| Field | Description | Spiral |
| Number (>=3) of leaf triangles | The number of leaf triangles determines how quickly the spiral goes "round". In case of 8 leaf triangles the spiral is as follows: | |
N.B. A similar dialog appears for the Triangle-side-to-base, and for Triangle-base-to-side. The only difference is that the lower limit for the number of leaf triangles is 4 and 5, respectively.
Detailed information about these kind of triangle spirals can be found in the paper "Logarithmic Spiral Tilings of Triangles" by Robert Fathauer.
| Field | Description | Spiral |
| Angle in degrees | Angle (in degrees) at bottom right corner of red triangle tile. |  |
| Length base edge | The length of the base edge (i.c. 0.55) is entered as a relative value compared to the right edge with length 1.0 . |
The above two parameters determine the configuration of this four armed mirror spiral. More background information can be found in my paper "Four-armed spiral tiling of scalene triangles".
| Field | Description | Spiral |
| Number of leaf quadrilaterals |
The number of leaf triangles determines how quickly the spiral goes "round". In case of 4 leaf triangles the spiral is as follows: |  |
| Angle in degrees | Angle (in degrees) at bottom right corner of orange quadrilateral tile. | |
| Length base edge |
The length of the base edge (i.c. 0.55) is entered as a relative
value compared to the right edge with length 1.0 . |
The above three parameters determine the configuration of this quadrilateral spiral. Some background information can be found in my paper "The computation of the spiral center of triangles and quadrilaterals".
| Field | Description | Spiral |
| Number of arms clockwise direction |
The design of this spiral consists of points that rotate and scale in clockwise direction, and of points that rotate and scale in counterclockwise direction. For the default parameters, the points rotate 60 degrees in clockwise direction and scale a ratio of 1.25 . After 5 such operations a point is back to its original spiral arm, indicated as A-B-C. |  |
| Number of arms counterclockwise direction |
For the default parameters, the points rotate in counterclockwise direction the remaining 360 - 300 = 60 degrees. So, 20 degrees per point, and the ratio can be computed as: 1.25^(5/3). After 3 such rotate and scale operations a point is back to its original spiral arm, indicated as 1-2-3-4-5. | |
| Angle range clockwise direction in degrees | In clockwise direction the points rotate (for the default parameter) each 60 degrees, so in total 5 * 60 = 300 degrees. This covers the blue part of the circle. | |
| Ratio clockwise direction | In clockwise direction the points scale (for the default parameter) a factor 1.25, or its reciprocal 0.8 . |
Once chosen, the number of arms can not be changed anymore for a spiral. By editing tile corners the angle range and the ratio can still be varied. Invalid spiral configurations are detected, skipped and reported in the Status Bar.
| Field | Description | Spiral |
| Number of hexagons per revolution | In this spiral configuration all hexagons are similar: having the same shape but different size. For the default parameters, after 7 rotations, indicated as 1-2-3-4-5-6-7 the hexagon lies "next" to the base hexagon that is grey outlined. |  |
| Number of spiral arms | In counterclockwise direction it takes 3 steps to arrive at that 7th hexagon, indicated as A-B-C. |
|
| Factor diameter in range [ 0.0 - 1.0 ] | The diameter factor affects the thickness of the hexagon. The rotation to the next hexagon can be considered as a rotation around the middle (=diameter) of a hexagon plus a rotation around the top edge of a (staggered) hexagon. The diameter can be varied, ranging from 0.0 to 1.0 . In case of 0.0 the hexagon becomes 2 triangles, whereas in case of 1.0 the hexagon becomes a quadrilateral. In the latter case, the form of the quadrilateral can not be changed back to a hexagon. | |
| Radial ratio | This ratio affects the height of the hexagon. | |
| Skew in degrees | The skew gives a twist to the spiral. In case of zero skew, the ray to the spiral center is a straight line. |
Once chosen, the number of arms can not be changed anymore for a spiral. The other parameters can be changed by moving the corners as follows:
| Corner[0] | Rotate whole hexagon. |
| Corner[1] | Change diameter with focus on top/bottom edge. (Angle alpha) |
| Corner[2] | Rotate whole hexagon. |
| Corner[3] | Change skew. |
| Corner[4] | Change ratio. |
| Corner[5] | Change diameter with focus on middle. (Angle beta) |
The corner index is shown in the Status Bar after selection. Invalid spiral configurations are detected, skipped and reported in the Status Bar.
