The homothetical-Hopf transformation of Pythagorean quadruples

Mircea Crasmareanu
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 32, 2026, Number 2, Pages 313–320
DOI: 10.7546/nntdm.2026.32.2.313-320
Full paper (PDF, 226 Kb)

Details

Authors and affiliations

Mircea Crasmareanu

Faculty of Mathematics, “Al. I. Cuza” University
700506, Iași, Romania

Abstract

This note introduces a transformation of Pythagorean quadruples by using the composition between the Hopf map and a homothety of the space ℝ³. Both the real algebra of complex numbers and the algebra of quaternions are used in this construction. Three examples are detailed, the first one concerning the well-known twin Pythagorean quadruple (1, 2, 2, 3). The trigonometric parametrization of the Euclidean unit sphere S² ⊂ 𝔼³ allows us to prove that this transformation does not produce twin Pythagorean quadruples. A matrix approach for our transformation is also presented.

Keywords

• Pythagorean quadruple
• Hopf fibration
• Homothety
• Sphere

2020 Mathematics Subject Classification

• 11D09
• 53C20
• 11R52

References

1. Crasmareanu, M. (2002). A new method to obtain Pythagorean triple preserving matrices. Missouri Journal of Mathematical Sciences, 14(3), 149–158.
2. Crasmareanu, M. (2019). Conics from symmetric Pythagorean triple preserving matrices. International Electronic Journal of Geometry, 12(1), 85–92.
3. Crasmareanu, M. (2021). Magic conics, their integer points and complementary ellipses. Annals of the ”Alexandru Ioan Cuza” University of Iași, 67(1), 129–148.
4. Crasmareanu, M. (2025). A quaternionic product of planes in E3. Romanian Journal of Mathematics and Computer Science, 15(1), 9–14.
5. Crasmareanu, M., & Munteanu, M. (2025). The circular sum of points of an affine segment. Turkish Journal of Mathematics, 49(4), 411–420.
6. Cushman, R. H., & Bates L. M. (2015). Global Aspects of Classical Integrable Systems (2nd edition). Basel: Birkhäuser/Springer.
7. Frisch, S., & Vaserstein, L. (2012). Polynomial parametrization of Pythagorean quadruples, quintuples and sextuples. Journal of Pure and Applied Algebra, 216(1), 184–191.
8. Rushall, J., Guttierez, M., & McCarty, V. (2020). On the complete tree of primitive Pythagorean quadruples. Integers, 20, Article ID A73.
9. Takloo-Bighas, R. (2018). A Pythagorean Introduction to Number Theory. Right Triangles, Sums of Squares, and Arithmetic. Undergraduate Texts in Mathematics, Cham: Springer.

Manuscript history

• Received: 8 October 2025
• Revised: 12 May 2026
• Accepted: 17 May 2026
• Online First: 21 May 2026

Copyright information

Ⓒ 2026 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).