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 $0<x<{\sqrt {2}}$ .; In this case $x = 1$ is valid for equilateral triangle.
case 2) $△ ABC$ is right triangle: The condition is $x={\sqrt {2}}$ .; In this case no value is valid.
case 3) $△ ABC$ is obtuse triangle: The condition is ${\sqrt {2}}<x<2$ .; In this case the Calabi triangle is valid for the largest positive root of $2x^{3}-2x^{2}-3x+2=0$ at $x=1.55138752454832039226...$ (OEIS: A046095).

Example of answer

Example figure of Calabi triangle 01

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

0<x<2.

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

{\begin{aligned}HB&=HC=\cos \theta ={\frac {x}{2}},\\AH&=\sin \theta ={\frac {x}{2}}\tan \theta ,\\0&<\theta <{\frac {\pi }{2}}.\end{aligned}}

Then $HB = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠x/2⁠$ and $HE = ⁠ a / 2 ⁠$ , so $EB = ⁠ x - a / 2 ⁠$ .

From △DEB ∽ △AHB,

{\begin{aligned}&EB:DE=HB:AH\\&\Leftrightarrow {\bigg (}{\frac {x-a}{2}}{\bigg )}:a=\cos \theta :\sin \theta =1:\tan \theta \\&\Leftrightarrow a={\bigg (}{\frac {x-a}{2}}{\bigg )}\tan \theta \\&\Leftrightarrow a={\frac {x\tan \theta }{\tan \theta +2}}.\\\end{aligned}}

case 1) $△ ABC$ is acute triangle

Example figure of Calabi triangle 02

Let $□ IJKL$ be a square on side $AC$ with its side length as $b$ . From △ABC ∽ △IBJ,

{\begin{aligned}&AB:IJ=BC:BJ\\&\Leftrightarrow 1:b=x:BJ\\&\Leftrightarrow BJ=bx.\end{aligned}}

From △JKC ∽ △AHC,

{\begin{aligned}&JK:JC=AH:AC\\&\Leftrightarrow b:JC={\frac {x}{2}}\tan \theta :1\\&\Leftrightarrow JC={\frac {2b}{x\tan \theta }}.\end{aligned}}

Then

{\begin{aligned}&x=BC=BJ+JC=bx+{\frac {2b}{x\tan \theta }}\\&\Leftrightarrow x=b{\frac {x^{2}\tan \theta +2}{x\tan \theta }}\\&\Leftrightarrow b={\frac {x^{2}\tan \theta }{x^{2}\tan \theta +2}}.\end{aligned}}

Therefore, if two squares are congruent,

{\begin{aligned}&a=b\\&\Leftrightarrow {\frac {x\tan \theta }{\tan \theta +2}}={\frac {x^{2}\tan \theta }{x^{2}\tan \theta +2}}\\&\Leftrightarrow x\tan \theta \cdot (x^{2}\tan \theta +2)=x^{2}\tan \theta (\tan \theta +2)\\&\Leftrightarrow x\tan \theta \cdot (x(\tan \theta +2)-(x^{2}\tan \theta +2))=0\\&\Leftrightarrow x\tan \theta \cdot (x\tan \theta -2)\cdot (x-1)=0\\&\Leftrightarrow 2\sin \theta \cdot 2(\sin \theta -1)\cdot (x-1)=0.\end{aligned}}

In this case, ${\frac {\pi }{4}}<\theta <{\frac {\pi }{2}},2\sin \theta \cdot 2(\sin \theta -1)\neq 0.$

Therefore $x=1$ , it means that $△ ABC$ is equilateral triangle.

case 2) $△ ABC$ is right triangle

Example figure of Calabi triangle 03

In this case, $x={\sqrt {2}},\tan \theta =1$ , so $a={\frac {\sqrt {2}}{3}},b={\frac {1}{2}}.$

Then no value is valid.

case 3) $△ ABC$ is obtuse triangle

Example figure of Calabi triangle 04

Let $□ IJKA$ be a square on base $AC$ with its side length as $b$ .

