| Hexagonal mosaic | |
|---|---|
 | |
| Type of | Right mosaic |
| Vertex figure | 6.6.6 (6 3 ) |
| Shlefly symbol | {6.3} t {3,6} |
| 3 | 6 2 2 6 | 3 3 3 3 | | |
| Coxeter Chart |       |
| Symmetry group | , [6.3], (* 632) |
| Rotational symmetry | , [6.3] + , (632) |
| Dual mosaic | Triangular mosaic |
| The properties | Vertically transitive , |
Hexagonal parquet ( hexagonal parquet [1] ) or hexagonal mosaic - tiling of a plane with equal regular hexagons located side to side.
A hexagonal mosaic is a dual triangular mosaic - if you connect the centers of adjacent hexagons, then the drawn segments will give a triangular mosaic [1] [2] . The Shlefli symbol of hexagonal parquet is {6,3} (which means that three hexagons converge at each vertex of the parquet), or t {3,6} if the mosaic is considered as truncated triangular.
The English mathematician Conway called the mosaic hextille (six-parquet).
The internal angle of the hexagon is 120 degrees, so three hexagons at one vertex together give 360 ββdegrees. This is one of the three regular mosaics of the plane . The other two mosaics are triangular parquet and square parquet .
Content
- 1 Applications
- 2 Homogeneous coloring
- 2.1 Hexagonal mosaic with chamfer
- 3 Related Mosaics
- 3.1 Symmetry options
- 4 Building Withoff from hexagonal and triangular mosaics
- 5 Monohedral convex hexagonal mosaics
- 6 Topologically equivalent mosaics
- 7 Packing circles
- 8 Associated regular complex infinite angles
- 9 See also
- 10 notes
- 11 Literature
- 12 Links
Applications
Tiling the plane with regular hexagons is the basis for hex , hexagonal chess and other games on the checkered field , polyhexes , variants of the Life model and other two-dimensional cellular automata , ring flexagons , etc.
Hexagonal mosaic is the most dense way of packing circles in two-dimensional space. states that a hexagonal mosaic is the best way to divide a surface into areas of equal area with the smallest total perimeter. The optimal three-dimensional structure for honeycombs (rather, soap bubbles) was investigated by Lord Kelvin , who believed that (or the body-centered cubic lattice) was optimal. However, the less regular slightly better.
This structure exists in nature in the form of graphite , where each layer of graphene resembles a wire mesh, where strong covalent bonds play the role of a wire. Tubular sheets of graphene were synthesized; they are known as carbon nanotubes . They have many potential applications due to their high tensile strength and electrical properties. Silicene is similar to graphene .
The densest packing of circles has a structure similar to a hexagonal mosaic
Chicken net
Graphene
Carbon nanotubes can be considered as a hexagonal mosaic on a cylindrical surface
Hexagonal mosaic appears in many crystals. In three-dimensional space, a face - centered cubic structure and a hexagonal close-packed structure are often found in crystals. They are the most dense spheres in three-dimensional space. Structurally, they consist of parallel layers of a hexagonal mosaic similar to the structure of graphite. They differ in how the levels are shifted relative to each other, while the face-centered cubic structure is more correct. Pure copper , among other materials, forms a face-centered cubic lattice.
Homogeneous Coloring
There are three different hexagonal mosaic, all obtained from the mirror symmetry of the Withoff constructions . The notation ( h , k ) represents the periodic repetition of a colored tile with hexagonal distances h and k .
| k-homogeneous | 1- homogeneous | 2- homogeneous | 3- homogeneous | ||||
|---|---|---|---|---|---|---|---|
| Symmetry | p6m, (* 632) | p3m1, (* 333) | p6m, (* 632) | p6, (632) | |||
| Picture | |||||||
| Colors | one | 2 | 3 | 2 | four | 2 | 7 |
| (h, k) | (1,0) | (1,1) | (2.0) | (2.1) | |||
| Loafs | {6.3} | t {3,6} | t {3 [3] } | ||||
| 3 | 6 2 | 2 6 | 3 | 3 3 3 | | |||||
| Coxeter | |||||||
| H | tΞ | cH | |||||
A 3-color mosaic is formed by a permutation polyhedron of order 3.
