cardandcube #2

DESCRIPTION

This game is a virtual version of the cardandcube #1 card game. With a bit of imagination, you can create numerous pictures.

The game's motto is “Enjoy your own creativity”. It promises peace and relaxation. You will be amazed at the variety of design options.

The virtual card of cardandcube #2 is identical to the front of the card of Hana Hasilik´s game cardandcube #1:

an inseparable pair of triangles on a square surface. Both rectangular and isosceles, one black and the other white. Tiles with a two-tone pattern of this type were used in France for decorative purposes three or more hundred years ago (Reference No. 3). Earlier, Romans created mosaics with patters based on such squares.

On the screen, the virtual card can be rotated clockwise by 90° by touching or by clicking the mouse. After four clicks, the initial design is back. With a double click, the card rotates in the opposite direction (to the left). The initial card and the results of 1 to 4 clicks look as follows:

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In pairs, the first and third (or the second and fourth) image from the left are color exchange symmetrical. One can be converted into the other by turning it through 180°.

The game cardandcube #2 is mobile - you can play alone or with friends and practically anywhere, if you have the Android or iOS version. You can also play in parallel, save and share the results - regardless of whether you are playing on a mobile phone, on a Windows PC, laptop or tablet.

INSTRUCTIONS

By tapping at the Information button "i", you get an access to Game rules and the card´s introductory performance. You have a choice between English and German languages.

At the beginning this app presents one virtual card on the screen. The card starts to introduce itself as soon as you tap at it. You may stop the introduction and begin to play any time by touching the Play now button. 

The Layout button offers a drop-down menu to choose the number of cards to play with from. You can play, depending on your device, with up to 864 (desk top) or 600 (mobile devices) virtual cards. Any layout presents a grid with squares to be filled with cards one after another by being clicked on or touched. An additional clicking turns the card of choice by 90°. Turning the card affects the combinations of black and white surfaces separating it from or joining it with its neighbours. By these simple means a plethora of black and white patterns can be constructed and enjoyed.

Before starting a new game, you have the option of storing the current picture. In the bottom left corner you can open a menue with these options:

 Save - shows 9 thumbnails of empty or filled files, any of which can be  overwritten with the current game

 Reopen - shows 9 thumbnails of games you have saved earlier and may wish to play with now

 Save as - to save the current game under a specified name on a file of your choice

 Import - to resume a filed or a received game

 Export - to save the image of the curent game as a .png file

Rotating all cards – activation of this button  makes each card to rotate by 90°. From your image you can create three others with a minimum effort.

Recording the moves and time taken: - Both counts start from the first move. Clicking or tapping this option stops the timer; counting continues as soon as a card is touched i.e. a move is made.

Have fun!

WE SHALL MAKE THIS APP AVAILABLE SOON - THAN YOU CAN GO TO THE SHOP AND SELECT YOUR DEVICE FOR THE DOWNLOAD

TIPPS

The beginning: You might get started as soon as you have seen the free version of the game. Give your imagination free rein! With the following explanations we would like to bring you closer to the possibilities of the game.

We recommend initially using the 2x2 layout. In front of you is an empty playing field that you can gradually fill with 4 cards and rotate them by 90° as many times as you like. We refer to the resulting images as quartets and give them names depending on objects they resemble. 

Symmetry: Among 256 possible card quartets on average every fourth is symmetrical in one way or another. The quartet shown below - we call it arrowhead pointing upwards - is an instance of reflection symmetry or mirror-image symmetry. It has a vertical line of symmetry:

In addition, the upper and the lower halves of this picture are an example of colour exchange symmetry. If you compare the two halves to each other, you see that the white colour is swapped for black and vice versa. On the top you see a white peak in front of black background and at the bottom a black peak in front of white background. 

Do you react differently from the visual stimuli of the quartets depending on their symmetry or asymmetry? If so, you may have preferences in deciding on the orientations of cards and their neighbours.

Humans are not the only beings that do not perceive aesthetic phenomena with indifference [1]. The reception of symmetrical images may be deeply rooted in evolution and related to the prey-hunter recognition [2].