More background information about the construction of this spiral can be found in my paper "Logarithmic spiral tiling of hexagons".
Note: The computation of a (multi) tessellation is a lengthy process. Please, be patient.
| Field | Description | Spiral |
| Number of hexagons per revolution | In this spiral configuration all hexagons are similar: having the same shape but different size. For the default parameters, after 8 rotations, indicated as 1-2-3-4-5-6-7-8 the hexagon lies "next" to the base hexagon that is grey outlined. |  |
| Number of spiral arms | In counterclockwise direction it takes 3 steps to arrive at that 8th hexagon, indicated as A-B-C. |
|
| Factor diameter in range [ 0.0 - 1.0 ] | The diameter factor affects the thickness of the hexagon. The rotation to the next hexagon can be considered as a rotation around the middle (=diameter) of a hexagon plus a rotation around the top edge of a (staggered) hexagon. The diameter can be varied, ranging from 0.0 to 1.0 . In case of 0.0 the hexagon becomes 2 triangles, whereas in case of 1.0 the hexagon becomes a quadrilateral. In the latter case, the form of the quadrilateral can not be changed back to a hexagon. | |
| Radial ratio | This ratio affects the height of the hexagon. | |
| Skew in degrees | The skew gives a twist to the spiral. In case of zero skew, the ray to the spiral center is a straight line. |
Once chosen, the number of arms can not be changed anymore for a spiral. The other parameters can be changed by moving the corners as follows:
| Corner[0] | Rotate whole hexagon. |
| Corner[1] | Change diameter with focus on middle. |
| Corner[2] | Rotate whole hexagon. |
| Corner[3] | Change skew. |
| Corner[4] | Change ratio. |
| Corner[5] | Change diameter with focus on left/right edge. |
The corner index is shown in the Status Bar after selection. Invalid spiral configurations are detected, skipped and reported in the Status Bar.
Looking from the center the hexagon of this spiral (IH1b) is more or less 90 degrees rotated compared to the previous spiral IH1a. In case of 1 spiral arm the corners of a IH1a hexagon are all on the same spiral line; for IH1b the top corner (far from center) and the bottom corner (close to center) are on different spiral lines: the top corner is on the same spiral line as the 2 lower (close to center) corners left and right, whereas the bottom corner is on the same spiral line as the 2 upper (far from center) corners left and right.Note: The computation of a spiral (multi) tessellation is a lengthy process. Please, be patient.
| Field | Description | Circle |
| Number of radials | In this circle configuration all hexagons are similar: having the same shape but different size. Abbreviate the number of radials by N. For each radial the hexagons are the same, but rotated by the angle obtained by dividing 360 degrees by N. So, N hexagons have the same size. |  |
| Radial ratio | The hexagons grow by this ratio. It is the quotient of the distance of P3 to center C divided by the distance of P2 to C. Note that P0 and P2 lie on the same circle around C, just like P3 and P5. | |
| Radial skew in degrees | The angle from P2 to P3 seen from center C. | |
| Top ratio | The quotient of the distance of P4 to center C divided by the distance of P3 to C. | |
| Top skew in degrees | The angle from P3 to P4 seen from center C. |
Once chosen, the number of radials can not be changed anymore for a circle. The other parameters can be changed by moving the corners as follows:
| Corner[0] | Rotate whole hexagon. |
| Corner[1] | Change top ratio and top skew. |
| Corner[2] | Rotate whole hexagon. |
| Corner[3] | Change radial ratio and radial skew. |
| Corner[4] | Change top ratio and top skew. |
| Corner[5] | Change radial ratio and radial skew. |
The corner index is shown in the Status Bar after selection. Invalid circle configurations are detected, skipped and reported in the Status Bar.
