Design and history of the cards: Design I shows the triangle duo as it presents itself at the face of the lay card (cardandcube #1) and of the virtual cards in the apps cardandcube #2 and #3 . The play card has been designed by Hana Hasilik while contemplating grid art sculptures such as “Straight line” (2004). Interested reader can find a reproduction of this work as well as a description of lentoids in chapter cardandcube #1.
In decorative arts the history of our designs I and II reaches into medieval and antique periods. In 1704, the Dominican priest, monk, mathematician and graphic artist Sébastien Truchet (1657-1729) reported: „During the last trip that I took to the canal D´Orléans by order of His Royal Highness, in a chåteau called Motte St. Lyé, 4 leagues this side of Orléans, I found several ceramic tiles that were divided by a diagonal line into two colored parts“ [3, 4]. Truchet defined 16 distinct pairs of such tiles that formed beautiful patterns when laid out regularly in varied sequences. In 1722, Father Dominique Douat presented a meticulous mathematical analysis  of the possibilities of creating graphic patterns with the tiles that had been reported on by Truchet. Further details on history of the related design can be found at the end of this chapter and with cardandcube #1.
In the 20th century, the original report and the patterns described therein were brought to the attention of art historian Ernst H. Gombrich. In his book The sense of order  he described and analysed more than the psychological impact and influence of geometric decorative art on handicrafts and various art forms such as op art. He was impressed by the almost “ad infinitum” permutation of the tiling pattern.
It has not escaped the attention of the art historian that the four orientations used in the square patterns reflect the central phenomenon of biology. He indicated that they could be seen as a symbol of the basic building blocks of the genetic material. There are 64 (43) triplets that allow the transfer of genetic information during protein synthesis and just as many options for placing three cards with the two-coloured triangle pair with varied orientations next to each other. (The number of possible patterns that can be built from more cards is rapidly raising with the number of cards in use. When card designs I, III or IV are used with the 8x8 layout, the number reaches 664 for each one. More variations can be achieved by adjusting the line width when playing with designs II or III.) In this chapter most examples shall deal with design I.
Tesselation: Square tiling is one of three regular tiling patterns. The other two are the triangular and hexagonal tilings. In these, defined shapes - squares, equilateral triangles or hexagons - are used in selected designs. With the figure of the triangle pair in a square, the virtual cards and the laying cards from cardandcube offer the option of designing extending white and black polygons. Their regular arrangements are executed by tessellation.
Examples of tessellation with square tiles that are decorated with two isosceles triangles have been described and mathematically analysed by Sébastien Truchet more than three hundred years ago. Now, cardandcube #2 and #3 present this and, in the latter case also other decors on virtual cards to play with. Let these applications unleash your creativity and bring you enjoyment and relaxation.
From four possible orientations of the tiles, four options for monotonous tiling are emerging. They differ from each other solely by the orientation of the triangle on the tiled surface. It has been pointed to by Truchet that sixteen different "card duos" can be formed from two cards that are joined either horizontally or vertically. Using such duos relatively simple tiling patterns can be designed. There are 16 such duos in cardandcube #2 and 36 in cardancube #3. Possibilities can quickly be explored by placing two cards next to each other and then lining up the same duos repeatedly side by side, placing the same rows flush or offset, or rotated by 180°.
Tiling with one or more card quartets as units is much more interesting than the use of card duos. Further below you may find ideas, how to use our games to design geometric patterns. Hints help you try the next step on your own.
As mentioned earlier above, with cardancube #3 you have a choice from 1.296 quartets. When tiling, such units of card quartets are placed next to or on top of each other without gaps. In the virtual cardandcube games, all quartets are created from the individual cards. Development of a copy-and-paste process for duplicating and relocating quartets is under consideration. A schematic representation of examples of tiling quartets can be found at the end of this text. You can place such quartets all over the chosen layout. Alternatively, you may construct tiles composed of more than 4 cards, before constructing multiples of these.
The use of card quartets or larger units makes it possible to design an undreamt-of number of geometric patterns through tiling. Regularity and variety expand the possibilities in the arrangement of the tiles, as do the horizontally, vertically or rotationally symmetrical arrangements.
At this point, two special suggestions are offered: First, card quartets, in which the diagonally adjacent placement cards are in the same orientation, as are the cards i and iii (referred to in scheme b) as shown in example a, can be linked diagonally overlapping (scheme j). Furthermore, several chains of this kind can be put together in parallel and interlocked.