Chamfered Hexagonal Mosaic
Chamfering a hexagonal mosaic replaces the edges with new hexagons and converts it to another hexagonal mosaic. In the limit, the original faces disappear, and the new hexagons are converted into rhombuses, turning the mosaic into a rhombic one .
| Hexagons (H) | Chamfered hexagons (cH) | Rhombi (daH) | ||
|---|---|---|---|---|
Related Mosaics
Hexagons can be divided into 6 triangles. This leads to two 2-homogeneous mosaics , and a triangular mosaic :
| Right mosaic | Breaking up | 2 homogeneous mosaics | Right mosaic | |
|---|---|---|---|---|
| Source | | broken 1/3 hexagons | broken 2/3 hexagons | full partition |
The hexagonal mosaic can be considered an elongated rhombic mosaic , in which each vertex of the rhombic mosaic is βstretchedβ with the formation of a new edge. This is similar to the connection between the tilings of the rhombododecahedron and the in three-dimensional space.
| Rhombic mosaic | Hexagonal mosaic | A grid showing this relationship |
You can also split the proto-tiles of some hexagonal mosaics into two, three, four, or nine identical pentagons:
| A pentagonal mosaic of the 1st type with overlapping regular hexagons (each hexagon consists of 2 pentagons). | A pentagonal mosaic of the 3rd type with overlapping regular hexagons (each hexagon consists of 3 pentagons). | A pentagonal mosaic of the 4th type with overlapping semi-regular hexagons (each hexagon consists of 4 pentagons). | A pentagonal mosaic of the 3rd type with overlapping regular hexagons of two sizes (hexagons consists of 3 and 9 pentagons). |
Symmetry Options
This mosaic is topologically related to a sequence of regular mosaics with hexagonal faces that begins with a hexagonal mosaic. Mosaics of infinite sequence have the Schleafly symbol {6, n} and the Coxeter diagram .
| * n 62 symmetry options for regular mosaics: {6, n } | ||||||||
|---|---|---|---|---|---|---|---|---|
| Spherical | Euclidean | Hyperbolic mosaics | ||||||
| {6,2} | {6.3} | {6.4} | {6.5} | {6.6} | {6.7} | {6.8} | ... | {6, β} |
The hexagonal mosaic is topologically connected (as part of the sequence) with regular polyhedra with vertex figure n 3 .
| Spherical | Euclidean | Compact hyperbolic. | Paracom pacty. | Noncompact hyperbolic. | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| {2,3} | {3,3} | {4.3} | {5,3} | {6.3} | {7.3} | {8.3} | {β, 3} | {12i, 3} | {9i, 3} | {6i, 3} | {3i, 3} |
Similarly, the mosaic is connected with homogeneous truncated polyhedra with the vertex figure n .6.6.
| * n 32 mutations of symmetries of truncated mosaics: n .6.6 | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Symmetry * n 32 [n, 3] | Spherical | Euclidean | Compact hyperbolic | Paracompact. | Noncompact hyperbolic | |||||||
| * 232 [2,3] | * 332 [3.3] | * 432 [4.3] | * 532 [5.3] | * 632 [6.3] | * 732 [7.3] | * 832 [8.3] ... | * β32 [β, 3] | [12i, 3] | [9i, 3] | [6i, 3] | ||
| Truncated figures | ||||||||||||
| Conf. | 2.6.6 | 3.6.6 | 4.6.6 | 5.6.6 | 6.6.6 | 7.6.6 | 8.6.6 | β.6.6 | 12i.6.6 | 9i.6.6 | 6i.6.6 | |
| n-kitty figures | ||||||||||||
| Conf. | V2.6.6 | V3.6.6 | V4.6.6 | V5.6.6 | V6.6.6 | V7.6.6 | V8.6.6 | Vβ.6.6 | V12i.6.6 | V9i.6.6 | V6i.6.6 | |
Mosaic is also part of truncated rhombic polyhedra and mosaics with symmetry of the Coxeter group [n, 3]. A cube can be considered as a rhombic hexagon, in which all rhombuses are squares. Truncated shapes have regular n-gons in place of truncated vertices and irregular hexagonal faces.
| Spherical | Euclidean | Hyperbolic | |||||
|---|---|---|---|---|---|---|---|
| * n32 | * 332 | * 432 | * 532 | * 632 | * 732 | * 832 ... | * β32 |
| Mosaic | |||||||
| Conf. | V (3.3) 2 | V (3.4) 2 | V (3.5) 2 | V (3.6) 2 | V (3.7) 2 | V (3.8) 2 | V (3.β) 2 |
Building Withoff from hexagonal and triangular mosaics
Like there are eight homogeneous mosaics based on regular hexagonal mosaics (or dual triangular mosaics ).
