細胞自動機不僅具有計算結構，還具有幾何結構︒傳統研究主要關注其計算特性︒本文定義了一種稱為「嵌入」的幾何操作，並證明該操作能保持圖靈完備性︑可逆性（或不可逆性）以及手徵性（或非手徵性）；這表明，高維細胞自動機所具備的這些特性，可以透過低維結構的嵌入來建構︒
我們證明：對於 n ≥ 1，存在圖靈完備的 n 維不可逆手徵細胞自動機；對於 n ≥ 2，存在圖靈完備的 n 維不可逆非手徵細胞自動機；對於 n ≥ 2，存在圖靈完備的 n 維可逆手徵細胞自動機︒
此外，本文深入探討了 Wolfram 提出的細胞自動機分類法及其局限性，並簡要介紹了利用柯爾莫哥洛夫複雜性處理不可計算序列的方法︒
最後，我們回答了陶哲軒提出的關於群論與圖靈完備性之間關係的問題︒

Cellular automata possess geometrical structures in addition to computational structures. Traditional studies focused on computational properties. Here we define a geometrical operation called embeddings and prove they preserve Turing-completeness, reversibility or irreversibility, and chirality or non-chirality, to show that the existence of such properties in higher dimensions can be obtained by embeddings of lower dimensional constructs.
We prove for n ≥ 1, there exists n-dimensional irreversible, chiral cellular automaton that’s Turing-complete, and for n ≥ 2, there exists n-dimensional irreversible, non-chiral cellular automaton that’s Turing-complete, and for n ≥ 2, there exists n-dimensional reversible, chiral cellular automaton that’s Turing-complete.
Wolfram’s classification of cellular automata is then extensively discussed to reveal its limitations, and a brief Kolmogorov complexity approach is provided to deal with situations where a sequence may not be computable.
In addition, we answer a question posted by Terence Tao about group theory and Turing-completeness.

[1] Serafino Amoroso, Yale Patt, Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures, Journal of Computer and System Sciences, Volume 6, Issue 5, Page 448–464, 1972
[2] Ibrahim Al-Dayel, Muhammad Faisal Nadeem, Meraj Ali Khan, Bahre-selam Sielu Abraha, An image encryption scheme using 4-D chaotic system and cellular automaton, Nature, Scientific Reports Volume 15, Article Number: 19499, 2025
[3] Serafino Amoroso, Gerald Cooper, Tessellation structures for reproduction of arbitrary patterns, Journal of Computer and System Sciences, Volume 5, Issue 5, Page 455–464, 1971
[4] Matthew Cook, Universality in Elementary Cellular Automata, Complex Systems, Volume 15, Issue 1, Page 1–40, 2004
[5] Edward Fredkin, Tommaso Toffoli, Conservative logic, International Journal of Theoretical Physics, Volume 21, pages 219–253 1982
[6] Jarkko Kari, Reversibility of 2D cellular automata is undecidable, Physica D: Nonlinear Phenomena, Volume 45, Issues 1–3, Page 379–385, 1990
[7] Edward F. Moore, Machines models of self-reproduction, Mathematical Problems in the Biological Sciences, Volume 14, Page 17–33, 1962
[8] John Myhill, The converse of Moore’s Garden-of-Eden theorem, Proceedings of the American Mathematical Society, 14, Page 685–686, 1963
[9] Suhaib Nazir, Bhargava Rama Chilukuri, Cellular Automata in Traffic Flow: Evolution, Current Trends, and Future Directions for Mixed Traffic Modelling, Transportation in Developing Economies, Volume 12, article number 7, 2026
[10] Turlough Neary, Damien Woods, P-completeness of Cellular Automaton Rule 110, Automata, Languages and Programming, ICALP 2006, Page 132– 143, 2006
[11] Thomas J. Ostrand, Pattern reproduction in tessellation automata of arbitrary dimension, Journal of Computer and System Sciences, Volume 5, Issue 6, Page 623–628, 1971
[12] Daniel Richardson, Tessellations with local transformations, Journal of Computer and System Sciences, Volume 6, Issue 5, Page 373–388, 1972
[13] Thomas C. Schelling, Dynamic models of segregation, The Journal of Mathematical Sociology, Volume 1, Issue 2, Page 143–186, 1971
[14] Tommaso Toffoli, Computation and construction universality of reversible cellular automata, Journal of Computer and System Sciences, Volume 15, Issue 2, Page 213–231, 1977
[15] Jesús Urías, Agustín Enciso, Internal symmetries of cellular automata, Chaos, Volume 7, Issue 3, Page 447–454 1997
[16] Carlos Valentim, José Rabi, Sergio David, Cellular-automaton model for tumor growth dynamics: Virtualization of different scenarios, Computers in Biology and Medicine, Volume 153, Article 106481, 2023
[17] John von Neumann, Arthur W. Burks, Theory of Self-Reproducing Automata, University of Illinois Press, 1966
[18] Stephen Wolfram, Universality and complexity in cellular automata, Physica D: Nonlinear Phenomena Volume 10, Issues 1–2, Page 1–35, 1984
[19] Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002