• Images
• Videos
• Documents
• Books
• Articles

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.³⁴ 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 < 2 {\displaystyle 0<x<{\sqrt {2}}} . In this case x = 1 is valid for equilateral triangle. case 2) △ABC is right triangle The condition is x = 2 {\displaystyle x={\sqrt {2}}} . In this case no value is valid. case 3) △ABC is obtuse triangle The condition is 2 < x < 2 {\displaystyle {\sqrt {2}}<x<2} . In this case the Calabi triangle is valid for the largest positive root of 2 x 3 − 2 x 2 − 3 x + 2 = 0 {\displaystyle 2x^{3}-2x^{2}-3x+2=0} at x = 1.55138752454832039226... {\displaystyle x=1.55138752454832039226...} (OEIS: A046095).

Root of Calabi's equation

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

2 x 3 − 2 x 2 − 3 x + 2 = 0 , 2 < x < 2 {\displaystyle 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 : R → R {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} } as follows:

f ( x ) = 2 x 3 − 2 x 2 − 3 x + 2 , f ′ ( x ) = 6 x 2 − 4 x − 3 = 6 ( x − 1 3 ) 2 − 11 3 . {\displaystyle {\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 R {\displaystyle \mathbb {R} } and

f ( 2 ) = 2 − 2 < 0 , f ( 2 ) = 4 > 0 , f ′ ( x ) > 0 , ∀ x ∈ [ 2 , 2 ] . {\displaystyle {\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 2 < x < 2 {\displaystyle {\sqrt {2}}<x<2} .

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

x 0 = 2 , x n + 1 = x n − f ( x n ) f ′ ( x n ) = 4 x n 3 − 2 x n 2 − 2 6 x n 2 − 4 x n − 3 . {\displaystyle {\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
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:

x = 1 3 ( 1 + − 23 + 3 i 237 4 3 + − 23 − 3 i 237 4 3 ) . {\displaystyle x={1 \over 3}{\Bigg (}1+{\sqrt[{3}]{-23+3i{\sqrt {237}} \over 4}}+{\sqrt[{3}]{-23-3i{\sqrt {237}} \over 4}}{\Bigg )}.} ⁵⁶⁷

Viète's method

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

x = 1 3 ( 1 + 22 cos ( 1 3 cos − 1 ( − 23 11 22 ) ) ) = 1.55138752454832039226195251026462381516359170380389 ⋯ . {\displaystyle {\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}}} ⁸

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 + 1 1 + 1 1 + 1 4 + 1 2 + 1 1 + 1 2 + 1 1 + 1 5 + 1 2 + 1 1 + 1 3 + 1 1 + 1 1 + 1 390 + ⋯ {\displaystyle 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 }}}}}}}}}}}}}}}}}}}}}}}}}}}}} .^9101112

base angle and apex angle

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

θ = cos − 1 ⁡ ( x / 2 ) = 39.13202614232587442003651601935656349795831966723206 ⋯ ∘ , ψ = 180 − 2 θ = 101.73594771534825115992696796128687300408336066553587 ⋯ ∘ . {\displaystyle {\begin{aligned}\theta &=\cos ^{-1}(x/2)\\&=39.13202614232587442003651601935656349795831966723206\cdots ^{\circ },\\\psi &=180-2\theta \\&=101.73594771534825115992696796128687300408336066553587\cdots ^{\circ }.\\\end{aligned}}}

See also

• Biggest little polygon
• Cubic equation
• Inscribed square in a triangle

Footnotes

Notes

Citations

• 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.

References

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. /wiki/Eugenio_Calabi ↩

2. Stewart 2004, p. 15. - Stewart, Ian (2004), Galois Theory (3rd ed.), Chapman and Hall/CRC, ISBN 978-1-58488-393-7 ↩

3. Weisstein, Eric W. "Calabi's Triangle". MathWorld. /wiki/Eric_W._Weisstein ↩

4. Conway, J.H.; Guy, R.K. (1996). "Calabi's Triangle". The Book of Numbers. New York: Springer-Verlag. p. 206. /wiki/John_Horton_Conway ↩

5. Weisstein, Eric W. "Calabi's Triangle". MathWorld. /wiki/Eric_W._Weisstein ↩

6. Stewart 2004, pp. 7–10. - Stewart, Ian (2004), Galois Theory (3rd ed.), Chapman and Hall/CRC, ISBN 978-1-58488-393-7 ↩

7. If we set the polar form of complex number, we can calculate the value of x as follows: α = r e i φ = r ( cos ⁡ φ + i sin ⁡ φ ) = − 23 + 3 i 237 4 , r = 1 4 ( − 23 ) 2 + 9 ⋅ 237 = 1 4 2 ⋅ 11 3 = ( 11 2 ) 3 , cos ⁡ φ = − 23 4 1 r = − 23 ⋅ 2 2 4 ⋅ 11 11 = − 23 11 22 , α 3 = r 3 e i φ 3 = r 3 ( cos ⁡ ( φ 3 ) + i sin ⁡ ( φ 3 ) ) , α 3 + α ¯ 3 = 2 r 3 cos ⁡ ( φ 3 ) = 22 cos ( 1 3 cos − 1 ( − 23 11 22 ) ) , x = 1 3 ( 1 + α 3 + α ¯ 3 ) = 1 3 ( 1 + 22 cos ( 1 3 cos − 1 ( − 23 11 22 ) ) ) . {\displaystyle {\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. /wiki/Polar_form ↩

