This work analyzes the rules governing the growth of the numbers of vertices, edges and faces in all possible periodic tessellations of the 2D Euclidean space, and encodes those rules in several types of polynomial growth functions. These encodings map the geometric, combinatorial and topological properties of the tessellations into sets of integer coefficients. Several general statements about these encodings are given with rigorous mathematical proof. The variation of the growth functions is represented graphically and analyzed in orphic diagrams, so named because of their similarity to orphic art. Several examples of 3D space groups are included, to emphasize the complexity of the growth functions in higher dimensions. A freely available Python library is presented to facilitate the discovery of the growth functions and the generation of orphic diagrams.

Bartosz Naskręcki

Jakub Malinowski

Zbigniew Dauter

Mariusz Jaskolski

Acta Crystallographica Section A Foundations and Advances

Acta Crystallographica Section A: Foundations and Advances publishes articles reporting advances in the theory and practice of all areas of crystallography in the broadest sense. As well as traditional crystallography, this includes nanocrystals, metacrystals, amorphous materials, quasicrystals, synchrotron and XFEL studies, coherent scattering, diffraction imaging, time-resolved studies and the structure of strain and defects in materials.

The journal has two parts, a rapid-publication Advances section and the traditional Foundations section. Articles for the Advances section are of particularly high value and impact. They receive expedited treatment and may be highlighted by an accompanying scientific commentary article and a press release. Further details are given in the November 2013 Editorial.

The central themes of the journal are, on the one hand, experimental and theoretical studies of the properties and arrangements of atoms, ions and molecules in condensed matter, periodic, quasiperiodic or amorphous, ideal or real, and, on the other, the theoretical and experimental aspects of the various methods to determine these properties and arrangements.

Acta Crystallographica Section A: Foundations and Advances publishes articles that report advances in the theory and practice of all areas of crystallography in the broadest sense. This includes not only traditional crystallography but also nanocrystals, metacrystals, amorphous materials, quasicrystals, synchrotron and XFEL studies, coherent scattering, diffraction imaging, time-resolved studies, and the structure of strain and defects in materials.

The journal consists of two parts: a rapid-publication Advances section and the traditional Foundations section. Articles in the Advances section are of particularly high value and impact, receiving expedited treatment and may be highlighted by an accompanying scientific commentary article and a press release. Further details can be found in the November 2013 Editorial.

The central themes of the journal focus on both experimental and theoretical studies of the properties and arrangements of atoms, ions, and molecules in condensed matter, whether periodic, quasiperiodic, or amorphous, ideal or real, as well as the theoretical and experimental aspects of various methods used to determine these properties and arrangements.

《Acta Crystallographica Section A: Foundations and Advances》期刊发表关于晶体学各个领域理论和实践进展的文章，涵盖传统晶体学、纳米晶体、元晶体、非晶材料、准晶体、同步辐射和XFEL研究、相干散射、衍射成像、时间分辨研究以及材料中应变和缺陷的结构等广泛主题。

该期刊分为两个部分：快速出版的“进展”部分和传统的“基础”部分。“进展”部分的文章具有特别高的价值和影响力，接受快速处理，并可能通过附带的科学评论文章和新闻稿进行突出展示。更多细节请参见2013年11月的社论。

该期刊的核心主题包括：对凝聚态物质中原子、离子和分子的性质及排列进行实验和理论研究，涵盖周期性、准周期性或非晶态的理想或实际情况；以及各种方法的理论和实验方面，以确定这些性质和排列。

Category	Quartile
材料科学	4
材料科学, 晶体学	3
材料科学, 化学综合	4

Growth functions of periodic space tessellations

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