From △AHC ∽ △JKC,

{\begin{aligned}&AH:HC=JK:KC\\&\Leftrightarrow \sin \theta :\cos \theta =b:(1-b)\\&\Leftrightarrow b\cos \theta =(1-b)\sin \theta \\&\Leftrightarrow b=(1-b)\tan \theta \\&\Leftrightarrow b={\frac {\tan \theta }{1+\tan \theta }}.\end{aligned}}

Therefore, if two squares are congruent,

{\begin{aligned}&a=b\\&\Leftrightarrow {\frac {x\tan \theta }{\tan \theta +2}}={\frac {\tan \theta }{1+\tan \theta }}\\&\Leftrightarrow {\frac {x}{\tan \theta +2}}={\frac {1}{1+\tan \theta }}\\&\Leftrightarrow x(\tan \theta +1)=\tan \theta +2\\&\Leftrightarrow (x-1)\tan \theta =2-x.\end{aligned}}

In this case,

\tan \theta ={\frac {\sqrt {(2+x)(2-x)}}{x}}.

So, we can input the value of $tan θ$ ,

{\begin{aligned}&(x-1)\tan \theta =2-x\\&\Leftrightarrow (x-1){\frac {\sqrt {(2+x)(2-x)}}{x}}=2-x\\&\Leftrightarrow (2-x)\cdot ((x-1)^{2}(2+x)-x^{2}(2-x))=0\\&\Leftrightarrow (2-x)\cdot (2x^{3}-2x^{2}-3x+2)=0.\end{aligned}}

In this case, ${\sqrt {2}}<x<2$ , we can get the following equation:

2x^{3}-2x^{2}-3x+2=0.

Root of Calabi's equation

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

2x^{3}-2x^{2}-3x+2=0,{\sqrt {2}}<x<2

we can calculate the value of $x$ by following methods.

Newton's method

We can set the function $f:\mathbb {R} \rightarrow \mathbb {R}$ as follows:

{\begin{aligned}f(x)&=2x^{3}-2x^{2}-3x+2,\\f'(x)&=6x^{2}-4x-3=6{\bigg (}x-{\frac {1}{3}}{\bigg )}^{2}-{\frac {11}{3}}.\end{aligned}}

The function $f$ is continuous and differentiable on $\mathbb {R}$ and

{\begin{aligned}f({\sqrt {2}})&={\sqrt {2}}-2<0,\\f(2)&=4>0,\\f'(x)&>0,\forall x\in [{\sqrt {2}},2].\end{aligned}}

Then $f$ is monotonically increasing function and by Intermediate value theorem, the Calabi's equation $f (x) = 0$ has unique solution in open interval ${\sqrt {2}}<x<2$ .

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

{\begin{aligned}x_{0}&={\sqrt {2}},\\x_{n+1}&=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}={\frac {4x_{n}^{3}-2x_{n}^{2}-2}{6x_{n}^{2}-4x_{n}-3}}.\end{aligned}}

Newton's method for the root of Calabi's equation
NO	itaration value
$x$ ₀	1.41421356237309504880168872420969807856967187537694...
$x$ ₁	1.58943369375323596617308283187888791370090306159374...
$x$ ₂	1.55324943049375428807267665439782489231871295592784...
$x$ ₃	1.55139234383942912142613029570413117306471589987689...
$x$ ₄	1.55138752458074244056538641010106649611908076010328...
$x$ ₅	1.55138752454832039226341994813293555945836732015691...
$x$ ₆	1.55138752454832039226195251026462381516359470986821...
$x$ ₇	1.55138752454832039226195251026462381516359170380388...

Cardano's method

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

x={1 \over 3}{\Bigg (}1+{\sqrt[{3}]{-23+3i{\sqrt {237}} \over 4}}+{\sqrt[{3}]{-23-3i{\sqrt {237}} \over 4}}{\Bigg )}.

