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FOREWORD INTRODUCTION CONVEX FIGURES FULLY MATCHED PATTERNS MISCELLANEA HIGHLY MATCHED PATTERNS SMALL CONVEX HULL AREA PATTERNS COUNTING TANGRAM PATTERNS MORE PATTERNS WRITE ME LINKS |
FOREWORD · Tangram is a world apart, with its own rules and characters. I found it fascinating since it can be appreciated from many points of view. It is a puzzle. It is a source of pictures. It is a companion to learn geometry. In any case, it is a way to activate the mind. · As in any world, there are people who try to explore and catalogue it. I used a geometrical key to make my own exploitation. · Of course, as well as in any source on Tangram, you can skip the written text and just look at the pictures. Still, I think you will feel the geometrical plot behind it. · I used a computer, and I have given an upper bound to the number of elements of a subset of Tangram silhouettes. The value I estimated is nearly ten millions. · I chose to let free the size of the Tangram pieces for each pattern, in order to properly fit the figures and get a nice graphical layout. Someone will be fooled by that variety, but to fully enjoy Tangram he needs to focus on one pattern at the time, transcending from the neighborhood. · see more than one billion of silhouettes |
    
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FOREWORD INTRODUCTION CONVEX FIGURES FULLY MATCHED PATTERNS MISCELLANEA HIGHLY MATCHED PATTERNS SMALL CONVEX HULL AREA PATTERNS COUNTING TANGRAM PATTERNS MORE PATTERNS WRITE ME LINKS |
INTRODUCTION · Tangram was born in China long time ago, but the first written sources date only back to the eighteenth century. · The name Tangram has not a fully accepted origin. Someone talks about Tang, a Cantones word for "Chinese", and Gram, meaning "something drawn". The Chinese have called Tangram the "Board of Wisdom". In any case, something mythical is still behind this puzzle, and I think is wise not to disperse the fog and mist that hide it. · Tangram consists of seven geometrical pieces, which can be obtained by cutting a square. They can be arranged on a plane in a virtually infinite number of configurations. · 
There exist many books containing hundreds of patterns obtained by arranging the Tangram pieces. The figures are printed in such a way you cannot detect the borders of matched pieces. The puzzle just consists in realizing how those silhouettes can be get using all the seven not-overlapping pieces. · The silhouettes often look like animals, humans, objects, and others. · Tangram tradition is focused onto pattern recognition, that is to say there is a sort of game in the game to consider silhouettes who represent something or someone. However, patterns are mainly of interest to a Tangram player as far as he is challenged to solve them. · |
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FOREWORD INTRODUCTION CONVEX FIGURES FULLY MATCHED PATTERNS MISCELLANEA HIGHLY MATCHED PATTERNS SMALL CONVEX HULL AREA PATTERNS COUNTING TANGRAM PATTERNS MORE PATTERNS WRITE ME LINKS |
CONVEX FIGURES · A hard-to-solve pattern has as few dangling pieces as possible. From this point of view, the convex figures should be the best, but well known. · A figure is called convex when every point on a line joining any two points of the figure lies within the figure itself. There exist exactly 13 convex figure, which can be obtained with the Tangram pieces. This was rigorously demonstrated by F. T. Wang and C. C. Hsung in the "American Mathematical Monthly", volume 19, 1942. · see more than one billion of silhouettes
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FOREWORD INTRODUCTION CONVEX FIGURES FULLY MATCHED PATTERNS MISCELLANEA HIGHLY MATCHED PATTERNS SMALL CONVEX HULL AREA PATTERNS COUNTING TANGRAM PATTERNS MORE PATTERNS WRITE ME LINKS |
FULLY MATCHED PATTERNS · It is easy to identify dangling pieces ( e.g. the head of the running man ), and also pieces with matched edges but not matched vertices ( e.g. the feet of the man ). As a rule of thumb, hard-to-solve patterns are built by matching as much edges and vertices of the pieces as possible. · I call fully matched patterns those configurations where each piece has at least one edge and one vertex matched on those of another piece. · All the previous figurative examples are not fully matched ones since at least one piece of the pattern is not fully matched. For instance, the pieces of the roof of the house have edges matched on those of the wall below, but not the vertices. The head of the cat has neither edges nor vertices matched on the body. The tail of the cat has a vertex matched onto a vertex of the body, but it has no matched edges. · On the contrary, all the convex figures are fully matched. · Fully matched patterns usually have pieces with more than just one matched