Geometry Defense: Infinite Mac OS
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The kuta software – infinite geometry is developing at a frantic pace. New versions of the software should be released several times a quarter and even several times a month.
Infinite Dreams Inc. Is a game development studio located in Gliwice, Poland. With over 10 years of experience in the industry, the company develops top quality games for a broad range of digital platforms such as iOS, Android, Mac OS, Windows Phone, Symbian and beyond.
Update for kuta software – infinite geometry.
There are several reasons for this dynamic:
First, new technologies are emerging, as a result, the equipment is being improved and that, in turn, requires software changes.
Secondly, the needs of users are growing, requirements are increasing and the needs are changing for kuta software – infinite geometry.
Therefore, it is necessary to monitor changes in the kuta software – infinite geometry and to update it in a timely manner.
/ If you do not update
There are many sources for obtaining information on software.
Software:
December 2001
June 2006
Named after Sebastien Truchet (1657 - 1729) who studied tiling of squares with a diagonal orientation.
2 Dimensions
Single cell| POVRay source, tile on the x-y plane |
2x2 tilings
Each unit is randomly rotated by 0 or 90 degrees or flipped verticallyor horizontally, there are only two unit states.
5x5 tilings
Larger tiling [POVRay source]
3 Dimensions
Single cell| POVRay source |
2x2 tilings
Each unit is randomly rotated by integer multiples of 90 degrees about eachaxis and randomly mirrored about each Cartesian axis plane (xy, xz, yz).
3x3x3 tilings
Larger tiling [POVRay source]
Contribution by Robert McGrego
Contribution by Robert McGrego
December 2006
2D tiling ... actually an orthographic projection of a 3D model
Perspective from the same position as above
Perspective projection from another position
POVRay scene file: blocktile.pov
Inspired by work by Quincy Kim
December 2006
Single tiling unit (Unit square in red).
Perspective view of single tiling unit.
Regular tiling.
Randomized tiling.
Perspective view of the randomized tiling.
POVRay scene file: weave.pov for the single unit.
Scene file for a random tiling of the plane: weave.pov.
December 1997, Updated February 2004 C libraries:hexlib.h, andhexlib.c |
Of the three 2 dimensional shapes (equilateral triangle,rectangle, and hexagon) that can be used to tilethe plane without holes, the hexagon is the most complexand has many interesting properties.In what follows, anindexing system will be described for a hexagonal tilingcalled a Spiral Honeycomb Mosaic (SHM). A SHM consists ofgroups of 7n hexagons (n > 0) called super-hexagons.It uses a base 7 numbering system for the hexagonal mesh, this isillustrated below for n = 1, 2, and 3.
SoftwareAn application has been developed to explore operations in SHM space.It is based upon X-Windows and OpenGL and is currently available for Mac OS-X and Linux (by request).Note that before running the Mac OS-X version check that X-Windows isrunning.Download: macosx.tar.gz
InterfaceThe interface is straightforward, to find out about any command line optionstype 'hex -h'. The left mouse button pans the image, the middle buttonrolls, and the right button presents a list of menu options. In order toposition/rotate/scale the hexagon to the image see the image mappingdialog box. To experiment with SHM operations see the transformationdialog box.
Addition and Multiplication
The two basic arithmetic operations can be defined for the SMH,addition and multiplication. These operations act on the addressesof the SHM and result in translation (addition) and rotation/scaling(multiplication) when applied to images represented with the 'pixels'of the SHM.The following two C snippets (HexAdd() and HexMul())implement addition and multiplication.An important characteristics of the operations defined in this wayis they are bijective, that is, it is a one to one mapping of each address. Every address (hexagonal pixel) maps uniquely to another address so thatno information is lost.
ExamplesThis is an example of repeatedlymultiplying an image on an order 5 SHM by 10hex or 7dec
This is the same example at two powers of 7 greater resolution showing the 'superduck'.
This example shows repeated addition by 6666hex.
The curve through powers of 7 (10hex) is an equiangular spiraldescribed by
a = sqrt(3)
b = loge(sqrt(7)) / arctan(sqrt(3) / 2) = 1.3632084.
Since the angle in the above caseis taken from the negative y (imaginary) axis, the curve would be traced by
Geometry Defense: Infinite Mac Os 7
y = - r cos(theta)
The angle between each successive multiples of 10hex isarctan(sqrt(3) / 2) = 40.893395o, the ratio of two successive radii is sqrt(7) = 2.6457513.
