Eisenstein triple explained

Similar to a Pythagorean triple, an Eisenstein triple (named after Gotthold Eisenstein) is a set of integers which are the lengths of the sides of a triangle where one of the angles is 60 or 120 degrees. The relation of such triangles to the Eisenstein integers is analogous to the relation of Pythagorean triples to the Gaussian integers.

Triangles with an angle of 60°

Triangles with an angle of 60° are a special case of the Law of Cosines:^{[1] [2] [3]}

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.^[4]

Examples of Eisenstein triples include:^[5]

Side a	Side b	Side c
3	8	7
5	8	7
5	21	19
7	15	13
7	40	37
8	15	13
9	24	21

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:^[6]

Side a	Side b	Side c
3	5	7
7	8	13
5	16	19

See also

• Loeschian number
• Integer triangles with a 60° angle
• Integer triangles with a 120° angle

References

1. Gilder, J., Integer-sided triangles with an angle of 60°," Mathematical Gazette 66, December 1982, 261 266
2. Burn, Bob, "Triangles with a 60° angle and sides of integer length," Mathematical Gazette 87, March 2003, 148–153.
3. Read, Emrys, "On integer-sided triangles containing angles of 120° or 60°", Mathematical Gazette, 90, July 2006, 299–305.
4. Web site: Archived copy . 2014-05-05 . https://web.archive.org/web/20060723174300/http://www2.edc.org/cme/showcase/slides.delaware.2006.pdf . 2006-07-23 . dead .
5. Web site: Integer Triangles with a 60-Degree Angle.
6. Web site: Integer Triangles with a 120-Degree Angle.

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