288028890098077267
  		3.569945671872
  11	211	3.5699456041110784381341172613816007521225000068803685272166043045405094665424281206359689997458308877734781457913815037809005914529093687189432067744502476
  		4.6692015095135523273469838873237076943956749664002544894697819836827055196618920101593119766059856014859044385293416576061172315989491243061763282
  		3.5699456718709449
  12	212	3.5699456573588564997296076145681267526604102684991967637705852002354713653820893854741375444174208995023400
  		4.66920158752238550757470344212514649449162641289165970962239842580163164026071686795808041677125498
  		3.5699456718709449
  13	213	3.569945668762899968347097669809200160065831076763823961161157531230505461262313467289941012104908314550386159320314
  		4.6692016045121851859714945959568992666067149916147887120687377861307128005390245855422457402100074
  		3.5699456718709449018420
  14	214	3.56994567120529685452891409409151815971268146867923538325188489117359807780036511698705615118783856867189259381328464079965
  		4.6692016081159352225771651388641873237572619453682727903711618877636658049316868079191083134962588
  		3.5699456718709449018420
  15	215	3.5699456717283834742050690192096973333697091450035687234702498642248558107577980776922112580271311050564027593348169476157459996739
  		4.66920160889206909737490206806375395283166615984691490783311611272498663683097155472781545785352264101556
  		3.56994567187094490184200515
  16	216	3.569945671840412609691834470086924203043324392655270183627732448881249291618000928932976044327232972606101781150973007047781426749448637876
  		4.6692016090577589360884198360692639923388764224050709425223794975802047253043020355424485531773740887756539423048
  		3.56994567187094490184200515
  17	217	3.56994567186440581985191574318149894373664287687425000121803137763086363436836487704811164166019855222403330892576577947141862253747052684997090126
  		4.66920160909331068907965159127987235700513634021570608027198676781282791811398117642475703827050593914231033854319310534
  		3.56994567187094490184200515138650
  18	218	3.569945671869544430725628114908079602302766997206856548715089125616937571570700213305548689349306358814489176014059143651386138091707501276364081036238034
  		4.669201609100916617031867535103088179904518167072883968893045971368093389678210140833999841827299918109632704181517170049380326
  		3.56994567187094490184200515138650
  19	219	3.5699456718706449638154168955016539851271981094979676818237388138845167213547758096336642220318698871046160487947041536625843398640326887138178014816160000
  		4.66920160910254658398034633352797212937120981896195060199171642222094287526638152393432555384368030515924269878504022316544833396648834
  		3.5699456718709449018420051513864989367
  20	220	3.569945671870880664301693440872144320183939122416360922551500755137189364100209334238092966937967896150205
  		4.6692016091028955480853525736039363228255966628462011085045843838361040244062941160334802876
  		3.5699456718709449018420051513864989367
  21	221	3.56994567187093114412801257624867419534500998282257762851335081021589260842642501311663052372743420197212165050
  		4.669201609102970300954675484286582569285190973417485835915652284786865541105829545422183904
  		3.56994567187094490184200515138649893676384
  22	222	3.56994567187094195536096955991606199564113617899402695670379311690673043086105308263537200792602280391703073254790327
  		4.6692016091029863088203986300609522560298319488438087216458961208440681589253745705223183797
  		3.56994567187094490184200515138649893676384
  23	223	3.569945671870944270795997232802598563057451448460475230687896220363618193105347151497746673059192594692067124825287531576
  		4.66920160910298973745117233308210663204846545823928682721815605261400807901122137155756471044750
  		3.569945671870944901842005151386498936763836911515
  24	224	3.56994567187094476669127473779543298333956279880066594388567932208629010605318140691082252773459394394622105050979884636329
  		4.6692016091029904717296895095887009764841225665564472613696607625629114690908768330484474766581779187
  		3.569945671870944901842005151386498936763836911515
  25	225	3.56994567187094487289685740918927697924105347326346859908782692959406898557138759534758431325019823688682280745134542981751
  		4.6692016091029906289932727584157566970916093120085001939210607930111252793971819647584460463941404788069
  		3.5699456718709449018420051513864989367638369115148323780
  26	226	3.569945671870944895642840890463113763785046551033922686155516901881889151370836863376334101691887987298055482616527870099
  		4.669201609102990662673869952616073857728872071059120829959925859220433964212418988932262897858215589021
  		3.5699456718709449018420051513864989367638369115148323780
  27	227	3.569945671870944900514334003317499924753205388837860560980964705972302849303979606066162736300811749356553592377789537882
  		4.66920160910299066988727767936710719068872179673817309944728111526369526804622921736731846979815596396
  		3.56994567187094490184200515138649893676383691151483237810797553
  28	228	3.569945671870944901557658648660937579518760408443690751065465066846957149767695153556804553788778939958270179090633576351
  		4.6692016091029906714321618922583559593660763451650219516660449699345313532832148691691379327573053514
  		3.56994567187094490184200515138649893676383691151483237810797553
  29	229	3.56994567187094490178110683802791433874364086692995699841913790055622208055364374955350057688487572617511988437902416292
  		4.6692016091029906717630296258749633720588530656335875966905204858109237750701590177565791820300588738
  		3.569945671870944901842005151386498936763836911514832378107975529921

• Rotation matrix for three dimensions

  In Cartesian coordinates, the image v' = (v_x', v_y', v_z') of vector v = (v_x, v_y, v_z) rotated by θ radians in a right-handed direction (counterclockwise when looking toward the origin) about the axis defined by the unit vector a = (a_x, a_y, a_z) is given by

  v' = R_a(θ) v,

  where

  0

  -azθ

  ayθ

  ax2 + (1 - ax2) cos θ

  axay(1 - cos θ) - az sin θ

  azax(1 - cos θ) + ay sin θ

  Ra(θ) = exp

  azθ

  0

  -axθ

  =

  axay(1 - cos θ) + az sin θ

  ay2 + (1 - ay2) cos θ

  ayaz(1 - cos θ) - ax sin θ

  .

  -ayθ

  axθ

  0

  azax(1 - cos θ) - ay sin θ

  ayaz(1 - cos θ) + ax sin θ

  az2 + (1 - az2) cos θ

  A Mathematica definition for this function is:

  RotationMatrix3D[{ax_, ay_, az_}, theta_] :=
      MatrixExp[{{0, -az theta, ay theta}, {az theta, 0, -ax theta},
                 {-ay theta, ax theta, 0}}]

  or, equivalently,

  RotationMatrix3D[{ax_, ay_, az_}, theta_] :=
      {{ax^2 + (1 - ax^2)*Cos[theta], ax*ay*(1 - Cos[theta]) - az*Sin[theta],
        az*ax*(1 - Cos[theta]) + ay*Sin[theta]},
       {ax*ay*(1 - Cos[theta]) + az*Sin[theta], ay^2 + (1 - ay^2)*Cos[theta],
        ay*az*(1 - Cos[theta]) - ax*Sin[theta]},
       {az*ax*(1 - Cos[theta]) - ay*Sin[theta],
        ay*az*(1 - Cos[theta]) + ax*Sin[theta], az^2 + (1 - az^2)*Cos[theta]}}