^[3]^[5]^[a]

Viète's method

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

{\begin{aligned}x&={1 \over 3}{\bigg (}1+{\sqrt {22}}\cos \!{\bigg (}{1 \over 3}\cos ^{-1}\!\!{\bigg (}\!-{23 \over 11{\sqrt {22}}}{\bigg )}{\bigg )}{\bigg )}\\&=1.55138752454832039226195251026462381516359170380389\cdots .\end{aligned}}

^[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, ...] =

1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{5+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{3+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{390+\cdots }}}}}}}}}}}}}}}}}}}}}}}}}}}}

.^[3]^[6]^[7]^[b]

base angle and apex angle

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

{\begin{aligned}\theta &=\cos ^{-1}(x/2)\\&=39.13202614232587442003651601935656349795831966723206\cdots ^{\circ },\\\psi &=180-2\theta \\&=101.73594771534825115992696796128687300408336066553587\cdots ^{\circ }.\\\end{aligned}}

Footnotes

Notes

^ If we set the polar form of complex number, we can calculate the value of $x$ as follows:
${\begin{aligned}\alpha &=re^{i\varphi }=r(\cos \varphi +i\sin \varphi )={\frac {-23+3i{\sqrt {237}}}{4}},\\r&={\frac {1}{4}}{\sqrt {(-23)^{2}+9\cdot 237}}={\frac {1}{4}}{\sqrt {2\cdot 11^{3}}}={\Bigg (}{\sqrt {\frac {11}{2}}}{\Bigg )}^{3},\\\cos \varphi &=-{\frac {23}{4}}{\frac {1}{r}}=-{\frac {23\cdot 2{\sqrt {2}}}{4\cdot 11{\sqrt {11}}}}=-{\frac {23}{11{\sqrt {22}}}},\\{\sqrt[{3}]{\alpha }}&={\sqrt[{3}]{r}}e^{\frac {i\varphi }{3}}={\sqrt[{3}]{r}}{\Big (}\cos {\Big (}{\frac {\varphi }{3}}{\Big )}+i\sin {\Big (}{\frac {\varphi }{3}}{\Big )}{\Big )},\\{\sqrt[{3}]{\alpha }}+{\sqrt[{3}]{\bar {\alpha }}}&=2{\sqrt[{3}]{r}}\cos {\Big (}{\frac {\varphi }{3}}{\Big )}={\sqrt {22}}\cos \!{\bigg (}{1 \over 3}\cos ^{-1}\!\!{\bigg (}\!-{23 \over 11{\sqrt {22}}}{\bigg )}{\bigg )},\\x&={\frac {1}{3}}{\bigg (}1+{\sqrt[{3}]{\alpha }}+{\sqrt[{3}]{\bar {\alpha }}}{\bigg )}={1 \over 3}{\bigg (}1+{\sqrt {22}}\cos \!{\bigg (}{1 \over 3}\cos ^{-1}\!\!{\bigg (}\!-{23 \over 11{\sqrt {22}}}{\bigg )}{\bigg )}{\bigg )}.\end{aligned}}$
Then this Cardano's method is equivalent as Viète's method.

^ If a continued fraction

[a 0, a 1, a 2, ...]

are found, with numerators

h

₁,

h

₂, ... and denominators

k

₁,

k

₂, ... then the relevant recursive relation is that of Gaussian brackets:

h n = a n h n - 1 + h n - 2

,

k n = a n k n - 1 + k n - 2

.

The successive convergents are given by the formula

⁠ h n / k n ⁠ = ⁠ a n h n - 1 + h n - 2 / a n k n - 1 + k n - 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 $h n$ and denominators $k n$ of continued fraction
$n$	$a n$	$h n$	$k n$
-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 ⁠ $h$ ₉₅/ $k$ ₉₅⁠ and an error bounds $ε$ is as follows:

{\begin{aligned}x&\approx {\frac {h_{95}}{k_{95}}}\\&={\frac {10264770284430115358350493989796951584352149694}{6616509493602288551995313304988866597070326335}}\\&=1.5513875245483203922619525102646238151635917038038871995280071201179267425542569572957604536135484903\cdots ,\\\varepsilon &={\frac {1}{k_{95}(k_{95}+k_{94})}}\\&=1.82761...\times 10^{-91}.\end{aligned}}

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.