vertex. That is to say, the vertices, which delimitate the matched edge, are often both matched. This relies on the fact that the pieces are not bricks but still they are quite modular, their side lengths being proportional to 1, Ö 2, 2 and 2 Ö 2. That Golden sequence is probably the core of Tangram charm. · In this site, I reported silhouettes obtained by random generation of fully matched patterns. Their solutions are always available. They are collected in three sets. The first one is a miscellany of figures which look like something, in the respect of Tangram tradition. The second and third sets are collections automatically selected by the computer, according to precise rules applied to very large sets of configurations, in order to obtain hard-to-solve silhouettes. · see more than one billion of silhouettes |
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FOREWORD INTRODUCTION CONVEX FIGURES FULLY MATCHED PATTERNS MISCELLANEA HIGHLY MATCHED PATTERNS SMALL CONVEX HULL AREA PATTERNS COUNTING TANGRAM PATTERNS MORE PATTERNS WRITE ME LINKS |
MISCELLANEA · I saw a strong resemblance of those figures with something, and I have collected them consequently. · It is worthy to be mentioned that the 53 patterns have been chosen among nearly 500 silhouettes randomly generated. Most of the patterns have, then, been rotated as a whole from the original configuration found by the computer. Therefore, in a statistical sense, nearly 10% of the fully matched patterns should have a silhouette looking like something. · see more than one billion of silhouettes |
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FOREWORD INTRODUCTION CONVEX FIGURES FULLY MATCHED PATTERNS MISCELLANEA HIGHLY MATCHED PATTERNS SMALL CONVEX HULL AREA PATTERNS COUNTING TANGRAM PATTERNS MORE PATTERNS WRITE ME LINKS |
HIGHLY MATCHED PATTERNS · Fully matched patterns have at least six pairs of matched edges. This minimum value is always encountered in snakelike patterns. · When wrapping occurs, we can have seven, eight, and nine pairs of matched edges. It is likelihood that the larger the number of pairs, the harder to solve is the figure. Therefore, the number of matched edges can be a measure of challenging of the pattern. · At my knowledge, only the square has ten pairs of matched edges. · To get an idea on the whole population of fully matched patterns, I measured a sample of 100000 random configurations. It turned out that nearly 78% of the patterns had just 6 pairs of matched edges, 20% had 7 pairs, about 1.5% had 8 pairs, and less than 0.03% had 9 pairs. · see more than one billion of silhouettes |
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FOREWORD INTRODUCTION CONVEX FIGURES FULLY MATCHED PATTERNS MISCELLANEA HIGHLY MATCHED PATTERNS SMALL CONVEX HULL AREA PATTERNS COUNTING TANGRAM PATTERNS MORE PATTERNS WRITE ME LINKS |
SMALL CONVEX HULL AREA PATTERNS · Another way to measure how much a pattern could be hard-to-solve is by its convex hull. · A Tangram patterns can be seen as the set of the vertices of the seven pieces. The convex hull of a set of points is defined as the smallest convex figure whose vertices are points of the set itself. · If you find this definition too tuff, don’t worry. The concept of convex hull is more intuitive than it should appear from its exact definition. Look at the silhouette of the teapot. The segmented orange line, which contours the silhouette, delimitates its convex hull. You can figure out that line, as it is a rubber band that holds the Tangram pieces. · The 13 convex Tangram patterns coincide with their own convex hulls. Any other pattern has the convex hull larger than the pattern itself. Let’s set to one the overall area of the Tangram pieces. Therefore, the closer to one the convex hull area is, the more compact the figure is, and the harder to solve it is likely to be. · To get an idea on the whole population of fully matched patterns, I have measured the convex hull area over a sample of 10000 configurations. It turned out that less than 1% of the patterns has the area smaller than 1.2, nearly 20% has the area smaller than 1.4, while almost all the patterns have the area smaller than 2. · see more than one billion of silhouettes |
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FOREWORD INTRODUCTION CONVEX FIGURES FULLY MATCHED PATTERNS MISCELLANEA HIGHLY MATCHED PATTERNS SMALL CONVEX HULL AREA PATTERNS COUNTING TANGRAM PATTERNS MORE PATTERNS WRITE ME LINKS |