Online SHM Calculator
| Base Conversion Addition/multiplication Other Functions Cartesian |
References
Alexander, D. and Sheridan, P.
Proceedings Australian Neuroscience Society (1995)
Algebraic Geometric Model of the Receptive Fields Propertiesof the Macaque Striate Cortex.
Sheridan, P.
PhD Thesis, University of Technology, Sydney (1996)
Spiral Architecture for Machine Vision
Sheridan, P., Alexander, D.M.
Proceedings of Vision, Recognition, Action: Boston. (1997)
Invariant transformations on a space-variant hexagonal grid
Sheridan, P., Hintz, T., Alexander, D.
Image and Vision Computing, 18 (2000)
Pseudo-invariant image transformations on a hexagonal lattice
Written by Paul Bourke October 1995 |
Tiling a plane with single geometric shapes is an old exercise ingeometry. The platonic solids were shapes that tile 3D space usingregular polyhedra, in 2 dimensions examples of regular polygons that tile (without gaps or overlap) the plane are the equilateral triangle, square, and hexagon.
Periodic tilingsThere are many examples of general polygons (not regular) that tile the plane,but in the most common cases they form periodic tiling. A periodic tiling isone where it is possible to make a parallelogram (generally larger than thetiles) that can be repeated to produce the same tiling. An example ofthe more common periodic tiling is shown below for two tiles (blue and yellow),the red boundary illustrates one possible region that can be replicatedto create the same effect as the tiles. Note that in general the periodforming shape need not be a rectangle but any parallelogram.Note also that there are generally a large, if not infinite, numberof periodic parallelograms for a particular periodic tiling.
Periodic and nonperiodic
An example of a single tile that can tile periodically and nonperiodicallyis a decagon (ten sided) with a 'bite' taken out of it. Tiles that canform both periodic and nonperiodic tilings are often referred to as 'versatile' tiles.
An example of this tiled periodically is given below. Note that for a tilingnot to have holes all the angles at a vertex must sum to 360 degrees. Inthis case each angle is either 72, 144, or 216 degrees. A number ofcombinations of these numbers can sum to 360.
The periodic tiling as shown above have the decagons orientated inmultiples of 180 degrees. If the decagons are rotated in multiples of 36degrees (10 different orientations)the tilings can become nonperiodic as shown below.
Some shapes that can be combined to form larger versions of themselveshave an obvious nonperiodic tiling, for example the following is calledthe sphinx. A periodic tiling with the same tiles is obvious.
Nonperiodic only
How about purely non periodic tiling, that is, a set of polygons that tile theplane but never form a periodic tiling.For quite a while it was assumedthat any set of tiles that formed a non periodic tiling could also beused to form a periodic tiling.In 1966 Robert Berger showed that it was not possible to determinewhether a particular set of tiles would tile non periodically!
This following set of six nonperiodic tilings were 'discovered' byRoger Penrose around 1973. It is illustrative to print and cut a fewsets of these tiles out and 'experience' how they tile the plane.
The following six tiles are attributed to Richard Robinson who in 1971reduced a 104 aperiodic tile set by Robert Berger(they were a reduction from a set oforiginally 20426) to just six tiles.
Around 1973/74 Roger Penrose found a set of two tiles thatonly tile non periodically. While these come in a number of formsthe most common one is illustrated below. Note that phi indicatedbelow is the golden ratio of (1 + sqrt(5))/2.
The start of the tiling of these two tiles is illustrated below.The two tiles are often referred to as the kite and dart. The rhombuswould of course tile periodically, the triangular pieces force aselect combination of allowed connections.
There are a number of elegant properties of this tiling. For example,the ratio of kits to darts in an infinite tiling is the golden ratio!
Another two tile non periodic tiling by Roger PenroseFloor of the Chemistry building, University of Western Australia.