If you draw tiles of the original faces in red, the original vertices (the polygons obtained in their place) are yellow, and the original edges (the polygons obtained in their place) are blue, there are 8 shapes, 7 of which are topologically different. (A truncated triangular mosaic is topologically identical to a hexagonal mosaic.)
| Homogeneous hexagonal / triangular mosaics | ||||||||
|---|---|---|---|---|---|---|---|---|
| Fundamental domains | Symmetry : [6.3], (* 632) | [6.3] + , (632) | ||||||
| {6.3} | t {6,3} | r {6,3} | t {3,6} | {3,6} | rr {6.3} | tr {6,3} | sr {6,3} | |
| Config. | 6 3 | 12/3/12 | (6.3) 2 | 6.6.6 | 3 6 | 3.4.6.4 | 4.6.12 | 3.3.3.3.6 |
Monohedral Convex Hexagonal Mosaics
There are 3 types of monohedral [3] convex hexagonal mosaics [4] . All of them are isohedral . Each has parametric options with fixed symmetry. Type 2 contains moving symmetries and keeps chiral pairs different.
| one | 2 | 3 | |
|---|---|---|---|
| p2, 2222 | pgg, 22 Γ | p2, 2222 | p3, 333 |
| b = e B + C + D = 360 Β° | b = e, d = f B + C + E = 360 Β° | a = f, b = c, d = e B = D = F = 120 Β° | |
| two-tile grid | four-tile grid | three-tile grid | |
Topologically equivalent mosaics
Hexagonal mosaics can be identical to the {6.3} topology of a regular mosaic (3 hexagons at each vertex). There are 13 options for hexagonal mosaics with isohedral faces. From the point of view of symmetry, all faces have the same color, the coloring in the figures represents the position in the grid [5] . One-color (1-tile) grids consist of hexagonal parallelogons .
| pg (Γ^) | p2 (2222) | p3 (333) | pmg (22 *) | |||
|---|---|---|---|---|---|---|
| pgg (22 Γ) | p31m (3 * 3) | p2 (2222) | cmm (2 * 22) | p6m (* 632) | ||
Other topologically isohedral hexagonal mosaics look like quadrangular and pentagonal, not touching side-to-side, but whose polygons can be considered as having collinear adjacent sides:
| pmg (22 *) | pgg (22 Γ) | cmm (2 * 22) | p2 (2222) | |||
|---|---|---|---|---|---|---|
| Parallelogram | Trapezoid | Parallelogram | Rectangle | Parallelogram | Rectangle | Rectangle |
| p2 (2222) | pgg (22 Γ) | p3 (333) |
|---|---|---|
2-homogeneous and 3-homogeneous tilings have a rotational degree of freedom that bends 2/3 of the hexagons, including the case of collinearity of the sides, which can be seen as mosaics of hexagons and large triangles with non-matching sides (not side-to-side) [6] .
Mosaic can be curved to chiral 4-color interlaced in three directions patterns, with the transformation of some hexagons into parallelograms . Interlaced patterns with 2 colored faces have a rotational symmetry of 632 (p6) .