The circle tessellation IH3 satisfies also the above description.
Note: The computation of a circle (multi) tessellation is a lengthy process. Please, be patient.
| Field | Description | Circle |
| Number of radials | In this circle configuration all hexagons are similar: having the same shape but different size. Abbreviate the number of radials by N. For each radial the hexagons are the same, but rotated by the angle obtained by dividing 360 degrees by N. So, N hexagons have the same size. Note that between 2 hexagons of the same size there are 2 other hexagons with smaller and bigger sizes. |  |
| Radial ratio | The hexagons grow by this ratio. It is the quotient of the distance of P3 to center C divided by the distance of P1 to C. Note that P0 and P1 lie on the same circle around C, just like P3 and P4, and also P2 and P5. | |
| Radial skew in degrees | The angle from P1 to P3 seen from center C. | |
| Middle skew in degrees | The angle from P0 to P5 seen from center C. |
Once chosen, the number of radials can not be changed anymore for a circle. The other parameters can be changed by moving the corners as follows:
| Corner[0], Corner[1] | Rotate whole hexagon. |
| Corner[3], Corner[4] | Change radial ratio and radial skew. |
| Corner[2], Corner[5] | Change middle skew, and radial ratio. |
The corner index is shown in the Status Bar after selection. Invalid circle configurations are detected, skipped and reported in the Status Bar.
Looking from the center the hexagon of this circle tessellation (IH1b) is more or less 90 degrees rotated compared to the previous circle tessellation IH1a.
Note: The computation of a circle (multi) tessellation is a lengthy process. Please, be patient.
| Field | Description | Circle |
| Number of radials | In this circle configuration all pentagons are similar: having the same shape but different size. Abbreviate the number of radials by N. For each radial the pentagons are the same, but rotated by the angle obtained by dividing 360 degrees by N. So, N pentagons have the same size. |  |
| Ratio | The pentagons grow by this ratio. It is the quotient of the distance of P3 to center C divided by the distance of P2 to C. Note that P0 and P2 lie on the same circle around C, just like P3 and P5. | |
| Skew in degrees | The angle from P2 to P3 seen from center C. |
Once chosen, the number of radials can not be changed anymore for a circle. The other parameters can be changed by moving the corners as follows:
| Corner[0], Corner[1], Corner[2] | Rotate whole pentagon. |
| Corner[3], Corner[5] | Change ratio and skew. |
| Corner[4] | Change ratio. |
The corner index is shown in the Status Bar after selection. Invalid circle configurations are detected, skipped and reported in the Status Bar.
Point P1 lies on the line from P0 to P2. Therefore, the "hexagon" of 6 points becomes a pentagon.
Note: The computation of a circle (multi) tessellation is a lengthy process. Please, be patient.
| Field | Description | Circle |
| Number of radials | In this circle configuration all hexagons are similar, or have mirror symmetry. Each radial has a ray of similar tiles and a ray of mirrored, similar tiles. |  |
| Radial ratio | The hexagons grow by this ratio. It is the quotient of the distance of P1 to center C divided by the distance of P5 to C. This quotient is the same for the pair P2 and P4. | |
| Skew in degrees | The angle from P0 to P1 seen from center C. | |
| Top ratio | The quotient of the distance of P2 to center C divided by the distance of P1 to C. This quotient is the same for the pair P4 and P5. | |
| Top skew in degrees | The default angle between P2 and P1 equals 180 degrees divided by the number of radials. This angle is lowered by the top skew angle. Note that it affects the position of P3. |
Once chosen, the number of radials can not be changed anymore for a circle. The other parameters can be changed by moving the corners as follows:
| Corner[0], Corner[3], Corner[4], Corner[5] | Rotate whole hexagon. |
| Corner[1] | Change radial ratio and skew. |
| Corner[2] | Change top ratio and top skew. |
The corner index is shown in the Status Bar after selection. Invalid circle configurations are detected, skipped and reported in the Status Bar.
