Home / Current Issue / Paper 1724234
The Eight Vertices of a Cube: A Connection Between Coordinate Geometry, Vector Algebra and The Algebraic Structure Z₂³
Subject area: Science,Engineering and Technology · Area of research: Mathematics
DOI: 10.64388/IREV10I4-1724234
Abstract
This paper investigates the correspondence between the eight vertices of a unit cube, their coordinate representations in three-dimensional Euclidean space, their associated vectors, and the finite algebraic structure Z₂ × Z₂ × Z₂. Each vertex has coordinates whose entries belong to {0,1}. The standard basis vectors i, j, and k represent the coordinate axes, while the eight binary triples form a vector space of dimension three over the field Z₂. The paper distinguishes ordinary vector addition over the real numbers from componentwise addition modulo 2, establishes basic algebraic properties, and discusses applications to digital electronics, coding theory, graph theory, and finite algebra.
Keywords
coordinate geometry; cube vertices; vector algebra; binary vectors; Z₂³; finite fields; Boolean algebra; digital electronics; coding theory
References
[1] Anton H, Rorres C (2013). Elementary Linear Algebra: Applications Version (11th ed.). Wiley.
[2] Axler S (2015). Linear Algebra Done Right (3rd ed.). Springer.
[3] Biggs N (1993). Algebraic Graph Theory (2nd ed.). Cambridge University Press.
[4] Hamming RW (1950). Error detecting and error correcting codes. Bell System Technical Journal, 29(2), 147–160.
[5] Lidl R, Niederreiter H (1997). Finite Fields (2nd ed.). Cambridge University Press.
[6] Mano MM, Ciletti MD (2017). Digital Design (6th ed.). Pearson.
How to cite this paper
@article{1724234,
author = {Dr. Kunapuli Venkata Rama Rao},
title = {The Eight Vertices of a Cube: A Connection Between Coordinate Geometry, Vector Algebra and The Algebraic Structure Z₂³},
journal = {Iconic Research And Engineering Journals},
year = {2026},
volume = {10},
number = {4},
pages = {3329-3335},
issn = {2456-8880},
url = {https://www.irejournals.com/formatedpaper/1724234.pdf},
abstract = {This paper investigates the correspondence between the eight vertices of a unit cube, their coordinate representations in three-dimensional Euclidean space, their associated vectors, and the finite algebraic structure Z₂ × Z₂ × Z₂. Each vertex has coordinates whose entries belong to {0,1}. The standard basis vectors i, j, and k represent the coordinate axes, while the eight binary triples form a vector space of dimension three over the field Z₂. The paper distinguishes ordinary vector addition over the real numbers from componentwise addition modulo 2, establishes basic algebraic properties, and discusses applications to digital electronics, coding theory, graph theory, and finite algebra.},
keywords = {coordinate geometry; cube vertices; vector algebra; binary vectors; Z₂³; finite fields; Boolean algebra; digital electronics; coding theory},
month = {October},
doi = {https://doi.org/10.64388/IREV10I4-1724234}
}