Exercise: Choose the 4x4 layout and build a quartet such as this one

in the lower left quarter. The pattern resembles an open book or a pecking bird. Next, build another quartet above the first one. Imagine building the same pattern turned by 90° clock-wise. (If you imagined a bird in the bottom corner, it will be heading to the left, now. Continue in the same manner building another quartet to fill  the top right quarter and, finally, the bottom right quarter of the 4x4 layout. The quarters added to the first one display patterns that are rotated by  90°, 180° and 270° as compared to it. Have fun!

The result of the exercise is a pattern with a fourfold rotational symmetry. At the beginning of this kind of construction, you may place the chosen quartet into any of the four corners of the 4x4 layout.  Depending on where you start, the result is different from the others. One of the four patterns that can be obtained using the quartet shown above can be viewed here.

You may explore the effects of turning the cards in layouts 2x2 or 4x4 further on. Having assembled a symmetrical pattern, tap each card once. Start at the top left and tap the cards one by one. Conceive of making the pattern and repeat the single taps through the layout.

Build a quartet in any of the four quarters of the 4x4 layout. Construct a mirror image at any free edge. Continue with the mirroring to fill out the remaining half of the layout.

Play: Choose a larger layout for everyday gaming. By turning one or another card you can reshape the picture according to your ideas. You can start with a quartet and turn the cards around until you have realized your design ideas on the entire layout.

Storage: The result can be saved before exiting the app. Caution, without saving, the pattern will be lost as soon as you choose a new layout.

Any pattern you have enjoyed and saved can be modified. You may reopen it. If you rearrange a portion, for example the central quartet of a rotationally symmetrical pattern, you have a choice. Either you discard the new version or you store it and perhaps replace the previous version. 

Photo: On most devices, you can capture any finished pattern that you like in jpeg format using the print screen command. You may use this, if you want to compete with a friend in building a pattern you have agreed on.

Design: With the patterns created with cardandcube, a special effect can be achieved by turning and cutting the image (an example). More about design and applications of virtual geometric cards can be found here.

SUPPLEMENTS

Decorative arts: The history of our design 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 [5] 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 [6] 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.  With the 8x8 layout, the number reaches 464.

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 presents this to play with using a pocket device while commuting, waiting a while or just relaxing. Let these applications unleash your creativity and bring you enjoyment.  

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. 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 in the preceding chapter, with cardancube #2 you have a choice from 256 quartets. When tiling,  card quartets are placed next to or on top of each other without gaps. Such horizontal or vertical rows of quartets or larger units like 3x3 (card nonets) are repeated next to each other without or with an offset.

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 seems not to be a must, because we want you to relax, rather than to hurry. 

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. A layout can itself be filled with seemingly irregular tiles such as a nonet (or a larger square formation) with an additional card leaning against one corner. 

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 further below) 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.

Second, if you find a pleasure in a pair of identical card quartets placed side by side or on top of each other, place 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 along the edge and in the second you swap the colours. This is easily done by turning each card image by 180° (by two clicks each). Use the image with 4x4 cards thus obtained to tile the remaining area of the layout and enjoy the result.

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 scheme 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 the card quartet B as the mirror image or colour exchange image of A. Eventually, you can do both: next to card quartet A build its mirror image and then turn each individual card by 180° (that is 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.

Examples: cardandcube #2 is a tool for your own design creations. Next, you can see a few examples out of the trillions of possible:

Create the panel-like image above by tessellating 4x4 card tile groups. Several different variations of 4x4 card tile groups can be used in this way. Rotate the whole layout by 45° and crop back to a rectangle to create a dramatic transformation in the look of this card tile group. See an example of this here.

Create fresh patterns by rotating all cards by 90°: cardandcube #2 has a particular tool for this (the icon is: ) which rotates all cards individually by 90°. Tapping this icon twice results in individually rotating all the cards 180°. This creates a version of the original pattern where the colours have been swapped. Tapping four times makes the original pattern reappear.

The illustrations below demonstrate this on one of the 4x4 units of tessellation previously mentioned.

It is an example of two-fold rotational symmetry.

After individually rotating all the cards, the symmetry of the pattern is still there. However, repeating these 90° rotations results in an alternation between reflectional and colour-swap reflectional symmetries. This is true in respect to orthogonal and to diagonal reflection planes as well. After each 90° rotation of the cards within this particular tile the reflection planes rotate by 45°.


The first and the third as well as the second and the fourth picture panel each form a colour-swapping symmetrical pair.

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. 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 [3] 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.

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 [8].

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 [9].