Note: The computation of a circle (multi) tessellation is a lengthy process. Please, be patient.
| Field | Description | Circle |
| Number of radials | In this circle configuration all quadrilaterals are similar: having the same shape but different size. Abbreviate the number of radials by N. For each radial the quadrilaterals are the same, but rotated by the angle obtained by dividing 360 degrees by N. So, N quadrilaterals have the same size. |  |
| Ratio | The quadrilaterals grow by this ratio. It is the quotient of the distance of P2 to center C divided by the distance of P1 to C. Note that P0 and P1 lie on the same circle around C, just like P2 and P3. | |
| Skew in degrees | The angle from P1 to P2 seen from center C. |
The circle tessellation IH43 satisfies also the above description.
| Field | Description | Circle |
| Number of radials | In this circle configuration all quadrilaterals are similar: having the same shape but different size. Abbreviate the number of radials by N. For each radial the quadrilaterals are the same, but rotated by the angle obtained by dividing 360 degrees by N. So, N quadrilaterals have the same size. |  |
| Ratio | The quadrilaterals grow by this ratio. It is the quotient of the distance of P1 to center C divided by the distance of P0 to C. Note that P0 and P2 lie on the same circle around C. | |
| Skew in degrees | The angle from the middle of P0 and P2 to P1 seen from center C. |
The circle tessellation IH44 satisfies also the above description.
| Field | Description |
| Number of iterations | Number of substitution iterations, starting from the initial chair configuration. |
| X-fraction (0.0 - 1.0) | Fraction of X-coordinate indicating the tile to be deformed. |
| Y-fraction (0.0 - 1.0) | Fraction of Y-coordinate indicating the tile to be deformed. |
| Configuration (1-3) | Select the initial configuration. |
The 3 initial configurations correspond to the 3 prototiles, refer to: Tilings Encyclopedia / Ammann A3.
The other non-periodic tessellations with multiple initial configurations have the same dialog.
The 3 initial configurations of this tessellation correspond to the 3
prototiles, refer to: Tilings Encyclopedia / Ammann A4.
Note that the third prototile is a mirror of the second prototile.
The 2 initial configurations of this tessellation correspond to the 2 prototiles, refer to: Tilings Encyclopedia / Ammann-Beenker.
The 9 initial configurations of this tessellation correspond to 7 special configurations, see image below, and the 2 prototiles, refer to: Tilings Encyclopedia / Penrose Kite Dart.
The 10 initial configurations of this tessellation correspond to 8 special configurations, see image below, and the 2 prototiles, refer to: Tilings Encyclopedia / Penrose Rhomb.
The 4 initial configurations of this tessellation correspond to a 7-star and the 3 prototiles, refer to: A Minimal 7-Fold Rhombic Tiling (Fig. 5).
The 8 initial configurations of this tessellation correspond to two 14-stars, the 3 prototiles, and the 3 mirrored prototiles, refer to: Tilings Encyclopedia / Danzer's 7-fold original.
The family of multigrid tessellations fills the plane in a non-periodic
way with rhombuses. The user must choose the number of directions, say N.
The angles of the rhombuses are then multiples of Pi/N. In each direction
there is a grid of parallel lines, crossing the parallel lines of another
direction. At each crossing a rhombus is constructed with edges
perpendicular to the lines. The user can choose for each direction the
value of a variable, named gamma, that represents the distance of the
first line closest to the origin. Suitable gamma sets are given below. At
a so called regular point exactly 2 lines cross, whereas at singular
points more than 2 lines cross. Singular points cause failures of the
tessellation: you will recognize this situation from uncovered areas and
overlapping tiles. No measures are taken to prevent this, so that in such
cases the gammas must be adapted for solving it.
The edges of the rhombuses can be deformed in different ways. Three
variations have been implemented.
In this grid the edges for each direction have their own deformation, yielding N * (N-1)/2 prototiles. So, the opposite edges of each rhombus have the same deformation.
In this grid the edges have the same deformation in all directions, yielding (N-1) prototiles.
This grid is limited to 5 directions, so that the rhombuses are like the
Penrose P3 tessellation. Hence, there are 2 kinds of edge deformation.