If you like a pair of identical card quartets placed side by side or on top of each other, try to create an inverse pattern from another 8 virtual cards along the longer edge. If the task is difficult for you, make it easy this way: in the first step you double the pattern and in the second you make a negative for each card of the copy one by one. On cards with triangles the images have to be turned by 180°. (In the six versions of the layout card shown in the description of cards, the third is the negative image of the first, the fourth of the second, and the sixth of the fifth). The black and the white cards have to be interchanged. Finished!
Examples of tiling with card quartets:
a) A common example of a card quartet
This quartet is reminiscent of an open book or a pecking bird. Let us take the letter A as a symbol of the quartet. If this picture is rotated by 90°, it results in a different quartet. We'll symbolize it with the letter A tilted sideways (see g below).
b) Assignment of symbols, i-iv, to the individual cards
c) Parallel arrangement showing a repetition of a card quartet, A, in rows and columns
If you try this arrangement with the 8x8 layout and the quartet example a, you will be surprised. In the finished picture you will hardly be able to find the template! The overall picture will be dominated by merged areas of the same colour.
Due to the subdivision, it is possible to move adjacent rows or columns of quartets by one card's width. The shift can lead to striking changes in the pattern formed at the seams of the rows (compare the scheme in paragraph g.)
d) Alternating arrangement of two different card quartets, A and B
With a little imagination, you can try out numerous variations of the arrangement d on larger layouts. Instead of two, you can “stack” three different quartets, A, B, C on top of each other and fill in the entire format with this formation of an elongated tile. The tiles lying next to each other can be laid either flush in height or offset by one or more cards. If the tiles are in straight stripes next to each other, the horizontal stripes or their quartets can be placed either flush or offset (like in j further below).
e) Vertical parallel arrangement of two different card quartets, A and B
With this arrangement, we recommend designing card quartet B as a negative of the mirror image of A (you can create its mirror image next to card quartet A and then turn each individual design by 180° or swap black and white).
f) Anti-parallel arrangement
g) Rotationally symmetrical arrangement of 4 quartets
Initially, a quartet of cards is laid out in the upper left quarter of the 8x8 layout. The next ones follow in a clockwise direction, built up as copies of the first and rotated by 90°, 180° and 270° respectively. The result is an image with 4th order rotational symmetry.
If we start in the upper right quarter of the layout and proceed counter-clockwise, with the quartet images also being rotated again by 90°, etc., we get an image with second-order rotational symmetry.
h) Arrangement of quartets in alternative grids A or X
When considering a tiling in several rows and several columns, a thought grid can be shifting between two alternatives. It may happen that the original card quartets in segments A attract less attention than the unintentionally created quartets in segment X (made of cards around the seam line).
i) Offset of strips of card quartets by one card width
j) Overlapping diagonal arrangement of card quartets
This arrangement is possible if in a quartet the diagonally adjacent cards, such as cards i and iii in scheme a, are oriented identically. Several diagonal chains constructed according to this principle can be put together in parallel and interlocked with one another.
Try to arrange 4 copies of the rather unprepossessing quartet a according to the pattern shown in g. Use the 4x4 layout and start in the upper left quarter. The correct result is shown here:
You can start with the same quartet at the bottom left corner of the 4x4 layout. The results will differ from each other.
Make a discovery. Open the 8x8 layout. Build a non-symmetrical quartet in the upper right corner. Follow the instructions under scheme g and create a two-fold rotationally symmetrical image. Copy the finished square into the free quarter in the top left. Your construction contains a pair of two-fold rotationally symmetrical patterns. Can you find a four-fold rotationally symmetrical pattern in it?
The character of the rotationally symmetrical images can be changed considerably by rearranging the central quartet. The symmetry of the whole is retained if you rotate the images of a group of cards by 90° or several times, as long as the contours of the group itself are rotationally symmetrical. If you want to try changes and don't want to lose an image that you think is beautiful, save it.
You can also play with larger patterns by using the four-pronged spiral previously shown as a template. Open the 8x8 layout and build two of these templates into the lower and upper left quarters. Place the next copy of the template at any height in the blank area on the right and fill in the remaining area as if the tiles were shifted through the width of one or more cards.
Finally, you can work with even larger layouts as long as the cards aren't too small to play with. Create the four-pointed spiral in the centre or anywhere on the edge of the selected format. Now place cards on the free edges of the copy to extend the white and black ribbons and fold them around each other as long as there is room. You will be amazed at the results. If this game takes longer than expected, please save the picture so you can continue later.