Inspiration and joy: Parquetting with cardandcube #2 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].


LITERATURE

  1. Mukherjee, R. und Kodandaramaiah, U. (2015): What makes eyespots intimidating - The importance of pairedness. MC Evol Biol  2015, 15:34.
  2. Anderson, J.R. et al. (2005): Are Monkeys Aesthetists? Rensch (1957) Revisited.  J Exper Psychol Animal Behav Processes 31:71-80
  3. Truchet, S. (1704): Memoir sur les Combinaisons. Memoires de l´ Académie Royale des Sciences, 363-372
  4. Smith, C.S. und Boucher, P. (1987): The Tiling Patterns of Sébastien Truchet and the Topology of Structural Hierarchy. Leonardo 20, 373-385
  5. 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
  6. Justi, K.: Das Marburger Schloss. Geschichte einer deutschen Burg. Veröffentlichungen der Historischen Kommission für Hessen und Waldeck XXI. N.G. Elwertsche Verlagsbuchhandlung, Marburg 1942
  7. Kier, H. (1970): Der mittelalterliche Schmuckboden. Rheinland Verlag, Düsseldorf
  8. Weisstein, Eric W."Truchet Tiling." From MathWorld--A Wolfram Web Resource.
  9. Lord, E.A. und Ranganathan, S. (2006): Truchet tilings and their generalisations. Resonance, June 2006, 42-50 
  10. Gombrich, E.H. (1994): The sense of order. A study in the psychology of decorative Art. 2nd ed. Phaidon Press Ltd., London

#2 Shop

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Mobile App

cardandcube #2 PREMIUM

ENJOY YOUR CREATIVITY
WHILE PLAYING
WITH THE APP
ON YOUR DEVICE

Android available / for iOS #3 is recommended

 

€ 3.99
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Desktop App

cardandcube #2 PREMIUM

ENJOY YOUR CREATIVITY
WHILE PLAYING
WITH THE APP
ON YOUR DESKTOP

currently not available

 

cardandcube #3

THE CARDS

cardandcube #3 is an enhanced alternative to cardandcube #2. It offers these four card designs to play with:

Designs I and II have been used for decorative purposes and in mathematical theses previously (SUPPLEMENTS).

In the present virtual game the design in use can be replaced in bulk by another one - back and forth at any time. In addition to the four orientations known from the analogue version or from cardandcube #2, black or white cards can be combined with the chosen figure.

By repeatedly touching or clicking the cards change their orientation. The designs I, III and IV can be rotated three times by 90°. The diagonal line appears in only two orientations. If the clicking is continued, any figure is replaced by black, then by white cards. Eventually the initial figure is restored and the moves can be continued. With a double click (<<) the preceding move is reversed:

The game cardandcube #3 is mobile - you can play alone or with friends and practically anywhere if you have the Android or iOS version. You can also play in parallel, save and share the results - regardless of whether you are playing on a mobile phone, on a Windows PC, laptop or tablet.

INSTRUCTIONS

The instructions (“i”) are available in English and German.

On the top bar you find four designs of the card to choose from (to tap on). When playing with lines, you can use the and buttons to adjust their width.

At the beginning this app presents one virtual card on the screen. The card gives information about itself as soon as you tap it. You may stop this and begin to play any time by selecting the Play now button that is on display through the introduction. Then choose the number of cards to play with from the Layout button. For beginners we recommend to start with a 2x2 layout as shown further below.

Tap at the cards and follow your own aesthetic feelings while playing with them. You can switch between the designs at any time. While the design is altered, the orientation of the cards is preserved. The introductory presentation of the game card is accessible from the end of the paragraph on the rules at “i”.

This button is a kind of reset. It clears up and sets the starting pattern. By clicking on, the orientation of the selected design is varied until only black or white cards are displayed on the entire play field. Finally, the design chosen is displayed in an unorderly manner throughout the field. (You may feel asked to make order.)

Before starting a new game, you have the option of storing the current picture. In the bottom left corner you can open a menue with these options:

 Save - shows 9 thumbnails of empty or filled files, any of which can be  overwritten with the current game

 Reopen - shows 9 thumbnails of games you have saved earlier and may wish to play with now

 Save as - to save the current game under a specified name on a file of your choice

 Import - to resume a filed or a received game

 Export - to save the image of the curent game as a .png file

To restart use the Layout button. Caution: If you select the Layout without having saved, the pattern made in the previous game will be lost.

