Constructing the 15th pentagon that tiles the plane

Lord, Nick

doi:10.1017/mag.2016.28

Cited by 4 publications

(5 citation statements)

References 2 publications

(3 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example, in the 2D tessellation, the sum of angles around a multi-cellular junction must equal 360°. Furthermore, the mathematical laws of 2D tessellation suggested that the geometry of individual cells is not random, especially when the tessellation was characterized by specific distributions of polygonal cells and the average number of sides approached 6 as the number of cells increased ( Aboav, 1980 ; Grünbaum & Shephard, 1987 ; Lewis, 1928 ; Lord, 2016 ; Weaire & Rivier, 1984 ). Therefore, if the geometry of individual cells were random or controlled only by each cell itself, then it would be quite possible that a group of cells would not tile a flat plane.…”

Section: Discussionmentioning

confidence: 99%

“…Two basic mathematical generalizations were found to underlie the tessellations in which only one kind of polygon was used to tile a flat plane (Grünbaum & Shephard, 1987; Lord, 2016): 1. Any kind of polygon with more than 6 sides would be unable to form a close tile pattern on a flat plane; 2.…”

Section: Introductionmentioning

confidence: 99%

“…Three laws were here generalized for the analysis of general topological properties of 2D tessellation: Euler’s law (faces − edges + vertex = 1), Lewis’ law (the relationship between mean area of a convex n-sided cell and n) and Aboav–Weaire law ( Aboav, 1980 ) (the relationship between the mean number of sides of neighboring cells of a convex n-sided cell and n) ( Aboav, 1980 ; Lewis, 1928 ; Sanchez-Gutierrez et al, 2016 ; Weaire & Rivier, 1984 ). Two basic mathematical generalizations were found to underlie the tessellations in which only one kind of polygon was used to tile a flat plane ( Grünbaum & Shephard, 1987 ; Lord, 2016 ): 1. Any kind of polygon with more than 6 sides would be unable to form a close tile pattern on a flat plane; 2.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Cells tile a flat plane by controlling geometries during morphogenesis of Pyropia thalli

et al. 2017

PeerJ

View full text Add to dashboard Cite

BackgroundPyropia haitanensis thalli, which are made of a single layer of polygonal cells, are a perfect model for studying the morphogenesis of multi-celled organisms because their cell proliferation process is an excellent example of the manner in which cells control their geometry to create a two-dimensional plane.MethodsCellular geometries of thalli at different stages of growth revealed by light microscope analysis.ResultsThis study showed the cell division transect the middle of the selected paired-sides to divide the cell into two equal portions, thus resulting in cell sides ≥4 and keeping the average number of cell sides at approximately six even as the thallus continued to grow, such that more than 90% of the cells in thalli longer than 0.08 cm had 5–7 sides. However, cell division could not fully explain the distributions of intracellular angles. Results showed that cell-division-associated fast reorientation of cell sides and cell divisions together caused 60% of the inner angles of cells from longer thalli to range from 100–140°. These results indicate that cells prefer to form regular polygons.ConclusionsThis study suggests that appropriate cell-packing geometries maintained by cell division and reorientation of cell walls can keep the cells bordering each other closely, without gaps.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Cells tile a flat plane by controlling geometries during morphogenesis of Pyropia thalli

et al. 2017

PeerJ

View full text Add to dashboard Cite

show abstract

“…Furthermore, the mathematical laws of 2D tessellation suggested that the geometry of individual cells is not random, especially when the tessellation was characterized by specific distributions of polygonal cells and the average number of sides approached 6 as the number of cells increased (Aboav 1980;Grünbaum & Shephard 1987;Lewis 1928;Lord 2016;Weaire & Rivier 1984). So, if the geometry of individual cells were random or controlled only by each cell itself, then it would be quite possible that a group of cells would not tile a flat plane.…”

Section: Discussionmentioning

confidence: 99%

“…Three laws were here generalized for the analysis of general topological properties of 2D tessellation: Euler's law (faces -edges + vertex = 1), Lewis' law (the relationship between mean area of a convex n-sided cell and n) and Aboav-Weaire law (Aboav 1980) (the relationship between the mean number of sides of neighboring cells of a convex n-sided cell and n) (Aboav 1980;Lewis 1928;Sanchez-Gutierrez et al 2016;Weaire & Rivier 1984). Two basic mathematical generalizations were found to underlie the tessellations in which only one kind of polygon was used to tile a flat plane (Grünbaum & Shephard 1987;Lord 2016): 1. Any kind of polygon with more than 6 sides would be unable to form a close tile pattern on a flat plane.…”

Section: Introductionmentioning

confidence: 99%

Peer Review #1 of "Cells tile a flat plane by controlling geometries during morphogenesis of Pyropia thalli (v0.1)"

Escudero¹

2017

View full text Add to dashboard Cite

Background.Pyropia haitanensis thalli, which are made of a single layer of polygonal cells, are a perfect model for studying the morphogenesis of multi-celled organisms because their cell proliferation process is an excellent example of the manner in which cells control their geometry to create a two-dimensional plane. Methods. Cellular geometries of thalli at different stages of growth revealed by light microscope analysis. Results. This study showed the cell division transect the middle of the selected paired-sides to divide the cell into two equal portions, thus resulting in cell sides ≥ 4 and keeping the average number of cell sides at approximately 6 even as the thallus continued to grow, such that more than 90% of the cells in thalli longer than 0.08 cm had 5-7 sides. However, cell division could not fully explain the distributions of intracellular angles. Results showed that cell-division-associated fast reorientation of cell sides and cell divisions together caused 60% of the inner angles of cells from longer thalli to range from 100-140 o. These results indicate that cells prefer to form regular polygons. Conclusions. This study suggests that appropriate cell-packing geometries maintained by cell division and reorientation of cell walls can keep the cells bordering each other closely, without gaps.

show abstract

Feedback

2016

Math. Gaz.

View full text Add to dashboard Cite

Constructing the 15th pentagon that tiles the plane

Cited by 4 publications

References 2 publications

Cells tile a flat plane by controlling geometries during morphogenesis of Pyropia thalli

Cells tile a flat plane by controlling geometries during morphogenesis of Pyropia thalli

Peer Review #1 of "Cells tile a flat plane by controlling geometries during morphogenesis of Pyropia thalli (v0.1)"

Feedback

Contact Info

Product

Resources

About