Calabi triangle

The Calabi triangle is a special triangle found by Eugenio Calabi and defined by its property of having three different placements for the largest square that it contains.[1] It is an isosceles triangle which is obtuse with an irrational but algebraic ratio between the lengths of its sides and its base.[2]

Definition

Consider the largest square that can be placed in an arbitrary triangle. It may be that such a square could be positioned in the triangle in more than one way. If the largest such square can be positioned in three different ways, then the triangle is either an equilateral triangle or the Calabi triangle.[3][4] Thus, the Calabi triangle may be defined as a triangle that is not equilateral and has three placements for its largest square.

Shape

The triangle ABC is isosceles which has the same length of sides as AB = AC. If the ratio of the base to either leg is x, we can set that AB = AC = 1, BC = x. Then we can consider the following three cases:

case 1) ABC is acute triangle
The condition is .
In this case x = 1 is valid for equilateral triangle.
case 2) ABC is right triangle
The condition is .
In this case no value is vaild.
case 3) ABC is obtuse triangle
The condition is .
In this case the Calabi triangle is valid for the value x of .
Example of answer
Example figure of Calabi triangle 01
Example figure of Calabi triangle 01

Consider the case of AB = AC = 1, BC = x. Then

Let a base angle be θ and a square be DEFG on base BC with it's side length as a. Let H be the foot of the perpendicular drawn from the apex A to the base. Then

Then HB = x/2 and HE = a/2, so EB = x - a/2.

From △DEB ∽ △AHB,

case 1) ABC is acute triangle

Example figure of Calabi triangle 02
Example figure of Calabi triangle 02

Let IJKL be a square on side AC with it's side length as b. From △ABC ∽ △IBJ,

From △JKC ∽ △AHC,

Then

Therefore if two squares are congruent,

In this case,

Therefore , it means that ABC is equilateral triangle..

case 2) ABC is right triangle

Example figure of Calabi triangle 03
Example figure of Calabi triangle 03

In this case, , so

Then no value is valid.

case 3) ABC is obtuse triangle

Example figure of Calabi triangle 04
Example figure of Calabi triangle 04

Let IJKA be a square on base AC with it's side length as b.

From △AHC ∽ △JKC,

Therefore if two squares are congruent,

In this case,

So, we can input the value of tanθ,

In this case, , we can get the following equation:

Root of Calabi's equation

If x is the largest positive root of Calabi's equation:

we can calculate the value of x by following methods.

Newton's method

We can set the function f : ℝ → ℝ as follows:

The function f is continuous and differentiable on and

Then f is monotonically increasing function and by Intermediate value theorem, the Calabi's equation f(x) = 0 has unique solution in open interval .

The value of x is calculated by Newton's method as follows:

Newton's method for the root of Calabi's equation
NOitaration value
x0 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
x1 1.5894336937532359661730828318788879137009030615937450512751148453786767135955334258126245234229958264
x2 1.5532494304937542880726766543978248923187129559278453297672175091495211072007305767791656852489509532
x3 1.5513923438394291214261302957041311730647158998768913729601250520981843884320426580607217084988450228
x4 1.5513875245807424405653864101010664961190807601032843229099732387605785283093834429009245515620370908
x5 1.5513875245483203922634199481329355594583673201569148697180882650868527691911329413038054294821923335
x6 1.5513875245483203922619525102646238151635947098682124174996287494182837608373482038378355220938329884
x7 1.5513875245483203922619525102646238151635917038038871995280071201179267425542569573083751082345375878
x8 1.5513875245483203922619525102646238151635917038038871995280071201179267425542569572957604536120254363
x9 1.5513875245483203922619525102646238151635917038038871995280071201179267425542569572957604536120254363

Cardano's method

The value of x can expressed with complex numbers by using Cardano's method:

[3][5][lower-alpha 1]

Viète's method

The value of x can also be expressed without complex numbers by using Viète's method:

[2]

Lagrange's method

The value of x has continued fraction representation by Lagrange's method as follows:
[1, 1, 1, 4, 2, 1, 2, 1, 5, 2, 1, 3, 1, 1, 390, ...] =

.[3][6][7][lower-alpha 2]

base angle and apex angle

The Calabi triangle is obtuse with base angle θ and apex angle ψ as follows:

See also

Footnotes

Notes

  1. If we set the polar form of complex number, we can calculate the value of x as follows:
    Then this Cardano's method is equivalent as Viète's method.
  2. If a continued fraction [a0, a1, a2, ...] are found, with numerators h1, h2, ... and denominators k1, k2, ... then the relevant recursive relation is that of Gaussian brackets:
    hn = anhn − 1 + hn − 2,
    kn = ankn − 1 + kn − 2.
    The successive convergents are given by the formula
    hn/kn = anhn − 1 + hn − 2/ankn − 1 + kn − 2.
    If the continued fraction is
    [1, 1, 1, 4, 2, 1, 2, 1, 5, 2, 1, 3, 1, 1, 390, 1, 1, 2, 11, 6, 2, 1, 1, 56, 1, 4, 3, 1, 1, 6, 9, 3, 2, 1, 8, 10, 9, 25, 2, 1, 3, 1, 3, 5, 2, 35, 1, 1, 1, 41, 1, 2, 2, 1, 2, 2, 3, 1, 4, 2, 1, 1, 1, 1, 3, 1, 6, 2, 1, 4, 11, 1, 2, 2, 1, 1, 6, 3, 1, 1, 1, 1, 1, 1, 4, 1, 7, 2, 2, 2, 36, 7, 22, 1, 2, 1, ...],[8]
    we can calculate the rational approxmation of x is as follows:
    The value of numerators hn and denominators kn of continued fraction
    n an hn kn
    -2
    0 1
    -1
    1 0
    0 1 1 1
    1 1 2 1
    2 1 3 2
    3 4 14 9
    4 2 31 20
    5 1 45 29
    6 2 121 78
    7 1 166 107
    8 5 951 613
    9 2 2068 1333
    10 1 3019 1946
    11 3 11125 7171
    12 1 14144 9117
    13 1 25269 16288
    14 390 9869054 6361437
    15 1 9894323 6377725
    16 1 19763377 12739162
    17 2 49421077 31856049
    18 11 563395224 363155701
    19 6 3429792421 2210790255
    20 2 7422980066 4784736211
    21 1 10852772487 6995526466
    22 1 18275752553 11780262677
    23 56 1034294915455 666690236378
    24 1 1052570668008 678470499055
    25 4 5244577587487 3380572232598
    26 3 16786303430469 10820187196849
    27 1 22030881017956 14200759429447
    28 1 38817184448425 25020946626296
    29 6 254933987708506 164326439187223
    30 9 2333223073824979 1503958899311303
    31 3 7254603209183443 4676203137121132
    32 2 16842429492191865 10856365173553567
    33 1 24097032701375308 15532568310674699
    34 8 209618691103194329 135116911658951159
    35 10 2120283943733318598 1366701684900186289
    36 9 19292174184703061711 12435432075760627760
    37 25 484424638561309861373 312252503578915880289
    38 2 988141451307322784457 636940439233592388338
    39 1 1472566089868632645830 949192942812508268627
    40 3 5405839720913220721947 3484519267671117194219
    41 1 6878405810781853367777 4433712210483625462846
    42 3 26041057153258780825278 16785655899121993582757
    43 5 137083691577075757494167 88361991706093593376631
    44 2 300208440307410295813612 193509639311309180336019
    45 35 10644379102336436110970587 6861199367601914905137296
    46 1 10944587542643846406784199 7054709006913224085473315
    47 1 21588966644980282517754786 13915908374515138990610611
    48 1 32533554187624128924538985 20970617381428363076083926
    49 41 1355464688337569568423853171 873711221013078025110051577
    50 1 1387998242525193697348392156 894681838394506388186135503
    51 2 4131461173387956963120637483 2663074897802090801482322583
    52 2 9650920589301107623589667122 6220831633998687991150780669
    53 1 13782381762689064586710304605 8883906531800778792633103252
    54 2 37215684114679236797010276332 23988644697600245576416987173
    55 2 88213749992047538180730857269 56861195927001269945467077598
    56 3 301856934090821851339202848139 194572232478604055412818219967
    57 1 390070684082869389519933705408 251433428405605325358285297565
    58 4 1862139670422299409418937669771 1200305946101025356845959410227
    59 2 4114350024927468208357809044950 2652045320607656039050204118019
    60 1 5976489695349767617776746714721 3852351266708681395896163528246
    61 1 10090839720277235826134555759671 6504396587316337434946367646265
    62 1 16067329415627003443911302474392 10356747854025018830842531174511
