Calabi triangle
![](http://upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Calabi_triangle.svg/300px-Calabi_triangle.svg.png)
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 valid.
- case 3) △ABC is obtuse triangle
- The condition is .
- In this case the Calabi triangle is valid for the largest positive root of at (OEIS: A046095).
![Example figure of Calabi triangle 01](http://upload.wikimedia.org/wikipedia/commons/thumb/0/07/Example_figure_of_Calabi_triangle_01.svg/220px-Example_figure_of_Calabi_triangle_01.svg.png)
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 its 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](http://upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Example_figure_of_Calabi_triangle_02.svg/220px-Example_figure_of_Calabi_triangle_02.svg.png)
Let □IJKL be a square on side AC with its 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](http://upload.wikimedia.org/wikipedia/commons/thumb/5/53/Example_figure_of_Calabi_triangle_03.svg/400px-Example_figure_of_Calabi_triangle_03.svg.png)
In this case, , so
Then no value is valid.
case 3) △ABC is obtuse triangle
![Example figure of Calabi triangle 04](http://upload.wikimedia.org/wikipedia/commons/thumb/2/23/Example_figure_of_Calabi_triangle_04.svg/475px-Example_figure_of_Calabi_triangle_04.svg.png)
Let □IJKA be a square on base AC with its 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 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:
NO | itaration value |
---|---|
x0 | 1.41421356237309504880168872420969807856967187537694... |
x1 | 1.58943369375323596617308283187888791370090306159374... |
x2 | 1.55324943049375428807267665439782489231871295592784... |
x3 | 1.55139234383942912142613029570413117306471589987689... |
x4 | 1.55138752458074244056538641010106649611908076010328... |
x5 | 1.55138752454832039226341994813293555945836732015691... |
x6 | 1.55138752454832039226195251026462381516359470986821... |
x7 | 1.55138752454832039226195251026462381516359170380388... |
Cardano's method
The value of x can expressed with complex numbers by using Cardano's method:
Viète's method
The value of x can also be expressed without complex numbers by using Viète's method:
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, ...] =
base angle and apex angle
The Calabi triangle is obtuse with base angle θ and apex angle ψ as follows:
See also
Footnotes
Notes
- ^ If we set the polar form of complex number, we can calculate the value of x as follows:
- ^ 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.
- hn/kn = anhn − 1 + hn − 2/ankn − 1 + kn − 2.
- [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]
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
- ^ 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.
- ^ a b Stewart 2004, p. 15.
- ^ a b c Weisstein, Eric W. "Calabi's Triangle". MathWorld.
- ^ Conway, J.H.; Guy, R.K. (1996). "Calabi's Triangle". The Book of Numbers. New York: Springer-Verlag. p. 206.
- ^ Stewart 2004, pp. 7–10.
- ^ 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.
- ^ 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.
- ^ (sequence A046096 in the OEIS)
References
- Stewart, Ian (2004), Galois Theory (3rd ed.), Chapman and Hall/CRC, ISBN 978-1-58488-393-7 - Galois Theory Errata.