Similar to a

Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...

, an Eisenstein triple (named after

Gotthold Eisenstein Ferdinand Gotthold Max Eisenstein (16 April 1823 – 11 October 1852) was a German mathematician. He specialized in number theory and mathematical analysis, analysis, and proved several results that eluded even Carl Friedrich Gauss, Gauss. Like ...

) is a set of

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s which are the lengths of the sides of a

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

where one of the angles is 60 or 120 degrees. The relation of such triangles to the

Eisenstein integer In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form :z = a + b\omega , where and are integers and :\omega = \f ...

s is analogous to the relation of Pythagorean triples to the

Gaussian integer In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...

Triangles with an angle of 60°

Triangles with an angle of 60° are a special case of the

Law of Cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...

: :

c^2 = a^2 - ab + b^2.

When the lengths of the sides are integers, the values form a set known as an Eisenstein triple. Examples of Eisenstein triples include:

Triangles with an angle of 120°

A similar special case of the Law of Cosines relates the sides of a triangle with an angle of 120 degrees: :

c^2 = a^2 + ab + b^2.

Examples of such triangles include: {, class="wikitable" , - ! Side ''a'' ! Side ''b'' ! Side ''c'' , - , 3 , 5 , 7 , - , 7 , 8 , 13 , - , 5 , 16 , 19

References

External links

*https://web.archive.org/web/20140505043056/http://161.200.126.13/download/2301499_Senior_Project/Report/Year_2555/MATH19%20-%20Eisenstein%20Triples%20and%20Inner%20Products.pdf *https://www.callutheran.edu/schools/cas/programs/mathematics/people/documents/honorsfinalpresentation.pdf Arithmetic problems of plane geometry Triangle geometry Diophantine equations

Triangles with an angle of 60°

Triangles with an angle of 120°

See also

References

External links