    63 1 26158169135904239270045858234063 16861144441341356265788898820776
    64 3 94541836823339721254048877176581 60940181178049087628209227636839
    65 1 120700005959243960524094735410644 77801325619390443893998126457615
    66 6 818741872578803484398617289640445 527748134894391750992197986382529
    67 2 1758183751116850929321329314691534 1133297595408173945878394099222673
    68 1 2576925623695654413719946604331979 1661045730302565696870592085605202
    69 4 12065886245899468584201115732019450 7777480516618436733360762441643481
    70 11 135301674328589808839932219656545929 87213331413105369763838978943683493
    71 1 147367560574489277424133335388565379 94990811929723806497199741385326974
    72 2 430036795477568363688198890433676687 277194955272552982758238461714337441
    73 2 1007441151529626004800531116255918753 649380722474829772013676664814001856
    74 1 1437477947007194368488730006689595440 926575677747382754771915126528339297
    75 1 2444919098536820373289261122945514193 1575956400222212526785591791342341153
    76 6 16106992538228116608224296744362680598 10382314079080657915485465874582386215
    77 3 50765896713221170197962151356033555987 32722898637464186273241989415089499798
    78 1 66872889251449286806186448100396236585 43105212716544844188727455289671886013
    79 1 117638785964670457004148599456429792572 75828111354009030461969444704761385811
    80 1 184511675216119743810335047556826029157 118933324070553874650696899994433271824
    81 1 302150461180790200814483647013255821729 194761435424562905112666344699194657635
    82 1 486662136396909944624818694570081850886 313694759495116779763363244693627929459
    83 1 788812597577700145439302341583337672615 508456194919679684876029589392822587094
    84 4 3641912526707710526382028060903432541346 2347519539173835519267481602264918277835
    85 1 4430725124285410671821330402486770213961 2855975734093515204143511191657740864929
    86 7 34656988396705585229131340878310824039073 22339349677828441948272059943869104332338
    87 2 73744701917696581130084012159108418292107 47534675089750399100687631079395949529605
    88 2 182146392232098747489299365196527660623287 117408699857329240149647322102661003391548
    89 2 438037486381894076108682742552163739538681 282352074804408879399982275284717956312701
    90 36 15951495901980285487401878097074422284015803 10282083392816048898549009232352507430648784
    91 7 112098508800243892487921829422073119727649302 72256935824516751169243046901752269970854189
    92 22 2482118689507345920221682125382683056292300447 1599934671532184574621896041070902446789440942
    93 1 2594217198307589812709603954804756176019949749 1672191607356701325791139087972654716760295131
    94 2 7670553086122525545640890034992195408332199945 4944317886245587226204174217016211880310031204
    95 1 10264770284430115358350493989796951584352149694 6616509493602288551995313304988866597070326335
    The rational approxmation of x is h95/k95 and an error bounds ε is as follows:

Citations

  1. Calabi, Eugenio (3 Nov 1997). "Outline of Proof Regarding Squares Wedged in Triangle". Archived from the original on 12 December 2012. Retrieved 3 May 2018.
  2. Stewart 2004, p. 15.
  3. Weisstein, Eric W. "Calabi's Triangle". MathWorld.
  4. Conway, J.H.; Guy, R.K. (1996). "Calabi's Triangle". The Book of Numbers. New York: Springer-Verlag. p. 206.
  5. Stewart 2004, pp. 7–10.
  6. Joseph-Louis, Lagrange (1769), "Sur la résolution des équations numériques", Mémoires de l'Académie royale des Sciences et Belles-lettres de Berlin, 23 - Œuvres II, p.539-578.
  7. Joseph-Louis, Lagrange (1770), "Additions au mémoire sur la résolution des équations numériques", Mémoires de l'Académie royale des Sciences et Belles-lettres de Berlin, 24 - Œuvres II, p.581-652.
  8. (sequence A046096 in the OEIS)

References

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