The recordings of  time and moves are displayed at the bottom right hand side. Counting starts from the first move. By clicking on, the timer is stopped; it continues as soon as a card is moved.                

Applications with tasks:

  • 1953!: "Liberation" of political prisoners
  • A game of winning: You can compete if you agree on a strategy for creating a picture with your game partners.

The primary goal: cardandcube are the plays to relax with. Give your creativity a free run, enjoy the varieties and savor the aesthetic appeal of new patterns.

Have fun!

YOU MAY START RIGHT AWAY, IF YOU GO TO THE SHOP AND SELECT YOUR DEVICE FOR THE DOWNLOAD

DEFINITIONS & TIPPS

The beginning: You might get started immediately after seeing the interactive presentation of the card. Give your imagination free rein! With the following explanations we would like to bring you closer to the possibilities of the game.

Mobility: You can play alone or with friends and practically anywhere. You can play in parallel and share results regardless of whether you are playing on a mobile phone (Android or iOS), on a Windows PC, laptop or tablet.

Card quartets: We recommend initially using 4 cards that can be accommodated on the smallest playing field (2x2 layout). This formation is called the card quartet or simply the quartet. With the 6 options for each card, 1.296 different quartets can be created. You can name the quartets that remind you of familiar shapes or objects.

Symmetrical and negative patterns: The simplest quartet, which consists of 4 identically oriented cards can simply be obtained by choosing the layout 4x4 and clicking on the clearance button. It is neither horizontally nor vertically symmetrical and using design I it looks as follows:

However, this pattern has a mirror axis that is the diagonal between the top left and the bottom right corners. The other diagonal divides the quartet into two triangles with mirrored patterns in inverted colours - positive and negative. In one moment the image can appear as four black triangles, in another as an inclined black and white double cone and thus has the attributes of an ambiguous image.

The next quartet shown here (a “tree”) is a symmetrical pattern mirrored along the central vertical line. It was created from the above quartet by rotating two card images through 90°:

Among the 1.296 quartets mentioned above, there are 256 that are symmetrical in one way or another. Interestingly, it is mostly the symmetrical patterns that catch our attention. Humans are not the only beings that do not perceive aesthetic phenomena with indifference [1]. The reception of symmetrical images may be deeply rooted in evolution and related to the prey-hunter recognition [2].

Exercises: Choose the 2x2 layout, create various card quartets with design I. Switch to other designs and back. Then switch to the 4x4 layout that will accommodate 4 quartets. You will be in for a surprise if you fill it with variants of the "tree" quartet as follows. Place a “tree” in the top right. To the left you create a horizontal variant, as if you would rotate the "tree" to be pointing horizontally to the left. Then in the lower left corner an upside down one and finally one in the lower right corner tipped on the right side. Look at the result. What do you discover in the middle of the picture?

Select design I and layout 2x2. Construct all point symmetrical quartets with the 4-fold axis of rotation. The number is 6. Four of them contain exclusively the design with the black-and-white rectangular triangles. Two of these show squares composed of four adjacent rectangular triangles. One is black and the other white.

Select design I and the layout 8x8. Assemble a quartet with a square as mentioned above in the middle of the screen. Then add a card to each of the corners of the quartet in such a way to preserve its rotational symmetry. This results in a kind of a paddle wheel. Continue the expansion according to the same principle until you get a square made up of 8x8 virtual cards.

Another possibility is to start with a disordered pattern that can be obtained by using the clearance button. Within the layout in a small area you may find an interesting pattern. Turn cards in its vicinity until the result meets your expectations or simply pleases. Alternatively, you can combine one or two of your favourite card quartets according to practice of tessellation.

An exercise for advanced players: Open cardandcube #3, select the rectangular triangle pair design I and choose the 8x8 layout. In the lower left corner of the field you create the card quartet reminiscent of an inclined white double cone:


In the next 4 stages you can build up a handsome picture while filling the entire layout. From the quartet picture and with the interim results taking up larger and larger areas, you create one after the other: 1. negative pattern of the horizontal reflection, 2. negative pattern of the vertical reflection, 3. horizontal reflection and 4. vertical reflection of the previous pattern. With the first two steps you quadruple the area of ​​the original quartet while maintaining its diagonal mirror symmetry. With the other two, you can create a 4-fold rotating and mirror-symmetrical pattern (point group C4v). It will take you a few minutes to place the 64 virtual cards correctly. Then you will see the double cone in numerous places. Let yourself be surprised: if you look at the finished picture several times, you will discover motifs that are not present in the quartet you have started with. Have fun without adrenaline! Here is another source of inspiration.