| Right | Rotated | Right | Intertwined |
|---|---|---|---|
| p6m, (* 632) | p6, (632) | p6m (* 632) | p6 (632) |
| p3m1, (* 333) | p3, (333) | p6m (*632) | p2 (2222) |
Π£ΠΏΠ°ΠΊΠΎΠ²ΠΊΠ° ΠΊΡΡΠ³ΠΎΠ²
Π¨Π΅ΡΡΠΈΡΠ³ΠΎΠ»ΡΠ½ΡΡ ΠΌΠΎΠ·Π°ΠΈΠΊΡ ΠΌΠΎΠΆΠ½ΠΎ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°ΡΡ Π΄Π»Ρ ΡΠΏΠ°ΠΊΠΎΠ²ΠΊΠΈ ΠΊΡΡΠ³ΠΎΠ² , ΡΠ°Π·ΠΌΠ΅ΡΡΠΈΠ² ΠΊΡΡΠ³ΠΈ ΠΎΠ΄ΠΈΠ½Π°ΠΊΠΎΠ²ΠΎΠ³ΠΎ ΡΠ°Π΄ΠΈΡΡΠ° Ρ ΡΠ΅Π½ΡΡΠ°ΠΌΠΈ Π² Π²Π΅ΡΡΠΈΠ½Π°Ρ ΠΌΠΎΠ·Π°ΠΈΠΊΠΈ. ΠΠ°ΠΆΠ΄ΡΠΉ ΠΊΡΡΠ³ ΡΠΎΠΏΡΠΈΠΊΠ°ΡΠ°Π΅ΡΡΡ Ρ 3 Π΄ΡΡΠ³ΠΈΠΌΠΈ ΠΊΡΡΠ³Π°ΠΌΠΈ ΡΠΏΠ°ΠΊΠΎΠ²ΠΊΠΈ ( ΠΊΠΎΠ½ΡΠ°ΠΊΡΠ½ΠΎΠ΅ ΡΠΈΡΠ»ΠΎ ) [7] . ΠΡΡΠ³ΠΈ ΠΌΠΎΠΆΠ½ΠΎ Π·Π°ΠΊΡΠ°ΡΠΈΡΡ Π΄Π²ΡΠΌΡ ΡΠ²Π΅ΡΠ°ΠΌΠΈ. ΠΡΠΎΡΡΡΠ°Π½ΡΡΠ²ΠΎ Π²Π½ΡΡΡΠΈ ΠΊΠ°ΠΆΠ΄ΠΎΠ³ΠΎ ΡΠ΅ΡΡΠΈΡΠ³ΠΎΠ»ΡΠ½ΠΈΠΊΠ° ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ ΠΏΠΎΠΌΠ΅ΡΡΠΈΡΡ ΠΎΠ΄ΠΈΠ½ ΠΊΡΡΠ³, ΡΠΎΠ·Π΄Π°Π²Π°Ρ Π½Π°ΠΈΠ±ΠΎΠ»Π΅Π΅ ΠΏΠ»ΠΎΡΠ½ΡΡ ΡΠΏΠ°ΠΊΠΎΠ²ΠΊΡ ΡΡΠ΅ΡΠ³ΠΎΠ»ΡΠ½ΠΎΠΉ ΠΌΠΎΠ·Π°ΠΈΠΊΠΈ , Π² ΠΊΠΎΡΠΎΡΠΎΠΉ ΠΊΠ°ΠΆΠ΄ΡΠΉ ΠΊΡΡΠ³ ΡΠΎΠΏΡΠΈΠΊΠ°ΡΠ°Π΅ΡΡΡ Ρ ΠΌΠ°ΠΊΡΠΈΠΌΠ°Π»ΡΠ½ΠΎ Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΡΠΌ ΡΠΈΡΠ»ΠΎΠΌ ΠΊΡΡΠ³ΠΎΠ² (6).
Π‘Π²ΡΠ·Π°Π½Π½ΡΠ΅ ΠΏΡΠ°Π²ΠΈΠ»ΡΠ½ΡΠ΅ ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ½ΡΠ΅ Π±Π΅ΡΠΊΠΎΠ½Π΅ΡΠ½ΠΎΡΠ³ΠΎΠ»ΡΠ½ΠΈΠΊΠΈ
Π‘ΡΡΠ΅ΡΡΠ²ΡΠ΅Ρ 2 , ΠΈΠΌΠ΅ΡΡΠΈx ΡΠ΅ ΠΆΠ΅ Π²Π΅ΡΡΠΈΠ½Ρ ΡΠ΅ΡΡΠΈΡΠ³ΠΎΠ»ΡΠ½ΠΎΠΉ ΠΌΠΎΠ·Π°ΠΈΠΊΠΈ. Π ΡΠ±ΡΠ° ΠΏΡΠ°Π²ΠΈΠ»ΡΠ½ΡΡ ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ½ΡΡ Π°ΠΏΠ΅ΠΉΡΠΎΠ³ΠΎΠ½ΠΎΠ² ΠΌΠΎΠ³ΡΡ ΡΠΎΠ΄Π΅ΡΠΆΠ°ΡΡ 2 ΠΈ Π±ΠΎΠ»Π΅Π΅ Π²Π΅ΡΡΠΈΠ½. ΠΡΠ°Π²ΠΈΠ»ΡΠ½ΡΠ΅ Π°ΠΏΠ΅ΠΉΡΠΎΠ³ΠΎΠ½Ρ p { q } r ΠΈΠΌΠ΅ΡΡ ΠΎΠ³ΡΠ°Π½ΠΈΡΠ΅Π½ΠΈΠ΅: 1/ p + 2/ q + 1/ r = 1. Π ΡΠ±ΡΠ° ΠΈΠΌΠ΅ΡΡ p Π²Π΅ΡΡΠΈΠ½ ΠΈ Π²Π΅ΡΡΠΈΠ½Π½ΡΠ΅ ΡΠΈΠ³ΡΡΡ ΡΠ²Π»ΡΡΡΡΡ r -ΡΠ³ΠΎΠ»ΡΠ½ΠΈΠΊΠ°ΠΌΠΈ [8] .