This tessellation suffers most from singular points, unlike the above two
multigrid tessellations. Therefore, predefined gamma sets are available.
Furthermore, for a particular class of gamma sets one of the edges must be
deformed symmetrical for a valid tessellation.
Examples of valid gamma sets:
| image description | gamma set | TIS file |
| no need for tile symmetry. | 0.40 0.40 -0.40 -0.40 0.00 | MgTwo_1.tis |
| tile symmetry needed; image has 180 degrees symmetry. A tessellation of size 5000 x 4000 pixels gives a correct image for this TIS file (with view 1280 x 700 pixels). |
0.45 0.45 0.45 0.45 0.00 | MgTwo_2.tis |
| tile symmetry needed; image has no symmetry. | 0.40 -0.40 -0.40 0.40 0.00 | MgTwo_3.tis |
| Field | Description |
| Number of directions | Number of directions. |
| X-fraction (0.0 - 1.0) | Fraction of X-coordinate indicating the tile to be deformed. |
| Y-fraction (0.0 - 1.0) | Fraction of Y-coordinate indicating the tile to be deformed. |
| Gamma's | Open another dialog to enter gamma values. |
MultiGridOne has the same dialog.
| Field | Description |
| X-fraction (0.0 - 1.0) | Fraction of X-coordinate indicating the tile to be deformed. |
| Y-fraction (0.0 - 1.0) | Fraction of Y-coordinate indicating the tile to be deformed. |
| 180 degrees rotation symmetry | Select this checkbox if one of the edges requires rotation symmetry. In case of (un)selecting the box, the gamma set gets predefined values (overwriting the previous values). |
| Gamma's | Open another dialog to enter gamma values. |
| Field | Description |
| Gamma[x] | The gamma value for direction 'x'. |
Recently David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim
Goodman-Strauss discovered an aperiodic monotile, nicknamed einstein.
Details are given in this paper.
Note that deforming the edges of the einstein causes two different tiles
due to the fact that the tessellation also includes the mirrored version
of the einstein.
| Field | Description |
| Number of iterations | Number of substitution iterations, starting from the initial configuration. |
| X-fraction (0.0 - 1.0) | Fraction of X-coordinate indicating the tile to be deformed. |
| Y-fraction (0.0 - 1.0) | Fraction of Y-coordinate indicating the tile to be deformed. |
| Configuration (1-4) | Select the initial configuration based on the metatile: T, H, P, F. |
| Shape (0.0 - 1.0) | The einstein tile has 2 edges with lengths called A and B. This
shape parameter determines the (relative) lengths as: A =
sqrt(shape) and B = sqrt(1-shape) . Common values are: 0.25 = hat, 0.4 = bird, 0.5 = turtle, 0.75 = penguin. |
The tessellations Vierstein and Spectre have a subset of the above parameters.
In all curve configurations the shape can be 'line' or 'tile'. The 'line' shape reveals the path of the curve. Therefore, this shape can not be deformed, unlike the 'tile' shape.
| Field | Description | ||||||
| Configuration (0-15) | Each of the 4 edges can be reversed, giving 2^4 = 16 choices. | ||||||
| Range |
|
| Field | Description | ||||||
| Configuration (1-8) | Index to the configuration, see overview below. | ||||||
| Range |
|
The generators of the Root-7-Triangle grid family are not self-avoiding, so that only Edge based tessellations are available. Index 1-8:
| Field | Description | ||||||
| Range |
|
The above dialog also applies for the Root-12-Triangle tessellation.
These plane-filling curves traverse all edges of a triangular grid once,
refer to the paper "Plane-Filling
curves on all uniform grids", by
Arndt.
| Field | Description | ||||||
| Order | The order equals the number of curve segments of the generator. Available orders: 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 28. For orders 12, 13, 16, 19 the duplicate indices have been removed. For order 25 only a subset of indices is available. For orders 25 and 28 only a single index is available. |
||||||
| Index (1-10) | Index to the configuration, depending on the order. | ||||||
| Range |
|
The Hilbert curve is one of the most famous space-filling curves.
| Field | Description | ||||||||||
| Number of iterations | Number of curve iterations. | ||||||||||
| X-fraction (0.0 - 1.0) | Fraction of X-coordinate indicating the tile to be deformed. | ||||||||||
| Y-fraction (0.0 - 1.0) | Fraction of Y-coordinate indicating the tile to be deformed. | ||||||||||
| Shape |
|
||||||||||
| Tessellation |
|
||||||||||
| Range |
|
This dialog is the same as for CircleHilbert.