Design: cardancube # 3 makes it easy to design novel patterns. We show this using the example of a "basic pattern", the image of which we created from a square card formation and a single card (n2 + 1), as described below.
First of all, the 8x12 format with design I is selected. Then a basic pattern is built according to the formula 9 + 1 (framed in red in panel I below). It consists of a square of 3x3 cards, a card nonet, and a single card that leans against one corner of this nonet. Furthermore, one can make many designs with imagination. We choose the single card in black. The same goes for the middle card of the nonet. In the four corners of the nonet we put cards with black and white triangles so that the blacks form its points. We choose the other 4 edge cards of the nonet in white. Thus, we have constructed a rotationally symmetrical nonet.
We put the first copy of this unit with its upper right corner under the black 10th card of the basic unit. Correspondingly offset, we built all the other units or their matching parts into the selected format. The result can be executed in designs II - IV with just a few clicks. By setting the line thickness, the result can be adapted to individual needs (see designs II and II d):
A reward for completing the selected format comes about when looking at the finished overall picture. In doing so, image motifs become visible that are formed beyond the basic pattern. Sometimes the entire format can be divided into different grids in which different motifs are created.
Not only by turning individual cards, but also by setting different colours of the figure and the background, cardandcube #3 can be used to create image variants in no time:
The colours can be set precisely in hexadecimal format. In the example below, variants of an image were created by changing the character design. In addition, the colours were varied:
The examples show the basic colours used by Piet Mondrian a century ago (the codes can be found here).
You can use cardandcube #3 for your own design creations. Just change the basic pattern to try it out. After that, you can change its shape and create new charming designs. Perhaps you would like to design a decor for a fabric, placemat, napkin, wallpaper, towel or whatever and have it produced in your desired colours.
Ambiguous pictures: On the internet we may find many ambiguous images. An example is the Necker cube that is known since 1832. There are also examples of ambiguous images for cardandcube #2 and #3. It can be pictures that are created by tiling with certain mirror or diagonally symmetrical quartets.
Open the 8x8 layout and build a quartet resembling hourglass using 4 cards in one of the quarters. Expand the picture further according to scheme g. Enjoy your first ambiguous image from cardandcube.
Take another quartet that may be called "inclined hourglass":
Select the 4x4 layout and fill it in with 4 quartets of this type according to the principle of the diagram g shown above. You get a four-fold symmetrical image. However, if you lay out the same quartet linearly - according to scheme c - you will be amazed: an ambiguous image will appear in front of you.
The resulting pattern can be perceived as ambiguous. What we see depends on how we look at it. Certain perceived image motifs are repeated in diagonal, others in orthogonal (vertically and horizontally arranged) grids.
Let us consider the present quartet as a combination of two vertical card duos as shown in scheme b: on the left side card i above card iv and on the right card ii above card iii. Now we move the right card duo to the left side of the other. In both cases, before and after the swap, we have striking symmetrical images. If the two card duos are lined up repetitively along a horizontal line, the interpretation of the image is bistable. Typical of these ambiguous patterns is the appearance of familiar symbols or regularities along the horizontal or vertical lines and at the same time along the diagonals (in the present example the “inclined hourglasses”).
Of course, a "standing hourglass" can also be created from 4 cards (support). This is twice as big as the "inclined hourglass" because its funnel-shaped parts consist of two triangles: two black or two white. The quartets with black and white hourglasses placed next to each other according to scheme d give an ambiguous picture, as described in the following paragraph.
Marburg's ambiguous pictures:
Ambiguous pictures on the ornamental floor in the sacristy of the palace chapel and in the sacristy of the Elisabeth church in Marburg. The illustration shows A) a preserved part of the decorative floor in the sacristy of the Marburg palace chapel. B) The corresponding geometric pattern, created with cardandcube #2P. The yellow frame in A and B enhances the identity of the two patterns. C) Another part of the decorative floor of the sacristy of the palace chapel. We would like to thank the Museum für Kulturgeschichte of the Philipps University of Marburg for the permission to show detailed photos of the decorative floor of the sacristy of the palace chapel (A and C). D) A replica of pattern A in the sacristy of the Elisabeth church. We thank the Ev. Elisabethkirchengemeinde Marburg for permission to publish this detailed picture.
