cardandcube #1

#1 wellcomes you with a


Geoff Hodbod Geoff Hodbod



This is a set of 44 lay cards with, amazingly, an identical front figure: a black and a white triangle. On the reverse 32 of the cards display patterns of card quartets. So of the remaining cards the reverse of 4 are black and 8 are white.
1: In this game, cards are laid out side by side with the triangles atop in a rectangular manner to create larger areas containing either the black or the white colour. The contrast between these colours is driving your game forward. Create new patterns by lifting and turning the cards around to satisfy your creativity. Make room on your desk, go for it!  

2: The reverse faces of the cards are suitable for a memory game. If two cards are turned over, the players see either quartet pictures, or a white or a black card. The whole set consists of 32, 8 and 4 cards of these types, respectively. Pairs of the same type are collected by those, who found them, while the uneven pair must be returned to their original positions. The winner is the one who collects the largest number of cards.

3: The aim of the third game is to recognize geometric patterns on the reverse of the cards. Spread out all 32 cards with the card quartet pictures facing upwards. Display the numbered ones in a row or two and put the others on a pile. The game begins with revealing the pattern of the upper card on top of the pile. The players call the number of the twin card that is among the displayed ones. If the call is correct, the two matching cards are collected in front of the player. The win goes to who has collected the highest number of cards

Have fun - enjoy these cards + your creativity! 

Here you can see how to put all cards back into the box.

Consider these cards for kids and for adults, too! For cooperative pairs and for singles - as a kind of solitaire - as well! 

Note: Currently this card game is available in Germany only. If you like the idea of assembling the uneven triangle twins to larger patterns, you may consider the virtual cards of cardandcube game #2  that will be made available soon.



The face of the cards shows two rectangular isosceles triangles, one black, the other white. We have got an inseparable couple combined onto a single square.

Within a rectangular grid layout the card can be turned up to four times by 90° to return to the original placing

By laying cards side by side the white or the black areas become extended while larger patterns assemble. Some structures may remember you of common items, some patterns may catch your eyes. Usually regularities and symmetries will please you. The cards invite you to explore the patterns and to construct larger images from smaller units. Alternatively, you can use all cards to assemble a complex asymmetric structure. The aim is to perform without any pressure, just for fun. Not for an adrenaline kick, but rather for play and relaxation.

To begin with it may be helpful to push two cards together in any orientations edge to edge to obtain card duos. Take two more cards and then examine card quartets. These consist of 4 cards in the 2x2 layout. In fact there are 256 possible card quartets. Images of these are displayed on the reverse of two sets of 16 cards. Among them, there are 64 symmetric quartets that hold observer´s attention and may be a source of inspiration. On the reverse of the cards you will easily find those with fourfold rotational symmetry: These are contained within four asterisks.

On any flat surface start with a few cards, then make and alter quartets and find out yourself, how to construct a few of the symmetrical kind. Try to assemble several quartets by a simple repetition.

Next, extend a quartet with copies of it. Alternatively you may want to assemble the second quartet as a mirror image of the first one. Or you make a copy and swap its colours. (To swap the colours of a quartet, lift the first card, turn it by 180° and place it back. Repeat this with the other three cards of the quartet. Continue making such pairs of quartets, or change the strategy.

At any time you may add cards at the periphery at your imagination. Eventually you may have assembled a geometrical picture from all 44 cards, one of more than 1026 possible variants. This appears never ending. However, most of possible assemblies of the cards you won´t ever try. Something will drive you to create pleasing and harmonic pictures. Your latest creation will be like a fresh air and if you feel there are too few cards to continue, where you are, just add cards from the other side of the play area and let it find its way to where you want.

Constructing a geometrical picture is something you may enjoy in solitude. Now, why not to invite a friend to join in on it? Beyond this, if you like to play against somebody, you may prefer an option using the reverse of the cards for a competition. All three alternatives to play with cardandcube #1 are outlined in the next chapter.

Play 1

Play 2

Play 3


The beginning: You might get started as soon as you have opened the cube with the cards. We recommend initially to build a symmetrical quartet. You can name some of the patterns created in this way. The pattern shown below, for example, may be called an “arrowhead pointing upwards”.


The picture may also be reminiscent of a gable roof. It's quite memorable and that may be due to its symmetry. Possibly also because its upper and lower halves are a colour-exchange symmetrical pair. In the upper half of the picture you can see a white peak on a black background and in the lower half a black peak on a white background.