The fields 'Number of iterations', 'X-fraction (0.0 - 1.0)', 'Y-fraction (0.0 - 1.0)', 'Shape', 'Tessellation' also hold for the dialogs below.
The Peano curve was the first space-filling curve, discovered in 1890. The curve in 'Line' shape was used by Giuseppe Peano to proof the space-filling property.
| Field | Description | ||||||||||||
| Configuration | A line is drawn recursively through a 3x3 grid of squares from top left to bottom right. On each iteration, a square is replaced by a smaller 3x3 grid, etc. In the original curve, a line only looks like a Z, or its horizontally/vertically mirrored version. (Although there are 8 ways due to rotation and flipping.) The Z can also be flipped diagonally, creating an upside-down N. In each of the 9 basic squares a choice can be made between Z and N. So, there are 2^9 = 512 variants. | ||||||||||||
| Tessellation |
|
||||||||||||
| Range |
|
This dialog is the same as for CirclePeano.
The space-filling PeanoMeander curve has an R-like shape, unlike the normal Peano curve with a Z-like or N-like shape.
| Field | Description | ||||||||||
| Configuration | See Peano parameters. | ||||||||||
| Tessellation |
|
||||||||||
| Range |
|
This dialog is the same as for CirclePeanoMeander.
The PeanoMix curve is a mixture of normal Peano curves and PeanoMeander curves. There are 128 combinations to fill a normal Peano 3x3 grid with Z-shapes and R-shapes, where the starting point and end point are diagonally opposite each other in a square. And, there are 80 combinations to fill a Meander Peano 3x3 grid, where the starting point and end point lie along the same side of a square. Each of the 9 shapes in a combination can be mirrored. So, there are 128 * 512 * 80 * 512 = 2.684.354.560 combinations (not counting the rotations and reflections).
| Field | Description | ||||||
| Index-Z (1-128) | Index to normal Peano curve combination. | ||||||
| Configuration-Z (0-511) | Refer to normal Peano curve configuration. | ||||||
| Index-R (1-80) | Index to PeanoMeander curve combination. | ||||||
| Configuration-R (0-511) | Refer to PeanoMeander curve configuration. | ||||||
| Tessellation |
|
||||||
| Range |
|
This dialog is for Root-5-Square, Root-13-Square, Root-17-Square,
Root-20-Square and Root-45-Square tessellations.
The Root-5-Square curve is also known as: Mandelbrot's Quartet. According
to McKenna it is the initial curve of 2 families, namely Eddy Curve
G(1,2), see [1, page 69] and Frenzy Curve G(1,2), see [1, page 70].
| Field | Description | ||||
| Range |
|
Four generators of the Root-13-Square grid family, shown with Edge based tessellation. Index 1-4:
 |
 |
 |
 |
The Root-13-Square curve with index 1 is Frenzy Curve G(2,3) according to McKenna, see [1, page 70].
Five generators of the Root-17-Square grid family, shown with Edge based tessellation. Index 1-5:
 |
 |
 |
 |
 |
The Root-17-Square curve with index 1 is the only cousin of Eddy Curve
G(1,4) according to McKenna, see [1, page 71]. It is used by Richard
Hassell in his artwork "Coral Geckos I" (2018).
The curve with index 2 is Eddy Curve G(1,4) according to McKenna, see [1,
page 69]. It can also be created by the Eddy Curve family.
The curve with index 3 is not self-avoiding.
Mandelbrot designed the curve with index 5, see [2, page 127]. The curve
with index 4 is a self-avoiding alternative.
