What is a “Tangram”?

A tangram is an ancient Chinese puzzle. The traditional tangram is made from seven geometric shapes: five triangles of various sizes, a square, and a parallelogram. These seven shapes can be arranged to form a perfect square. They can be rearranged to make literally thousands of other shapes that resemble people, animals and objects.

There are tangram puzzles that are made of other shapes, as well. Some have rounded edges and form an egg shape when assembled correctly. Each piece that constitutes the tangram puzzle is called a “tan” while the overall puzzle is called a “tangram.”

What is a polygon?

A polygon is an enclosed, two-dimensional shape with more than two straight sides. The sides must not be curved or rounded. Triangles, squares, and rectangles are polygons. Because they have rounded edges, circles, ovals and ellipses are not polygons. Polygons can have many sides.

Simple polygons have sides or edges that do not cross over each other, such as is the case with triangles and squares. Complex polygons have edges that do cross over each other, such as a star.

Are all tangrams polygons?

This one could start arguments, as opinions vary. Because proper tangrams should consist of all seven tans of the tangram puzzle touching and not overlapping, then, yes, all tangram shapes should be considered as polygons. Some are simple polygons, while others are complex polygons. It may seem odd that a “house,” or a “bunny,” or a “person running” could be considered a polygon, but keep in mind we’re not literally referring to “houses,” “bunnies,” or “people.” Tangrams are only shapes that suggest those things. Try as they might, nobody has ever managed to get a stomach as flat as those represented by tangram images.

I see that the Tangram Fury game includes several shapes that are listed as “convex tangrams.” I have to ask…

What is a “convex tangram?”

Cool question, and thanks for noticing! There are a couple of ways to answer that question: complicated and slightly less complicated. Let’s see if I can keep this down to the less-complicated side of things.

First, there are some terms that need to be understood in order to understand the overall answer. I could throw out a kabillion terms, but no one wants to read that (and, I don’t want to type all that), so I’ll stick to the following:

Polygon – A polygon is a shape. Technically, it’s a “closed shape”. By “closed” it means that all of the lines that make up its edges eventually come back together to touch end to end, so that if you dragged your finger along the outside of the shape, your finger would eventually come back to the starting point.

An “O” or a “D” would be closed shapes. A “U” or an “S” would be considered “open” shapes, because their ends never touch. BUT, although an “O” or a “D” are closed shapes, they are not polygons. Polygons only have flat sides, never rounded ones.

Polygons can have many sides, but must have at least 3 sides, because it is impossible for a single straight line, or even 2 straight lines to form an enclosed shape. A triangle is a polygon. Squares and rectangles are also polygons. A stop sign is a fine example of an 8-sided polygon. A circle is NOT a polygon. Why? Because it doesn’t have flat sides (unless you get really, really picky and claim that the circle is made up of an infinite number of very short flat lines, but then, is it truly a circle?).

Convex – This is the tough term that loses people. Mathematicians will explain that a convex shape is one in which the interior angle of each set of adjoining lines is less than 180 degrees.

Huh? What’s that mean?

Let’s try that again. A shape is considered “convex” if a straight line can be drawn from any two points within the shape’s edge, and that line never goes outside of the shape.

Um, that’s a little more helpful, but, what…?

Think of it as rowing a canoe across a lake. If you can start anywhere on the lake’s shore and paddle in a straight line directly to any other part of the lake’s shore, without having to turn the canoe, then that lake is convex. If any part of that lake’s shore pokes into the lake so that you have to steer around it, then that lake is not “convex.” (Keep in mind, that because lakes don’t have straight edges, they aren’t polygons….)

Concave – This is what you call a polygon that is NOT a convex polygon. ;^) Take the “cave” part of that word and think of it as a polygon where one of its sides has “caved” in. The polygon is “dented.” One of its angles comes in toward its center area. (Actually, it’s the reverse. At least one of its angles is greater than 180 degrees….)

OK, let’s try one last example to make sure this is clear. Picture a deflated beach ball. When it’s deflated, you can grab it because parts of it are tucked into itself, or there are dents in its surface, or however you want to describe that. Now, just picture the deflated beach ball’s shadow (because polygons aren’t 3-dimensional). Because of the dents in the edges of the shadow, the deflated beach ball’s shadow is concave. Fill that ball with air, and the shadow will be convex. (But, it won’t be a polygon, because it’s round….) Using this analogy, one could say that a convex polygon is a flat-sided shape that is essentially bursting at its seams.

Back to the question…

Convex Tangrams represent convex polygons. Mathematicians have studied tangram images and tangram polygons for decades (if not centuries). In 1942, Fu Traing Wang and Chuan-Chin Hsiung proved mathematically that of the thousands of potential tangram shapes that can be made with the seven tans, only 13 can be considered convex polygons.

That means that only a fraction of a percent of tangrams are convex polygons or convex tangrams. In honor of that rarity, all 13 convex tangrams have been included in the Tangram Fury set of cards. Some have been given different names (such as a pyramid, shed, tent or square), but most are identified simply as being a “Convex Tangram.”