Playing with cardandcube #3 contributes to a good mood; it promotes serenity, inner peace and contentment - as if a good mood hormone like serotonin were released.

However, if it has to be, competition is also possible: 

GULAG 1948 and To Freedom! With these two games we want to support the will to perform and defend democracy. The political reality of many countries makes indicates democracy needs more protection and support than it is experiencing. In the game, we hope, young people can be made aware of these problems. We play on labyrinth-like virtual game templates with one or more enclosed areas that symbolize prisons. These are generated with the figure II (diagonal) and the random function. For instructions see the next chapter.

Gulag is a symbol of oppression and destruction of political opponents. Although the term refers to the no longer existent Soviet Union, political resistance is violently suppressed by all dictators. Their number unfortunately is not low. More information, examples from the history of dictators' dealings with dissidents, examples from the history of the anti-imperial Czechoslovak resistance as well as links to important sources can be found here.

A game of winning: Arrange a task with one or more people who have installed this app on their devices to multiply a quartet and thereby systematically convert it (see "An exercise for advanced users"). They agree on a starting quartet, the size of the field and the type of conversions. In order to define the starting pattern and to compare the results, take screen prints (see further below) and exchange them with the other players. After the agreement everyone can start - no matter when. If the result is correct, the first place is determined by comparing the time required and if the time is the same, by the smaller number of steps. Exchange views on the visual impressions and perception of the result.

Storage: The result can be saved before exiting the app or starting a new project. Caution, without saving, the pattern will be lost as soon as you choose a new layout.

Any pattern you have enjoyed and saved can later be modified. You may reopen it. If you rearrange a portion, for example the central quartet of a rotationally symmetrical pattern, you have a choice. Either you discard the new version or you store it and perhaps replace the previous version. 

Photo: On most devices, you can capture any finished pattern that you like in jpeg format using the print screen command. You may use this, if you want to compete with a friend in building a pattern you have agreed on.

Design: With the patterns created with cardandcube, a special effect can be achieved by turning and cutting (an example).

FREEDOM vs GULAG

Two games, Gulag 1948 and To Freedom! are described in this chapter. In each case you select the line icon and choose a playing area (layout) to play with. Then you can use this button    to align all the cards in the same order or out of order. In this way, maze-like images can be made. In game Gulag 1948 you start with such an image and play on the screen directly as described below. In the other game, To Freedom!, the initial image must be modified to contain only one prison. Then, in the periphery a continuous line is created, representing a border of the country with a few passes. This is a game template that must be exported and printed. The task is to identify the pass of the border that can be reached by the shortest route from the prison. Several game templates are available for download.

Play Gulag 1948 alone or in a group. As described above, you create a maze-like playing area for yourself or as a game leader for a group. In the latter case, save the file naming it and share it with your fellow players. Create a complex maze in a multi-card format, but the size of the cards should not be below a level that can easily be touched with fingers on the displays used.

An example is shown in Figure 1A. Within the playing area enclosed rectangular spaces represent Gulag camps:

Figure 1A   A random picture as game template Gulag 1948

The task is to open all of these prisons in as little time as possible with as few card turns as possible. At the bottom bar of the game there is a counter that keeps track of the number of steps (card turns) and the time. Touching the counter pauses it until you touch a card again. All players can record their results or the score with "Screenshot" and share it with others. Can be played at different times. A gentlemen's agreement can be reached that only one play attempt is allowed.

Figure 1B.   The image of the game Gulag 1948 after opening all "prisons"

The screen capture shown in Figure 1B represents the result of game 1A along with the score achieved.

The Game To Freedom! is best to play with a group. In a labyrinth-like image, the shortest route from a prison to an escape place in the "state border" is sought. A playmaker is required. The task of the playmaker is to prepare the game materials and the participants for the game. Depending on the age of the players then freedom, democracy, despotism, political prisoners and prison conditions can be discussed with them at an appropriate level.