The generators of the Root-20-Square grid family are not self-avoiding,
so that only Edge based tessellations are available. The family has 3
polyomino types.
A curve starts at the blue dot and ends at the red dot.
Index 1-3:
 |
 |
Index 4-8:
 |
 |
Index 9-17:
 |
 |
The generators of the Root-45-Square grid family are all self-avoiding. The family has 2 polyomino types.
Index 1-41:
 |
 |
Index 42-68:
 |
 |
| Field | Description | ||||||
| Index configuration (1-14) | Index to the configuration, see overview below. | ||||||
| Range |
|
The generators of the Root-16-Square grid family are not self-avoiding,
so that only Edge based tessellations are available. The polyomino is a
square of 4 x 4 tiles.
Index 1-14:
| Field | Description | ||||
| Index configuration (1-18) | Index to the configuration, see overview below. | ||||
| Range |
|
All these tessellations are edge based. The paths of the possible curves
are shown below with their polyominoes.
This tessellation is not self-avoiding. Index 1 (top-left) to 15
(bottom-right).
 |
 |
This tessellation with index 16 is self-avoiding. It is named Frenzy Curve G(3,4) according to McKenna, see [1, page 70]. It can also be created by the Frenzy Curve family.
 |
 |
This tessellation is self-avoiding. Index 17-18:
 |
 |
This self-avoiding tessellation with index 19 is called E-curve by McKenna, see [1, page 60]. It can also be created by the SquaRecurve family.
 |
 |
This self-avoiding tessellation with index 20 is a perturbed order 5 square grid according to McKenna, see [1, page 67].
 |
 |
| Field | Description | ||||
| Index Straight Tile (1-38) | Index to the tile with straight path, see overview below. | ||||
| Index Corner Tile (1-28) | Index to the tile with corner in path, see overview below. | ||||
| Range |
|
The configurations of the possible path elements are shown below, starting from top-left (1) to bottom-right (38 c.q. 28).
| Straight Tile | Corner Tile |
 |
 |
| Field | Description | ||||||||
| Family |
|
||||||||
| Index Configuration (1-99) | Size index of a curve. The number of tiles
of the curve is about the square of the index! Note that some indexes are rounded to an even or an odd integer. |
||||||||
| Range |
|
N.B. The number of iterations can best be started with 1 or 2.
These plane-filling curves traverse all edges of a square grid once,
refer to the paper "Plane-Filling
curves on all uniform grids", by
Arndt.
| Field | Description | ||||||
| Order | The order equals the number of curve
segments of the generator. Available orders: 5, 9, 13, 17, 25. For order 25 the duplicate indices have been removed. |
||||||
| Index (1-5) | Index to the configuration, depending on the order. | ||||||
| Range |
|
Order 13 has 5 configurations:
Order 17 has 13 configurations:
Order 25 has 33 unique configurations:
These plane-filling curves are constructed by the folding principle,
refer to the paper "Plane-Filling
Folding Curves on the Square Grid", by
Arndt and Julia Handl. The
well known example of this type of curves is the Heighway-Harter
dragon.
| Field | Description | ||||||
| Order | The order equals the number of curve segments of the generator. Available orders: 2, 4, 5, 8, 9, 10, 13, 16, 17, 18. For orders 17 and 18 the duplicate indices have been removed. |
||||||
| Index (1-31) | Index to the configuration, depending on the order. | ||||||
| Range |
|
| Field | Description | ||||||||||||
| Configuration |
|
||||||||||||
| Range |
|
| Field | Description | ||||||||||||
| Tessellation |
|
||||||||||||
| Range |
|
The curve has 7 configurations:
The curve has 8 configurations:
These plane-filling curves traverse all edges of a tri-hexagonal grid
once, refer to the paper "Plane-Filling
curves on all uniform grids", by
Arndt.
| Field | Description | ||||||
| Order | The order equals the number of curve segments of the generator. Available orders: 7, 13, 19, 25, 31. For orders 31 the duplicate indices have been removed. |
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| Index (1-4) | Index to the configuration, depending on the order. | ||||||
| Range |
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