Graphical Multiplication Using Straightedge When a Conic Is Given (A Generalization of Möbius’s Observation)

Blažek, Jiří; Blažek, Josef

doi:10.3390/math14111902

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Graphical Multiplication Using Straightedge When a Conic Is Given (A Generalization of Möbius’s Observation)

by

Jiří Blažek

^1,*

and

Josef Blažek

²

¹

Faculty of Science, Humanities and Education, Technical University of Liberec, 461 17 Liberec, Czech Republic

²

Faculty of Education, University of South Bohemia, 371 15 České Budějovice, Czech Republic

^*

Author to whom correspondence should be addressed.

Mathematics 2026, 14(11), 1902; https://doi.org/10.3390/math14111902

Submission received: 24 April 2026 / Revised: 14 May 2026 / Accepted: 27 May 2026 / Published: 29 May 2026

(This article belongs to the Section B: Geometry and Topology)

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Abstract

Around 1840, A.F. Möbius discovered a remarkable property of the parabola: the curve drawn in the Cartesian coordinate system can be used to graphically multiply any integers. In this article, we generalize this algorithm in two regards: (1) we consider any conic in the plane without a Cartesian coordinate system; (2) we use an unmarked straightedge (i.e., only lines and intersections may be drawn). At the end of the article, we will discuss a special case in which the general conic section is replaced by a circle. We will determine some metric properties of this model and pose an open question as to whether this model represents a certain form of non-Euclidean geometry.

Keywords: straightedge; conic; graphical multiplication; euclidean constructions

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MDPI and ACS Style

Blažek, J.; Blažek, J. Graphical Multiplication Using Straightedge When a Conic Is Given (A Generalization of Möbius’s Observation). Mathematics 2026, 14, 1902. https://doi.org/10.3390/math14111902

AMA Style

Blažek J, Blažek J. Graphical Multiplication Using Straightedge When a Conic Is Given (A Generalization of Möbius’s Observation). Mathematics. 2026; 14(11):1902. https://doi.org/10.3390/math14111902

Chicago/Turabian Style

Blažek, Jiří, and Josef Blažek. 2026. "Graphical Multiplication Using Straightedge When a Conic Is Given (A Generalization of Möbius’s Observation)" Mathematics 14, no. 11: 1902. https://doi.org/10.3390/math14111902

APA Style

Blažek, J., & Blažek, J. (2026). Graphical Multiplication Using Straightedge When a Conic Is Given (A Generalization of Möbius’s Observation). Mathematics, 14(11), 1902. https://doi.org/10.3390/math14111902

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Graphical Multiplication Using Straightedge When a Conic Is Given (A Generalization of Möbius’s Observation)

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