COUNTING TANGRAM PATTERNS · In the previous sections we have exploited the set of the fully matched patterns. If you are not yet bored to hear about it, you could find interesting to know that it is countable and finite. It has, more or less, ten millions of distinguishable elements. If you don’t trust me, read the following. · Let’s build a snakelike pattern, as in the movie frames on the left, and count how many choices should be available at each step, neglecting the occurrence of overlap. · I started from the square, the choice of the first element being unessential for counting. In the second step, we can assemble the six remaining pieces, which reduce to four, since the two small and two large triangles are paired, and cannot be distinguished. · You can match one of the four independent piece on one side of the square. The sides of the square are undistinguishable. On the contrary, each side of the triangles gives rise to different patterns, while only two sides of the rhomboid can be distinguished, which double by flipping. Therefore, there are 3 pieces x 3 sides + 1 piece x 2 sides x 2 = 13 choices. For the sake of illustration, let's chose the medium triangle, and match it to the square, as in the reported sequence. · In the third step we can match the five remaining pieces (three of them being independent, one small triangle, one large triangle and the rhomboid) either on a side of the square or on one of the medium triangle, with the exclusion of the two shared sides. Now, the three free sides of the square give rise to independent configurations. · Therefore, there are (1 piece x 4 sides + 1 piece x 3 sides – 2 shared sides) x ( 2 pieces x 3 sides + 1 piece x 2 sides x 2) = 50 choices. We can continue in this way. · For the sequence reported in the table, at the end of the seven steps, we found more than 1 billion of alternative possibilities, and exactly 13 x 50 x 42 x 48 x 27 x 30 = 1061424000 choices. · You can make the exercise of choosing a different sequence of pieces, building another snakelike pattern. You will always get the same order of magnitude of different choices, i.e. 1000000000. · Indeed snakelike patterns are not the rule, but the exception. In fact many choices evolve in wrapped configurations, which should give rise to overlapping. For instance, if you match the last large triangle as in the last but one frame of the sequence, rather than as in the last frame, you get a forbidden pattern, since the triangle overlaps the rhomboid and the square. · The occurrence of overlapping pieces can be statistically checked. I did it by generating 10000 random sequences on the computer. It randomly tries to match the pieces, checking for overlapping at each choice, and, in case, rejecting it. · I found that 5 choices are rejected as an average, to assemble all the 7 pieces. This event nearly halves the number of allowed choices at each step, from the third one onward. Therefore, the number of independent patterns turns out to be nearly 1000000000 / 25 » 30000000 (thirty millions). · A second source of reduction on the independent choices is the fact that there exist twin and multiples configurations. They are patterns whose silhouettes are the same, while the internal compositions are not. However, as in mankind, twins are not the rule in the Tangram population. · A third source of reduction on the independent choices is the occurrence of ring-like patterns. Such kind of configurations has not to be counted twice. · As a whole, the value of ten millions of distinguishable silhouettes seem a conservative estimate for the fully matched Tangram patterns. · This number is large enough to challenge Tangram players for a long time. In fact, according to the sampling done in the previous sections, there should exist about one million of silhouettes looking like something, nearly 3000 with nine pairs of matched edges, and about 100000 with the convex hull area less than 1.2. · see more than one billion of silhouettes |
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FOREWORD INTRODUCTION CONVEX FIGURES FULLY MATCHED PATTERNS MISCELLANEA HIGHLY MATCHED PATTERNS SMALL CONVEX HULL AREA PATTERNS COUNTING TANGRAM PATTERNS MORE PATTERNS WRITE ME LINKS |
MORE PATTERNS · Fully matched patterns are a small subset of the Tangram configurations. A larger set is that of fully aligned patterns. · An example of a fully aligned, but not matched, piece is the tail of the cat silhouette. Any matched piece is also aligned. Therefore, the set of fully matched patterns is a subset of that of fully aligned ones. Those patterns give rise to nearly one billion of distinguishable figures. · I have selected 48 figures randomly generated by the computer. They have been selected according to having less than 4 matched pairs of edges and lot of dangling pieces. The solution of those ideogram-like silhouettes is easy. · Beginners and inpatient people could start from these patterns the long journey to the Tangram world. · see more than one billion of silhouettes |
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The text and figures are for individual or classroom use, not to be distributed or sold without the written consent of Franco Cocchini.