8. Stewart 2004, p. 15. - Stewart, Ian (2004), Galois Theory (3rd ed.), Chapman and Hall/CRC, ISBN 978-1-58488-393-7 ↩

9. Weisstein, Eric W. "Calabi's Triangle". MathWorld. /wiki/Eric_W._Weisstein ↩

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. /wiki/Joseph-Louis_Lagrange ↩

11. 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. /wiki/Joseph-Louis_Lagrange ↩

12. 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 fractionnanhnkn-201-11001111121213234149423120514529621217871166107859516139220681333101301919461131112571711211414491171312526916288143909869054636143715198943236377725161197633771273916217249421077318560491811563395224363155701196342979242122107902552027422980066478473621121110852772487699552646622118275752553117802626772356103429491545566669023637824110525706680086784704990552545244577587487338057223259826316786303430469108201871968492712203088101795614200759429447281388171844484252502094662629629625493398770850616432643918722330923332230738249791503958899311303313725460320918344346762031371211323221684242949219186510856365173553567331240970327013753081553256831067469934820961869110319432913511691165895115935102120283943733318598136670168490018628936919292174184703061711124354320757606277603725484424638561309861373312252503578915880289382988141451307322784457636940439233592388338391147256608986863264583094919294281250826862740354058397209132207219473484519267671117194219411687840581078185336777744337122104836254628464232604105715325878082527816785655899121993582757435137083691577075757494167883619917060935933766314423002084403074102958136121935096393113091803360194535106443791023364361109705876861199367601914905137296461109445875426438464067841997054709006913224085473315471215889666449802825177547861391590837451513899061061148132533554187624128924538985209706173814283630760839264941135546468833756956842385317187371122101307802511005157750113879982425251936973483921568946818383945063881861355035124131461173387956963120637483266307489780209080148232258352296509205893011076235896671226220831633998687991150780669531137823817626890645867103046058883906531800778792633103252542372156841146792367970102763322398864469760024557641698717355288213749992047538180730857269568611959270012699454670775985633018569340908218513392028481391945722324786040554128182199675713900706840828693895199337054082514334284056053253582852975655841862139670422299409418937669771120030594610102535684595941022759241143500249274682083578090449502652045320607656039050204118019601597648969534976761777674671472138523512667086813958961635282466111009083972027723582613455575967165043965873163374349463676462656211606732941562700344391130247439210356747854025018830842531174511631261581691359042392700458582340631686114444134135626578889882077664394541836823339721254048877176581609401811780490876282092276368396511207000059592439605240947354106447780132561939044389399812645761566681874187257880348439861728964044552774813489439175099219798638252967217581837511168509293213293146915341133297595408173945878394099222673681257692562369565441371994660433197916610457303025656968705920856052026941206588624589946858420111573201945077774805166184367333607624416434817011135301674328589808839932219656545929872133314131053697638389789436834937111473675605744892774241333353885653799499081192972380649719974138532697472243003679547756836368819889043367668727719495527255298275823846171433744173210074411515296260048005311162559187536493807224748297720136766648140018567411437477947007194368488730006689595440926575677747382754771915126528339297751244491909853682037328926112294551419315759564002222125267855917913423411537661610699253822811660822429674436268059810382314079080657915485465874582386215773507658967132211701979621513560335559873272289863746418627324198941508949979878166872889251449286806186448100396236585431052127165448441887274552896718860137911176387859646704570041485994564297925727582811135400903046196944470476138581180118451167521611974381033504755682602915711893332407055387465069689999443327182481130215046118079020081448364701325582172919476143542456290511266634469919465763582148666213639690994462481869457008185088631369475949511677976336324469362792945983178881259757770014543930234158333767261550845619491967968487602958939282258709484436419125267077105263820280609034325413462347519539173835519267481602264918277835851443072512428541067182133040248677021396128559757340935152041435111916577408649298673465698839670558522913134087831082403907322339349677828441948272059943869104332338872737447019176965811300840121591084182921074753467508975039910068763107939594952960588218214639223209874748929936519652766062328711740869985732924014964732210266100339154889243803748638189407610868274255216373953868128235207480440887939998227528471795631270190361595149590198028548740187809707442228401580310282083392816048898549009232352507430648784917112098508800243892487921829422073119727649302722569358245167511692430469017522699708541899222248211868950734592022168212538268305629230044715999346715321845746218960410709024467894409429312594217198307589812709603954804756176019949749167219160735670132579113908797265471676029513194276705530861225255456408900349921954083321999454944317886245587226204174217016211880310031204951102647702844301153583504939897969515843521496946616509493602288551995313304988866597070326335 The rational approxmation of x is ⁠h95/k95⁠ and an error bounds ε is as follows: x ≈ h 95 k 95 = 10264770284430115358350493989796951584352149694 6616509493602288551995313304988866597070326335 = 1.5513875245483203922619525102646238151635917038038871995280071201179267425542569572957604536135484903 ⋯ , ε = 1 k 95 ( k 95 + k 94 ) = 1.82761... × 10 − 91 . {\displaystyle {\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}}} /wiki/Continued_fraction ↩