Exercises: Build four identical quartets of your choice - or as shown in the scheme below - at some distance from each other in a circle. Next, proceeding clock-wise turn three of them by 90°, 180° and 270°. Finally, shift the quartets towards the centre to obtain a layout of 4x4 cards.

Next, you may attempt to assembly all four quartets with 4-fold rotational symmetry using a layout of 2x2 cards. This may be achieved with the help of the scheme shown just above while starting with four cards instead of four quartets.

Have fun!


History of the lay card: In the 90s of the last century, Hana Hasilik, the author of the laying card game, began to build abstract sculptures under the influence of constructivism. She developed square modules from which she constructed large-format floor sculptures. One of these modules consisted of 10 lentoids, which were mounted on a square base in the manner of a Pythagorean tetractys. One half of the module was covered with the lentoids, the other remained free.

A square module with 10 lentoids (fired white clay 22x22x7 cm). The Tetractys arrangement of the lentoids corresponds to that of Blaise Pascal


One of such grid art sculptures made of these modules is shown next: Linie / Trace (fired white clay, 140 x 140 x 8 cm - presented at FormsacheN, Packhof in Hannoversch Münden 2006).

In this sculpture an impression of a path emerged along modules that have been laid next to each other in opposite orientations such that a distinct imbalance challenged a beholder. Contemplating on this kind of module assemblies, the sculptor has taken squares of paper, halved them diagonally and shaded one of the triangles thus formed. Thus a pair of triangles emerges as a symbol of sculpture modules.

The cards lend themselves for experimentation and sparked an idea to use them for playing. In design, however, there seems to be nothing new on the Earth. For thousands of years constructions of triangles and squares have been known.

Tiles with two coloured rectangular triangles have caught the attention of the Dominican monk, mathematician and designer Sébastien Truchet (1657-1729) centuries ago on a countryside walk near Orléans. Truchet presented an account of this story and described decorative effects of such tiles in tesselation in his Memoir sur les Combinaisons [1, 2]. The astonishingly high number of permutations of possible patterns of diagonally divided two-tone squares has been recognized by the Carmelite order priest Dominique Douat. In 1722 he published a monograph with 60 patterns [3]. Half a century later, the versatile Henri-Louis Duhamel Du Monceau described the production of decorative tiles and showed 90 panels with beautiful patterns from the diagonally divided squares [4].


From antiquity to Marburg: the use of two-tone, diagonally divided squares in medieval arts and crafts can be traced back to antiquity south of the Alps [5]. One of the earliest examples is the floor mosaic shown below. This remarkable floor ornament lay beneath the ash and slag of the Vesuvius eruption for almost two millennia:

The picture, published in 1938* [6], shows a portion of the neat mosaic floor of a room in the Cassa delle nozze d`argento in Pompeii, which was probably used as a bedroom. The same pattern was executed a good 12 hundred years later in medieval Marburg with glazed tiles. Partly in the original, partly reproduced, it can be found in the sacristies of the palace chapel and the Elisabeth church*:

You can create this and thousands of other patterns with cardandcube. Show your patterns to friends or take a photo. We wish you a lot of joy, fun and relaxation!

The back of the cardandcube #1 cards shows graphic images of card quartets. In the guide above we have two games based on the backs of the cards.

Check on internet whether these cards (cardandcube #1) are available in your country. Nevertheless, cardancube #2 and #3, virtual game variants with more possibilities, will be offered and accessible from practically anywhere, soon.

* cardandcube thanks the de Gruyter publishing group and the Ev. Elisabethkirchengemeinde Marburg for generous permissions to present the respective photographs on this page.


  1. Truchet, S. (1704): Memoir sur les Combinaisons. Memoires de l´ Académie Royale des Sciences, 363-372
  2. Smith, C.S. und Boucher, P. (1987): The Tiling Patterns of Sébastien Truchet and the Topology of Structural Hierarchy. Leonardo 20, 373-385
  3. 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. L´Imprimerie de Jacques Quillau, Paris
  4. Duhamel Du Monceau, H.-L. (1773): L´art du potier de terre, Pl. VI - XIV. Nachdruck {BnF Gallica
  5. Kier, H. (1970): Der mittelalterliche Schmuckboden. Rheinland Verlag, Düsseldorf, S. 172 und 173
  6. Pernice, E. (1938): Pavimente und Figürliche Mosaiken. In: Winter und Pernice, E. (Hrsg.) Die Hellenistische Kunst in Pompeji, Band IV, Tafel 17. 2. Walter de Gruyter, Berlin



Available in: Kunstmuseum Marburg, Biegenstrasse 11, 35032 Marburg (no online service)