Hyperbolic Pascal Simplex

Year 2017, Volume: 10 Issue: 2, 46 - 55, 29.10.2017

László Németh

Abstract

In this article we introduce a new geometric object called hyperbolic Pascal simplex. This new

object is presented by the regular hypercube mosaic in the 4-dimensional hyperbolic space. The

definition of the hyperbolic Pascal simplex, whose hyperfaces are hyperbolic Pascal pyramids and

whose faces are hyperbolic Pascals triangles, is a natural generalization of the definition of the

hyperbolic Pascal triangle and pyramid.We describe the growing of the hyperbolic Pascal simplex

considering the numbers and the values of the elements. Further figures illustrate the stepping

from a level to the next one.

Keywords

Hiperbolic Pascal triangle, Pascal simplex, hypercube honeycomb, regular hypercube mosaic in 4-dimensional hyperbolic space, recurrence sequence

References

• [1] Anatriello, G. and Vincenzi, G., Tribonacci-like sequences and generalized Pascal’s pyramids. Internat. J. Math. Ed. Sci. Tech., 45 (2014), 1220-1232.
• [2] Belbachir, H., Németh, L. and Szalay, L., Hyperbolic Pascal triangles. Appl. Math. Comp., 273 (2016), 453-464.
• [3] Belbachir, H. and Szalay, L., On the arithmetic triangles. ˘ Siauliai Math. Sem., 9 (2014), 15-26.
• [4] Bondarenko, B. A., Generalized Pascal triangles and pyramids, their fractals, graphs, and applications. Translated from the Russian by Bollinger, R. C. (English) Santa Clara, CA: The Fibonacci Association, vii, 253 p. 1993. www.fq.math.ca/pascal.html
• [5] Coxeter, H. S. M., Regular honeycombs in hyperbolic space. Proc. Int. Congress Math., Amsterdam, Vol. III. (1954), 155-169.
• [6] Fiorenza, A. and Vincenzi, G., Limit of ratio of consecutive terms for general order-k linear homogeneous recurrences with constant coefficients. Chaos, Solitons & Fractals, 44 (2011), 147-152.
• [7] Fiorenza, A. and Vincenzi, G., From Fibonacci Sequence to the Golden Ratio. Journal of Mathematics, (2013), Article ID 204674, 3 pages. [8] Harris, J, M., Hirst, J. L. and Mossinghoff, M. J., Combinatorics and Graph Theory. Springer, 2008.
• [9] Németh, L. and Szalay, L., Alternating sums in hyperbolic Pascal triangles. Miskolc Mathematical Notes, 17 (2016), no. 2, 989-998.
• [10] Németh, L. – Szalay, L., Recurrence sequences in the hyperbolic Pascal triangle corresponding to the regular mosaic f4; 5g. Annales Mathematicae et Informaticae, 46 (2016), 165–173.
• [11] Németh, L. and Szalay, L., Power sums in hyperbolic Pascal triangles. Analele Univ. “Ovidius”, Math Series., (2018) (to appear).
• [12] Németh, L., Fibonacci words in hyperbolic Pascal triangles. Acta Universitatis Sapientiae Mathematica, 9 (2017) no. 2, (to appear).
• [13] Németh, L. On the hyperbolic Pascal pyramid. Beitr Algebra Geom., 57 (2016), 913–927.
• [14] Németh, L., On the 4-dimensional hypercube mosaics. Publ. Math. Debrecen, 70 (2007), no. 3–4, 291–305.
• [15] Németh, L., Pascal pyramid in space H2R. Mathematical Communications, 22 (2017), 211-225.
• [16] Németh, L., The growing ration of hyperbolic regular mosaics with bounded cell, Armenian Journal of Mathematics, 9 (2017), 1-19.
• [17] Siani, S. and Vincenzi, G., Fibonacci-like sequences and generalized Pascal’s triangles. Internat. J. Math. Ed. Sci. Tech., 45 (2014), 609-614.