Documentation

This documentation describes the user interface of Tissellator. The steps on the homepage and this page help you to create your tessellations. Also have a look at the Transformations page and the Animations page. If something is unclear, or if you want to report comments and/or bugs, please mail me.

The user interface of Tissellator consists of the menu bar, menu icons, status bar, the view, and dialog windows.
N.B. Mac users should read Command+X instead of Ctrl+X.

Menu bar

N.B. After loading a new (multi) background image, a new (multi) tessellation must be created before being able to perform a transformation operation of the tessellated image.

Menu icons

	Delete the selected point.
	Unselect the selected point.
	Move a point to the neighbour point that is very close to it.
	The added point is a cubic point, that gives a curved segment. A segment is curved if one or both endpoints are cubic points. Toggle this button for switching to linear mode.
	The added point is a linear point, that takes care of a straight segment. A segment is straight if both endpoints are either linear points, or corners, or intermediate points, or mirror points.Toggle this button for switching to cubic mode.
	Zoom in by 5%.
	Zoom out by 5%.
	Zoom slider and zoom value. You can zoom in or out the tessellation by a maximum factor of 10. Note that the background image does not zoom or pan.
	Edit points or corners. Toggle this button for switching to pan mode.
	Pan the tiles and the outline. Toggle this button for switching to edit mode.
	Edit the points of the outline. Toggle this button for switching to editing corners.
	Edit the corners of the tile. Straight edges between the corners appear instead of the points. Each corner can be moved in a way that depends on the type of tessellation. Just try and see how the tile changes. Toggle this button for switching to editing outline points.
	This icon is intended for non-periodic tessellations. Show a dialog for changing the number of substitution iterations, and the fractions of the deformed tiles.

N.B. The icons have tooltips.

Status bar

The red box of the status bar shows information, warnings and errors about your actions. For example:

indicates starting a new project. In this case, the numbers 3 (for pentagon) and 23 (isohedral type) are codes for the tesselation.

View

Editing of the tessellation happens in the central part of the window. For example the figure below shows a triangle tessellation of isohedral type IH79 :

The grey outline contains different kind of points with a certain color, indicating their nature:

	Cubic point. This point causes a curved segment in the outline, made of a third degree polynomial. A segment is curved if one or both endpoints are cubic points.
	Linear point. This point takes care of a straight segment in the outline. A segment is straight if both endpoints are either linear points, or corners, or intermediate points, or mirror points.
	Corner point. The basic form of a tile is a polygon with a number of corners (e.g., three for the triangle above, labeled C1, C2, C3). Corners can be moved in "corner mode" : select icon to enable this. For moving a corner, you first have to select it with the mouse. Then, click at another location. Often, not only the corner itself moves, but also other corners of the tile. That is needed to maintain the characteristics of a tile: for example, it must still be a rectangle. Moving another corner may change the tile in a different way. Just try and see what happens. In this way you can also scale and/or rotate a tile. Some moves may lead to an invalid configuration, as indicated in the status bar, and cancel the move action. This happens especially for spiral and circle configurations. Just try a smaller move, or in another direction. Non-periodic tessellations scale and rotate with respect to the corner with index 0.
	Intermediate point. This point lies exactly in the middle of an edge between two corners, that has rotation symmetry (e.g., I1 in the triangle above lies between C1 and C2). An arrow outside the outline indicates (when activated by Show transformations) such a symmetry. An intermediate point can not be edited.
	Mirror point. This point lies on the perpendicular bisector of an edge between two corners, that has mirror symmetry. It occurs in tessellations IH12, IH13, IH66, and IH79MCE. An arrow outside the outline indicates (when activated by Show transformations) such a symmetry. A mirror point can only be edited by moving it along the perpendicular bisector.
	Square around selected point.

Clicking with the mouse in the view adds a cubic or linear point. However, when you click on an existing point, then that point becomes the selected point. Click on another location in the view to move the selected point. Also, a selected point can be processed with an appropriate menu edit operation. (Caveat: sometimes a point is deleted instead of selected; this unwanted behavior can be solved by pressing the Shift-button twice. It seems to be an internal bug that I can't fix, sorry.)
Two mouse shortcuts exist. When you press the control button and then click on an existing point, then that point is deleted. When you press the shift button and then click on an existing point, then the mode of that point is toggled from Cubic to Linear, or vice versa.