To prepare for this game, cardandcube #3 is opened and - as described at the beginning of this chapter – starting from a random layout (Figure 2A) a fenced labyrinth (Figure 2B) is created, but with only one or a few exits (F), to which two or more paths lead from the edge of a centrally located gulag (G). The enclosed areas in the original image, which symbolize prisons, are all opened by rotating cards, except for the square closest to the centre of the image. The paths are laid out as corridors between the outside of the prison and the exits:

Figure 2A.   Random layout

Figure 2B.   Game template derived from the random layout shown above

Depending on the number of players, the template is printed and distributed together with pencils. Now the participants can consider which of the four "walls" of the prison are designed to overcome. They determine a location of the "breakout." From that moment on, everyone plays a dissident or a political prisoner who is looking for freedom. Freedom is not achieved until one has left the entire territory of the dictatorship. There are two escape routes from the agreed escape point: to the left or to the right and it must be briefly considered which is better. The path taken is traced in pencil. The aim of the game is to decide as quickly as possible for the shorter way and to find the way "to freedom". If you think you've made it, tell the others. The winner is the first to find the shortest route to freedom.

During the discussion, consider advantages and disadvantages the different escape routes may have. This website offers some downloadable examples that can be imported into cardandcube #3. In some cases there is an "unfortunate" alternative to choose from among the escape routes, whose "escape route" leads back to prison.

Depending on the size of the playing area that has been prepared, the "fugitives" may be given a deadline in which they cannot be caught. After the comparative evaluation of the escape successes, a liberation can be celebrated together. The opera fans may choose to listen to Beethoven's Fidelio or its Ouverture while discussing or playing. In the game, all adversaries are only abstractly present - you can deal with them appropriately verbally.

After the escape lines have been erased, the same template can be used to attempt a breakout on a different side of the prison. Depending on the mood, the deadline for drawing in the chosen escape route can be shortened.

If a template with multiple openings in the outer border is used for play, the game can be carried out without agreeing on a breakout side. The players should find both the most favorable exit side and the shortest path.

Dedication and present. We dedicate our Freedom Games to support political prisoners and to commemorate the Third Czechoslovak Resistance. Don't miss reading the story of the family of resistance fighter Václav Švéda by Jáchym Topol zu lesen!

Please consider how to instill a love of freedom or a determination to defend freedom when you are dealing with children and teenagers! Pass on to the next generation your squadron of defending freedom, tolerance and human rights and opposing dictatorships. You may tell the story in the style of the Fidelio-crash-course while playing our games!
____________________________________
TEMPLATES FOR DOWNLOAD: 2F 12x12 | 2F 12x12 +S | 2F 24x16 A | 2F 24x16 B | 4F 24x16 A | 4F 24x16 B and  abbreviations legend.
 

SUPPLEMENTS

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 [5] 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 [6] 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.

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 [3] 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.

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 [8].

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 [9].

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].


LITERATURE

  1. Mukherjee, R. und Kodandaramaiah, U. (2015): What makes eyespots intimidating - The importance of pairedness. MC Evol Biol  2015, 15:34.
  2. Anderson, J.R. et al. (2005): Are Monkeys Aesthetists? Rensch (1957) Revisited.  J Exper Psychol Animal Behav Processes 31:71-80
  3. Truchet, S. (1704): Memoir sur les Combinaisons. Memoires de l´ Académie Royale des Sciences, 363-372
  4. Smith, C.S. und Boucher, P. (1987): The Tiling Patterns of Sébastien Truchet and the Topology of Structural Hierarchy. Leonardo 20, 373-385
  5. 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
  6. Justi, K.: Das Marburger Schloss. Geschichte einer deutschen Burg. Veröffentlichungen der Historischen Kommission für Hessen und Waldeck XXI. N.G. Elwertsche Verlagsbuchhandlung, Marburg 1942
  7. Kier, H. (1970): Der mittelalterliche Schmuckboden. Rheinland Verlag, Düsseldorf
  8. Weisstein, Eric W."Truchet Tiling." From MathWorld--A Wolfram Web Resource.
  9. Lord, E.A. und Ranganathan, S. (2006): Truchet tilings and their generalisations. Resonance, June 2006, 42-50 
  10. Gombrich, E.H. (1994): The sense of order. A study in the psychology of decorative Art. 2nd ed. Phaidon Press Ltd., London

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