ΠΠ΅ΡΠ²ΡΠΉ Π°ΠΏΠ΅ΠΉΡΠΎΠ³ΠΎΠ½ ΡΠΎΡΡΠΎΠΈΡ ΠΈΠ· 2-ΡΡΠ±Π΅Ρ, ΠΏΠΎ ΡΡΠΈ Π²ΠΎΠΊΡΡΠ³ ΠΊΠ°ΠΆΠ΄ΠΎΠΉ Π²Π΅ΡΡΠΈΠ½Ρ, Π²ΡΠΎΡΠΎΠΉ ΠΈΠΌΠ΅Π΅Ρ ΡΠ΅ΡΡΠΈΡΠ³ΠΎΠ»ΡΠ½ΡΠ΅ ΡΡΠ±ΡΠ°, ΠΏΠΎ ΡΡΠΈ Π²ΠΎΠΊΡΡΠ³ ΠΊΠ°ΠΆΠ΄ΠΎΠΉ Π²Π΅ΡΡΠΈΠ½Ρ. Π’ΡΠ΅ΡΠΈΠΉ ΠΊΠΎΠΌΠΏΠ»Π΅ΠΊΡΠ½ΡΠΉ Π°ΠΏΠ΅ΠΉΡΠΎΠ³ΠΎΠ½, ΠΈΠΌΠ΅ΡΡΠΈΠΉ ΡΠ΅ ΠΆΠ΅ ΡΠ°ΠΌΡΠ΅ Π²Π΅ΡΡΠΈΠ½Ρ, ΠΊΠ²Π°Π·ΠΈΠΏΡΠ°Π²ΠΈΠ»Π΅Π½ ΠΈ Π² Π½ΡΠΌ ΡΠ΅ΡΠ΅Π΄ΡΡΡΡΡ 2-ΡΡΠ±ΡΠ° ΠΈ 6-ΡΡΠ±ΡΠ°.
| 2{12}3 or | 6{4}3 or |
|---|
See also
- Π¨Π΅ΡΡΠΈΡΠ³ΠΎΠ»ΡΠ½Π°Ρ ΡΠ΅ΡΡΡΠΊΠ°
- Π’ΡΠ΅ΡΠ³ΠΎΠ»ΡΠ½ΡΠ΅ ΠΏΡΠΈΠ·ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΡΠΎΡΡ
- ΠΠΎΠ·Π°ΠΈΠΊΠΈ ΠΈΠ· Π²ΡΠΏΡΠΊΠ»ΡΡ ΠΏΡΠ°Π²ΠΈΠ»ΡΠ½ΡΡ ΠΌΠ½ΠΎΠ³ΠΎΡΠ³ΠΎΠ»ΡΠ½ΠΈΠΊΠΎΠ² Π½Π° Π΅Π²ΠΊΠ»ΠΈΠ΄ΠΎΠ²ΠΎΠΉ ΠΏΠ»ΠΎΡΠΊΠΎΡΡΠΈ
- Π‘ΠΏΠΈΡΠΎΠΊ ΠΏΡΠ°Π²ΠΈΠ»ΡΠ½ΡΡ ΠΌΠ½ΠΎΠ³ΠΎΠΌΠ΅ΡΠ½ΡΡ ΠΌΠ½ΠΎΠ³ΠΎΠ³ΡΠ°Π½Π½ΠΈΠΊΠΎΠ² ΠΈ ΡΠΎΠ΅Π΄ΠΈΠ½Π΅Π½ΠΈΠΉ
Notes
- β 1 2 ΠΠΎΠ»ΠΎΠΌΠ±, 1975 , Ρ. 147.
- β Weisstein, Eric W. Dual Tessellation (Π°Π½Π³Π».) Π½Π° ΡΠ°ΠΉΡΠ΅ Wolfram MathWorld .