Editing an edge of the outline causes changes in related edges, depending on the transformation between those edges. Possible transformations are translation, rotation, mirror (or reflection), and glide ( kind of "upside down"). Also within an edge 2 transformations are possible: rotation or mirror. When choosing a new tessellation colored arrows in the dialog indicate the transformation. Here is an overview:

	Translation. The edge is often shifted to the opposite side of the tile, keeping the same orientation. It is also possible that the translated edge is rotated a bit, especially for the spiral tessellations. An edge and its related, translated edge have no corner in common (otherwise it would be a rotation).
	Rotation. An edge and its related, rotated edge have a corner in common. The edges rotate around the common corner. The rotation angle is often 60 degrees, or 90 degrees.
	Glide. This arrow is twisted upside down, and that is what also more or less happens with an edge. Instead of a normal translation, the translated corners are swapped together with the part of the outline between the 2 corners.
	Glide rotation. This transformation is a combination of a glide and a rotation, and it occurs between two neighbour edges with a common corner.
	Mirror (or reflection). The vertical bar in the middle of the arrow stands for a mirror, indicating an edge and its mirrored counterpart.
	Mirror rotation. This transformation is a combination of a mirror and a rotation, and it occurs between two neighbour edges with a common corner.
	This rare transformation is a combination of a translation and a 180 degrees rotation; therefore, I call it a twirl.
	Rotation of 180 degrees inside an edge. In the middle of the edge is an intermediate point. This arrow is drawn outside the tile.
	Mirror inside an edge. In the middle of the edge is a mirror point. This arrow is drawn outside the tile.

The following figures illustrate the transformations and the corresponding arrows, and also an example tessellation. The arrows have the same color within a tessellation if they refer to the same edge.

Dialog windows

Dialog: File / New

Create a new project and choose the tessellation type. Double click on one of the eight icons, or single click and select OK.

Dialog: File / Export Outline as SVG

Dialog: Edit / Move Point To ..

N.B. The coordinates mentioned are internal coordinates, independent of the zoom factor and the offset. The origin (0,0) of these coordinates lies in the middle of the screen.

Dialog: Tessellation / Load Multi Background

N.B. In case of spiral tessellations and circle tessellations, the number of background images is freely selectable. A dialog appears to ask for the number of images.

Dialog: Tessellation / Load Multi Background / Select color scheme circle/spiral tessellation

The color index is the sum of the values in the tangential direction and the radial direction, taken modulo the number of colors. The radial direction goes outward from the center of a spiral/circle; for a circle along a radial, and for a spiral it has the range of its number of arms. The tangential direction is (almost) perpendicular to the radial direction; for a circle the range equals the number of radials, and for a spiral the range is unlimited.

Dialog: Tessellation / Load Multi Background / Select color scheme non-periodic tessellation

Dialog: Tessellation / Create Tessellation

N.B. The same dialog appears for Create Multi Tessellation.

Dialog: Tessellation / Export Tiles as SVG

Dialog: Transform / Transform into circle

N.B. The same dialog appears for Tessellated image.
In case of a Background image the input image is always a rectangle; in case of a Tessellated image the input image is a repeating part of that tessellated image: it can be a rectangle or a parallelogram depending on the isohedral type. For example, the pentagon tessellation of isohedral type IH21 has always a corner of 60 degrees, so that the parallelogram has corners of 60 degrees and 120 degrees. Nevertheless, the transformation is conformal, meaning that the input image resembles the parts of the circular transformation very well.

Dialog: Transform / Transform into spiral

N.B. The same dialog appears for Tessellated image.
In case of a Background image the input image is always a rectangle; in case of a Tessellated image the input image is a repeating part of that tessellated image: it can be a rectangle or a parallelogram depending on the isohedral type. A parallelogram is deformed into a rectangle, so that the transformation is non-conformal. This remark holds also for the other spiral transformations below.

Dialog: Transform / Transform into double spiral

N.B. The same dialog appears for Tessellated image.

Dialog: Transform / Transform into multi spiral

N.B. The same dialog appears for Tessellated image.

Dialog: Transform / Transform into poly spiral

N.B. The same dialog appears for Tessellated image.

Dialog: Transform / Transform into sphere spiral

N.B. The same dialog appears for Tessellated image.
The transformation is based on the paper "Loxodromic Spirals in M. C. Escher's Sphere Surface" by James Marcotte and Matthew Salomone.

Dialog: Transform / Animation

Credits: The encoder of the animated GIF software comes from Kevin Weiner, refer to github.

Dialog: Transform / Background image / Transform into Circle Inversion

Dialog: Transform / Tessellated image / Transform into hyperbolic disk

Before transforming to a hyperbolic disk a (multi) tessellation must be executed via Tessellation/Create Tessellation or Tessellation/Create Multi Tessellation. The size of that created tessellation must be sufficiently high. Tissellator reports if the width or height is not sufficient (sometimes in several steps); in that case, create another tessellation with bigger size.