Written by Paul Bourke The three tiles making up the so called 'Danzer' aperiodic tileare shown on the right. The rules for coincident edges areshown as edge direction markers, these must match before two pieces canbe put together along a shared edge. | |
| Angles | |
| Mapping - First iteration This tiling was initially thought interesting because it wasone of the first found with a seven fold symmetry, somethingthat for many years earlier was considered unlikely. Eachof the tiles can be constructed from the three pieces,the first iteration of this is shown on the right. Allthree tiles are required in each case. | |
| Second iteration The second iteration (reduction scaling applied)is shown on the right. Note that with correct tilingno edges are bisected by other edges. | |
| Third iteration, piece 1 | Third iteration, piece 2 |
| Third iteration, piece 3 | |
| Fourth iteration, piece 3 | |
June 2000
| Seed The so called 'chair tile' is formed by starting with the seed tileas shown on the right. |
| Mapping - first iteration The first iteration is formed by replicating the seed 4 times,one orientated the same way, one rotated 90 degrees clockwiseand the other rotated 90 degrees anticlockwise. |
| Second iteration Subsequent iterations are formed by replicating and applying therotations and translations to the last iteration. The size ofthe tile increases by a factor of two along each axis. |
| Third iteration |
| Fifth iteration |
| Coloured The tiles can be coloured in order to bring out some featuressuch as the diagonal structure. The tile below is the fifthiteration using the coloured arrangement on the right. |
June 2002
| Trilobite This tiling consists of two tiles called the trilobite and thecross. The tiling is only possible as an aperiodic tile, that is,there are no periodic tilings of infinite extent on the planeusing these two tiles and the joining rule. | |
| Cross The second tile, the cross, got its name because it and the trilobiteis often drawn rotated by 45 degree to those shown here. | |
| Constraint The tips of two tiles can only be connected if the black and whitestrips align. It is this constraint that forces the only infinitetiling to be aperiodic. | |
| Stage 1 Four trilobites can be connected as follows without the need for anycross tiles. At each iteration the total bounds of the tiling increasesby a factor of two. | |
| Stage 2 The next stage of the tiling can be constructed as in thechair tile. The result shown on theright is made up of 4 copies of the earlier stage, 4 crosselements are required to fill in the gaps.Note that one of the copies is only translated, one other isrotated clockwise by 90 degrees and the other is rotatedanticlockwise by 90 degrees. The same applies for the fillingcross tiles. | |
| Stage 3 The next stage uses 4 copies of the last stage with the same rotationaland translation relationships as in the last iteration. The cross filesrequired to fill in the holes are again shown in blue below. | |
| Stage 4 And finally the fourth iteration scaled down by a factor of 2. | |
Attributed to Charles Radin Written by Paul Bourke An iterative procedure for creating the pinwheel tiling is givenbelow. This infinite tiling of the plane is known to beaperiodic, that is, there is no parallelogram shapedregion that can be repeated to form the same tiling. |
| Seed The pinwheel tiling can be constructed by starting with a rightangle triangle with side lengths of 1, 2 and square root of 5. | |
| Mapping - first iteration Five of these triangles can be rotated, mirrored, and translated to form a larger similar triangle as shown in red on the right.The rotation angles are one of the angles of the triangle,that is, atan(0.5), pi/2-atan(0.5), or pi/2 degrees. | |
| Second iteration This process is repeated, on each iteration the global triangleincreases in size by the square root of 5. Five copies of thelast iteration are packed into the new triangle.The second iteration is shown on the right. | |
| Third iteration From the right and top down the transformation for each triangleis as follows (leaving out the final translation).
| |
| Fourth iteration Locally one can pick out what seems to be regular structuresbut of course they don't form a larger regular tiling. For examplethe groups of 8 triangles that share a single vertex can bereadily identified, it is these 'pinwheel' shapes that give the tiling its name. The properties of this tiling were discovered and named by Charles Radin of the University of Texas in 1991 and first published in 1994 in the Annals of Mathematics number 139, page 661-702. | |
| Fifth iteration In this case the tile has been scaled down (clickon the image for a larger version) and another triangle is usedto form a rectangular region. A periodic tiling is one where it is possible to make a parallelogram (generally larger than the tiles) thatcan be repeated to produce the same tiling. Continuing this iterative process results in an infinite tilingon the plane that is never periodic. Note that forming a larger tilingby repeating this rectangular shape obviously isn't appropriate, the tilingwould then no longer be aperiodic! [Click on the image on the right for a higher resolution version] | |
| Coloured (fourth iteration) The lack of periodicity can be highlighted by colouring each triangleas shown below. The human visual system and brain are good at picking out patterns, and while there often seem to be small scale patternsthey don't persist at larger scales. |
Geometry Defense: Infinite Mac Os Download
Contribution by Dave Smith
March 2015The basic tiling unit is shown below, the constraint is that two semicircles cannot be placed adjacent to each other.All angles are integer multiples of 36 degrees, similarly rotations of the tiling unit are also multiples of 36 degrees.All edge lengths are equal.
A legal tiling might be as follows. Question: can this tile the plane indefinitely?
Witch hatThe basic tiling unit is shown below, the internal angles are integer multiples (as shown) of30 degrees. All edge lengths are equal.