In Marburg's Elisabeth Church there are several floor decorations, the design of which is based on two-tone, diagonally divided squares. Under the shrine of St. Elisabeth in the sacristy, the decorative floor is particularly attractive. In its reconstructed part there is a remarkable pattern of black and yellowish triangular tiles. The interpretation of the visual perception of the pattern changes (tilts) between several alternatives. The repetitive motifs of the decorative base appear either as a wind turbine (propeller), which can appear in different overlapping grids, or as an hourglass image. These structures sometimes appear black on a light background, sometimes light on black. The model of the pattern goes back to antiquity. It has been found in Cassa delle nozze d`argento in Pompeii)
According to Karl Justi [6, p. 21], the floor of the castle chapel and its sacristy, originally laid with stone, was covered with a "uniquely artistic clay tile mosaic" in the early 14th century. The renovation was arranged by the Münster bishop Ludwig II, for whom the "simple flooring was not enough". He was a great-grandson of Elisabeth of Thuringia and half-brother of the Marburg Landgrave Otto. After 1311 he owned Marburg. It is therefore likely that at least parts of the decorative floors preserved in the castle are 700 years old.
Marburg´s ambiguous patterns consist of diagonally divided two-tone squares, which in turn were created from triangular black and yellow glazed triangular tiles. The two-tone square tiles described by Truchet in 1704  were not available to the Marburg tilers. The ceramic floors of the older St. Viktor Church in Xanten [7, Fig. 67] and some Roman mosaics in Cologne [7, Fig. 269] could have served as models for their patterns. Instead of stone, more modern and cheaper (unfortunately also less durable) glazed triangular tiles were used, the production method of which came to us from Flanders.
Pretty: That's what we called one of the most beautiful flip images created with cardandcube. More than two manifestations can be seen in Pretty. We will show it here as soon as the new version of the game with colors is released.
Arts: A square with a pair of black and white isosceles triangles has been used as a module by Horst Schwitzki (1932-2016), a member of the Kassel minimum art group. An examle from a series he painted in 1980ies is shown here.
Classics: Houndstooth and checkerboard are two-tone textile patterns that have been in use for thousands of years. In the game cardandcube #3 you can imitate these classic patterns with black and white right triangles and squares. Houndstooth created in the 8x12 layout will enchant you. First, place the basic pattern as a tile with 4x4 cards in one of the corners of the layout (support). Then add copies of this tile linearly according to scheme c above.
Mathematics: Patterns that can be built up according to simple rules are called rule-based pictures. Algorithms and their efficiency have been studied in depth by mathematicians, so has logic and recursion. Both of these are eminently suited to establishing rule-based pictures. A demonstration program that includes Truchet tiles was developed by Eric W. Weisstein .
Regarding the interest of mathematics in the findings of S. Truchet and the possibility of designing 3D tiling analogous to his patterns, we refer to an article by E.A. Lord and S. Ranganathan and the original work annotated there .
Inspiration and joy: Parquetting with cardandcube #2 or #3 may inspire those who love, organize and maintain order as well as others who are always looking for something new and who like to place elements of asymmetry around themselves or who are able to gain something positive from any disorder. The spice lies between the extremes, as stated by the art historian E.H. Gombrich his elegant way: „we must ultimately be able to account for the most basic fact of aesthetic experience, the fact that delight lies somewhere between boredom and confusion” [10, p.9].
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- Truchet, S. (1704): Memoir sur les Combinaisons. Memoires de l´ Académie Royale des Sciences, 363-372
- Smith, C.S. und Boucher, P. (1987): The Tiling Patterns of Sébastien Truchet and the Topology of Structural Hierarchy. Leonardo 20, 373-385
- Douat, D. (1722): Méthode pour faire une infinité de Desseins différents avec des Carreaux mi- partis de deux Couleurs par une Ligne diagonale, ou Observations du Pere Dominique Douat Religieux Carme de la Province du Toulouse, Sur un Mémoire inseré dans l´Histoire de l´Académie Royale des Sciences de Paris l´année 1704, présenté par le Reverend Pere Sébastien Truchet, Religieux du même Ordre, Académicien honoraire. Paris
- Justi, K.: Das Marburger Schloss. Geschichte einer deutschen Burg. Veröffentlichungen der Historischen Kommission für Hessen und Waldeck XXI. N.G. Elwertsche Verlagsbuchhandlung, Marburg 1942
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- Weisstein, Eric W."Truchet Tiling." From MathWorld--A Wolfram Web Resource.
- Lord, E.A. und Ranganathan, S. (2006): Truchet tilings and their generalisations. Resonance, June 2006, 42-50
- Gombrich, E.H. (1994): The sense of order. A study in the psychology of decorative Art. 2nd ed. Phaidon Press Ltd., London