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This research aims to extend the Pythagorean theorem into quaternion and octonion spaces, particularly focusing on Lipschitz quaternions.

May 27, 2026Acta Universitatis Sapientiae Mathematica0 citationsOpen Access

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The pythagorean theorem in quaternion and octonion spaces

MAMichael Aristidou GCGeorge Chailos

Key Points

This research aims to extend the Pythagorean theorem into quaternion and octonion spaces, particularly focusing on Lipschitz quaternions.
Examined the Pythagorean theorem in various subrings including quaternions (H) and octonions (O).
Provided geometric interpretations for the theorem in Lipschitz quaternion subrings.
Presented findings for rings H/Z_p and O/Z_p.
Demonstrated the applicability of the Pythagorean theorem in Lipschitz quaternion spaces with specific examples.
Presented corresponding results for octonion subrings showing analogous properties under certain conditions.

Abstract

The Pythagorean theorem x^2+y^2=z^2 is usually stated over the integers Z, a subring of the reals R, and it holds true for infinitely many solutions. We explore the theorem over subrings of other number spaces, such as H (quaternions) and O (octonions). We present several results mainly for L (Lipschitz quaternions) and provide a geometric interpretation of the theorem in the subring L. Some results for the corresponding subring G of O are also presented. Finally, we also present some results for the rings H/Z₏ and O/Z₏.

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The pythagorean theorem in quaternion and octonion spaces

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