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. It is an isosceles triangle which is obtuse with an irrational but algebraic ratio between the lengths of its sides and its base.
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. 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 is isosceles which has the same length of sides as . If the ratio of the base to either leg is , we can set that . Then we can consider the following three cases: case 1) is acute triangle The condition is . In this case is valid for equilateral triangle. case 2) is right triangle The condition is . In this case no value is vaild. case 3) is obtuse triangle The condition is . In this case the Calabi triangle is valid for the value of .
Consider the case of . Then
Let a base angle be and a square be on base with it's side length as . Let be the foot of the perpendicular drawn from the apex to the base. Then
Then and , so .
From △DEB ∽ △AHB,
case 1) is acute triangle
Let be a square on side with it's side length as . From △ABC ∽ △IBJ,
From △JKC ∽ △AHC,
Then
Therefore if two squares are congruent,
In this case,
Therefore , it means that is equilateral triangle..
case 2) is right triangle
In this case, , so
Then no value is valid.
case 3) is obtuse triangle
Let be a square on base with it's side length as .
From △AHC ∽ △JKC,
Therefore if two squares are congruent,
In this case,
So, we can input the value of ,
In this case, , we can get the following equation:
Root of Calabi's equation
If is the largest positive root of Calabi's equation:
we can calculate the value of by following methods.
Newton's method
We can set the function as follows:
The function is continuous and differentiable on and
Then is monotonically increasing function and by Intermediate value theorem, the Calabi's equation has unique solution in open interval .
The value of is calculated by Newton's method as follows:
Cardano's method
The value of can expressed with complex numbers by using Cardano's method:
Viète's method
The value of can also be expressed without complex numbers by using Viète's method:
Lagrange's method
The value of 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 and apex angle as follows:
See also
Biggest little polygon
Cubic equation
Footnotes
Notes
Citations
References
- Galois Theory Errata.