Dialog: Transform / Droste effect

In the Droste effect a part of the image is replaced by a smaller version of itself, and this process repeats yielding a fractal structure. Also, parts of the image can be rotated and scaled so that a spiral arises. The size and shape of the replaced part determine what pixels of the original image are used for rendering.

Dialog: Extra / Settings

Dialog: Extra / Register

Dialog: Menu icon / Substitution parameters

After determining the minimum and maximum X-coordinates and Y-coordinates of the substituted image, the fractions are used to calculate a target location between the minima/maxima around which the deformed tiles are selected.
Note that a change of any of these 3 parameters affects the selection of the deformed tiles, and the orientation of the whole image. As a consequence the background image(s) may not match the deformed tiles anymore! In that case, change the fractions a little bit.
For zero iterations the (special) initial configuration is shown.

New Project Dialogs

Dialog: File / New / Triangle

Choose one of the triangle tessellation types. Double click an icon, or single click and select OK.
The tessellation type IH79MCE is a variation of IH79, having the additional property of mirror symmetry on 2 edges. M.C. Escher used this configuration in his famous artworks Angels and Devils and Circle Limit IV (Heaven and Hell).

Dialog: File / New / Quadrilateral

Choose one of the quadrilateral tessellation types. Double click an icon, or single click and select OK.

Dialog: File / New / Pentagon

Choose one of the pentagon tessellation types. Double click an icon, or single click and select OK.

Dialog: File / New / Hexagon

Choose one of the hexagon tessellation types. Double click an icon, or single click and select OK.

Dialog: File / New / Spiral

Choose one of the spiral tessellation types. Double click an icon, or single click and select OK.
The differences between IH1a , and IH1b are explained at their dialogs.

Dialog: File / New / Circle

Choose one of the circle tessellation types. Double click an icon, or single click and select OK.
The differences between IH1a , IH1b , and IH1c are explained at their dialogs.

Dialog: File / New / Non-periodic

Choose one of the non-periodic tessellation types. Double click an icon, or single click and select OK.
The different configurations of some types are explained at their dialogs.

Dialog: File / New / Fractal

Choose one of the fractal tessellation types. Double click an icon, or single click and select OK.
The five fractal types open their own menu for further selection. The different configurations of some types are explained at their dialogs.

Dialog: File / New / Fractal / SquareLimitBasic

Choose one of the fractal tessellation types. Double click an icon, or single click and select OK.
These 4 tessellations have types of the same shape without reflections, unlike Escher's Square Limit. In SquareLimitBasic2 the scheme at the diagonals has been made consistent so that an octagon is formed. The latter 2 tessellations have also fractal structure in the center.
The height (or width) of the total image equals 6 times the height of the triangle to be deformed for the first 2 tessellation types, and 12 times its height for the last 2 tessellation types.

Dialog: File / New / Fractal / SquareLimitEscher

Choose one of the fractal tessellation types. Double click an icon, or single click and select OK.
The first tessellation is Escher's Square Limit. The other 3 are variations. In SquareLimitEscher2 the scheme at the diagonals has been made consistent so that an octagon is formed. The latter 2 tessellations have also fractal structure in the center.
The height (or width) of the total image equals 6 times the height of the triangle to be deformed for the first 2 tessellation types, and 12 times its height for the last 2 tessellation types.

Dialog: File / New / Fractal / SquareLimitDuo

Choose one of the fractal tessellation types. Double click an icon, or single click and select OK.
In these tessellations 2 edges can be deformed independently. Hence, 2 different tile shapes are created. In SquareLimitDuo2 the scheme at the diagonals has been made consistent so that an octagon is formed. The latter 2 tessellations have also fractal structure in the center.
The height (or width) of the total image equals 6 times the height of the triangle to be deformed for the first 2 tessellation types, and 12 times its height for the last 2 tessellation types.

Dialog: File / New / Fractal / SquareLimitMirror

Choose one of the fractal tessellation types. Double click an icon, or single click and select OK.
In these tessellations the two edges around the top of the deformed triangle have mirror symmetry, and the two edges at the base have point symmetry. In SquareLimitMirror2 the scheme at the diagonals has been made consistent so that an octagon is formed. The latter 2 tessellations have also fractal structure in the center.
The height (or width) of the total image equals 6 times the height of the triangle to be deformed for the first 2 tessellation types, and 12 times its height for the last 2 tessellation types.

