Some geometric properties of the Padovan vectors in Euclidean 3-space

Serdar Korkmaz and Hatice Kuşak Samancı
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 4, Pages 842–860
DOI: 10.7546/nntdm.2023.29.4.842-860
Full paper (PDF, 743 Kb)

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Authors and affiliations

Serdar Korkmaz
Graduate Education Institute, Bitlis Eren University
Bitlis, Turkey

Hatice Kuşak Samancı
Department of Mathematics, Science and Art Faculty, Bitlis Eren University
Bitlis, Turkey

Abstract

Padovan numbers were defined by Stewart (1996) in honor of the modern architect Richard Padovan (1935) and were first discovered in 1924 by Gerard Cordonnier. Padovan numbers are a special status of Tribonacci numbers with initial conditions and general terms. The ratio between Padovan numbers is one of the important algebraic numbers because it produces plastic numbers. Up to now, various studies have been conducted on Padovan numbers and Padovan polynomial sequences. In this study, Padovan vectors are defined for the first time by using the Padovan Binet-like formula and reduction relation. Then, geometric properties of Padovan vectors such as inner product, norm, and vector products are analyzed. In the last part of the study, Padovan vectors were calculated with Binet formulas in the Geogebra program. In addition, the first ten Padovan numbers and Padovan vectors were calculated using the Binet formulas and shown as points and vectors in three-dimensional space. According to the Padovan vectors found, the Padovan curve was drawn in space for the first time by using the curve fitting feature of the Geogebra program. Thus, with our study, a geometric approach to Padovan number sequences was brought for the first time.

Keywords

• Padovan numbers
• Padovan vectors
• Inner product
• Vectorial product
• Geogebra

2020 Mathematics Subject Classification

• 11A99
• 11B99
• 11H99

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Manuscript history

• Received: 28 April 2023
• Revised: 16 November 2023
• Accepted: 24 December 2023
• Online First: 27 December 2023

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Ⓒ 2023 by the Authors.
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).