- β ΠΠΎΠ·Π°ΠΈΠΊΠ° Π½Π°Π·ΡΠ²Π°Π΅ΡΡΡ ΠΌΠΎΠ½ΠΎΡΠ΄ΡΠ°Π»ΡΠ½ΠΎΠΉ, Π΅ΡΠ»ΠΈ ΠΎΠ½Π° ΡΠΎΡΡΠΎΠΈΡ ΠΈΠ· ΠΊΠΎΠ½Π³ΡΡΡΠ½ΡΠ½ΡΡ ΠΏΠ»ΠΈΡΠΎΠΊ .
- β GrΓΌnbaum, Shephard, 1987 , Ρ. Sec. 9.3 Other Monohedral tilings by convex polygons.
- β GrΓΌnbaum, Shephard, 1987 , Ρ. 473β481, list of 107 isohedral tilings.
- β GrΓΌnbaum, Shephard, 1987 , Ρ. uniform tilings that are not edge-to-edge.
- β Critchlow, 1987 , Ρ. 74β75, pattern 2.
- β Coxeter, 1991 , Ρ. 111-112, 136.
Literature
- Π‘.Π. ΠΠΎΠ»ΠΎΠΌΠ±. ΠΠΎΠ»ΠΈΠΌΠΈΠ½ΠΎ = Polyominoes / ΠΠ΅Ρ. Ρ Π°Π½Π³Π». Π. Π€ΠΈΡΡΠΎΠ²Π°. ΠΡΠ΅Π΄ΠΈΡΠ». ΠΈ ΡΠ΅Π΄. Π. Π―Π³Π»ΠΎΠΌΠ°. β Π. : ΠΠΈΡ, 1975. β Π‘. 147. β 207 Ρ.
- HSM Coxeter . Regular Complex Polytopes. β 2ed. β New York, Port Chester, Melbourne, Sydney: Cambridge University Press, 1991. β ISBN 0-521-39490-2 .
- HSM Coxeter . Table II: Regular honeycombs // . β 3rd. β Dover, 1973. β Π‘. 296. β ISBN 0-486-61480-8 .
- B. GrΓΌnbaum , GC Shephard. Tilings and Patterns. β New York: WH Freeman & Co., 1987. β Π‘. 58β65. β ISBN 0-7167-1193-1 .
- R. Williams. The Geometrical Foundation of Natural Structure: A Source Book of Design. β New York: Dover Publications , 1979. β Π‘. 35. β ISBN 0-486-23729-X .
- John H. Conway , Heidi Burgiel, Chaim Goodman-Strass. The Symmetries of Things . β 2008. β ISBN 978-1-56881-220-5 . ΠΡΡ ΠΈΠ²ΠΈΡΠΎΠ²Π°Π½ΠΎ 19 ΡΠ΅Π½ΡΡΠ±ΡΡ 2010 Π³ΠΎΠ΄Π°. ΠΡΡ ΠΈΠ²Π½Π°Ρ ΠΊΠΎΠΏΠΈΡ ΠΎΡ 19 ΡΠ΅Π½ΡΡΠ±ΡΡ 2010 Π½Π° Wayback Machine
- Keith Critchlow. Order in Space: A design source book. β New York: Thames & Hudson, 1987. β ISBN 0-500-34033-1 .
Links
- Weisstein, Eric W. Hexagonal Grid (Π°Π½Π³Π».) Π½Π° ΡΠ°ΠΉΡΠ΅ Wolfram MathWorld .
- Weisstein, Eric W. Regular tessellation (Π°Π½Π³Π».) Π½Π° ΡΠ°ΠΉΡΠ΅ Wolfram MathWorld .
- Weisstein, Eric W. Uniform tessellation (Π°Π½Π³Π».) Π½Π° ΡΠ°ΠΉΡΠ΅ Wolfram MathWorld .
- Klitzing, Richard. 2D Euclidean tilings o3o6x β hexat β O3
- Amit Patel. Grid Math: Square, Hexagon, Triangle . β ΠΠ»Π³ΠΎΡΠΈΡΠΌΡ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΈΡ ΡΠ΅ΡΡΠΈΡΠ³ΠΎΠ»ΡΠ½ΠΎΠΉ ΠΈ ΡΡΠ΅ΡΠ³ΠΎΠ»ΡΠ½ΠΎΠΉ ΡΠ΅ΡΠΎΠΊ Π² ΠΊΠΎΠΌΠΏΡΡΡΠ΅ΡΠ½ΡΡ ΡΡΡΠ°ΡΠ΅Π³ΠΈΡΠ΅ΡΠΊΠΈΡ ΠΈΠ³ΡΠ°Ρ .