Dialog: File / New / Fractal / FtilingSquare

Choose one of the fractal tessellation types. Double click an icon, or single click and select OK.
These tessellations resemble the above Square Limit tessellations. The big difference is that here fractal structures occur not only at the edges but also halfway the picture, and also in the center for the latter 4 tessellations.
FtilingSquare1 has one tile without reflections. FtilingSquare2 has a tile with reflections of that tile. In FtilingSquare3 two edges can be deformed resulting in two different tile shapes. The edges in FtilingSquare4 have mirror symmetry causing also two different shapes.
The next 4 tessellations have the same deformations as the first 4, but now also with fractal structure in the center.
For the first 4 tessellation types, the height (or width) of the total image equals 8 times the height of the triangle to be deformed. The height and width of the latter 4 tessellation types are twice as big as the former 4.
N.B. The basic structure of this fractal image has been discovered by Robert Fathauer in 1993, see his “Bats and Owls”. The triangle scheme of these tessellations are described as the f(4,4) tiling in the paper “Self-Similar Fractal Drawings Inspired by M.C. Escher’s Print Square Limit”.

Dialog: File / New / Fractal / FtilingHexagon

Choose one of the fractal tessellation types. Double click an icon, or single click and select OK.
These tessellations are analoguous to the above Ftiling Square tessellations, but now with hexagons instead of squares.
FtilingHexagon1 has one tile without reflections. FtilingHexagon2 has a tile with reflections of that tile. In FtilingHexagon3 two edges can be deformed resulting in two different tile shapes. The edges in FtilingHexagon4 have mirror symmetry causing also two different shapes.
The next 4 tessellations have the same deformations as the first 4, but now also with fractal structure in the center.
For the first 4 tessellation types, the height of the total image equals 4 times the height of the triangle to be deformed, and the width is a factor 2/sqrt(3) bigger than the height. The height and width of the latter 4 tessellation types are twice as big as the former 4.
The triangle scheme of these tessellations are based on the f(6,3) tiling in the paper “Self-Similar Fractal Drawings Inspired by M.C. Escher’s Print Square Limit”. N.B. The fractal in the center of the latter 4 tessellations was added by myself.

Dialog: File / New / Fractal / Vshaped

Choose one of the fractal tessellation types. Double click an icon, or single click and select OK.
Vshaped1 has one tile without reflections. Vshaped2 has a tile with reflections of that tile. In Vshaped3 two edges can be deformed resulting in two different tile shapes. The edges in Vshaped4 and Vshaped5 have mirror symmetry causing also two different shapes. You can choose the number of directions, being 5 and 6 in the shown icons ! For the latter 2 types this number must be even due to the mirror symmetry.
The name of these tessellations comes from the shape of the tile. Every parent tile has 3 child tiles, but neighbor parents share one child.

New Spiral Dialogs

Dialog: New / Spiral / Triangle-side-to-side

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.

Dialog: New / Spiral / Four-armed-mirror spiral

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".

Dialog: New / Spiral / Quadrilateral spiral

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".

Dialog: New / Spiral / Bi-Logarithmic spiral

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.

Dialog: New / Spiral / IH1a

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:

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.

Dialog: New / Spiral / IH1b

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:

The corner index is shown in the Status Bar after selection. Invalid spiral configurations are detected, skipped and reported in the Status Bar.

Note: The computation of a spiral (multi) tessellation is a lengthy process. Please, be patient.

New Circle Dialogs

Dialog: New / Circle / IH1a

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:

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.

Dialog: New / Circle / IH1b

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:

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.

Dialog: New / Circle / IH1c

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:

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.

Dialog: New / Circle / IH2

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:

The corner index is shown in the Status Bar after selection. Invalid circle configurations are detected, skipped and reported in the Status Bar.

Dialog: New / Circle / IH41a

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.

Dialog: New / Circle / IH41b

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.

Dialog: New / Non-periodic / Chair

Dialog: New / Non-periodic / AmmannA3

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.

Dialog: New / Non-periodic / AmmannA4

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.

Dialog: New / Non-periodic / Ammann-Beenker

The 2 initial configurations of this tessellation correspond to the 2 prototiles, refer to: Tilings Encyclopedia / Ammann-Beenker.

Dialog: New / Non-periodic / Penrose P2

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.

Dialog: New / Non-periodic / Penrose P3

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.

Dialog: New / Non-periodic / Minimal7

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).

Dialog: New / Non-periodic / Danzer7Original

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.

Dialog: New / Non-periodic / MultiGrid

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.

MultiGridAll

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.

MultiGridOne

In this grid the edges have the same deformation in all directions, yielding (N-1) prototiles.

MultiGridTwo

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:

Dialog: New / Non-periodic / MultiGridAll

MultiGridOne has the same dialog.

Dialog: New / Non-periodic / MultiGridTwo

Dialog: New / Non-periodic / MultiGrid gamma values

Dialog: New / Non-periodic / Einstein

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.

The tessellations Vierstein and Spectre have a subset of the above parameters.