External links

Weisstein, Eric W. "Calabi's Triangle". MathWorld.

[6] If we set the polar form of complex number, we can calculate the value of $x$ as follows:
${\begin{aligned}\alpha &=re^{i\varphi }=r(\cos \varphi +i\sin \varphi )={\frac {-23+3i{\sqrt {237}}}{4}},\\r&={\frac {1}{4}}{\sqrt {(-23)^{2}+9\cdot 237}}={\frac {1}{4}}{\sqrt {2\cdot 11^{3}}}={\Bigg (}{\sqrt {\frac {11}{2}}}{\Bigg )}^{3},\\\cos \varphi &=-{\frac {23}{4}}{\frac {1}{r}}=-{\frac {23\cdot 2{\sqrt {2}}}{4\cdot 11{\sqrt {11}}}}=-{\frac {23}{11{\sqrt {22}}}},\\{\sqrt[{3}]{\alpha }}&={\sqrt[{3}]{r}}e^{\frac {i\varphi }{3}}={\sqrt[{3}]{r}}{\Big (}\cos {\Big (}{\frac {\varphi }{3}}{\Big )}+i\sin {\Big (}{\frac {\varphi }{3}}{\Big )}{\Big )},\\{\sqrt[{3}]{\alpha }}+{\sqrt[{3}]{\bar {\alpha }}}&=2{\sqrt[{3}]{r}}\cos {\Big (}{\frac {\varphi }{3}}{\Big )}={\sqrt {22}}\cos \!{\bigg (}{1 \over 3}\cos ^{-1}\!\!{\bigg (}\!-{23 \over 11{\sqrt {22}}}{\bigg )}{\bigg )},\\x&={\frac {1}{3}}{\bigg (}1+{\sqrt[{3}]{\alpha }}+{\sqrt[{3}]{\bar {\alpha }}}{\bigg )}={1 \over 3}{\bigg (}1+{\sqrt {22}}\cos \!{\bigg (}{1 \over 3}\cos ^{-1}\!\!{\bigg (}\!-{23 \over 11{\sqrt {22}}}{\bigg )}{\bigg )}{\bigg )}.\end{aligned}}$
Then this Cardano's method is equivalent as Viète's method.

[10] If a continued fraction $[a 0, a 1, a 2, ...]$ are found, with numerators $h$ ₁, $h$ ₂, ... and denominators $k$ ₁, $k$ ₂, ... then the relevant recursive relation is that of Gaussian brackets:
$h n = a n h n - 1 + h n - 2$ ,

$k n = a n k n - 1 + k n - 2$ .
The successive convergents are given by the formula
$⁠ h n / k n ⁠ = ⁠ a n h n - 1 + h n - 2 / a n k n - 1 + k n - 2 ⁠$ .
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 $h n$ and denominators $k n$ of continued fraction
$n$ $a n$ $h n$ $k n$

-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 ⁠ $h$ ₉₅/ $k$ ₉₅⁠ and an error bounds $ε$ is as follows:

${\begin{aligned}x&\approx {\frac {h_{95}}{k_{95}}}\\&={\frac {10264770284430115358350493989796951584352149694}{6616509493602288551995313304988866597070326335}}\\&=1.5513875245483203922619525102646238151635917038038871995280071201179267425542569572957604536135484903\cdots ,\\\varepsilon &={\frac {1}{k_{95}(k_{95}+k_{94})}}\\&=1.82761...\times 10^{-91}.\end{aligned}}$

[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.

[FOOTNOTEStewart200415-2] Stewart 2004, p. 15.

[Wolfram-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.

[FOOTNOTEStewart20047–10-5] Stewart 2004, pp. 7–10.

[7] 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.

[8] 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.

[9] (sequence A046096 in the OEIS)

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[2]

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[5]

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Calabi triangle

Contents

Definition