Depending on your definition, they can exist. One definition of polygram is the general case into which all polygons and some non-polygons fall into.

But I’m assuming you mean certain types of star polygons, like the traditional pentagram star. Polygrams are usually described by their [Schlafli symbol](https://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol#Regular_polygons_(plane)), which for a pentagon is {5} and for a standard pentagram is {5/2}.

What would a four-sided polygram look like? If it’s {4/1} or {4/3}, you just get a quadrilateral. If it’s {4/2}, you have two pairs of vertices, connected by crossing line segments. If it’s {4/4}, you have four unconnected vertices, each alone in space. To get something interesting in the Schlafli notation, your second number has to be greater than or equal to 2, less than vertices – 1, and relatively prime to the number of vertices. When vertices = 4, there is no integer that has that property.

The suffix “-gram” in mathematics refers to a star polygon. For example pentagram is 5 pointed star.

The way you make a star, in mathematics, is by drawing a number of points evenly spaced around a circle and then connecting each point to the point more than one but less than the number of points/2 away. So for example a pentagram is formed by drawing five points in a circle and connecting each point to the the point 2 in front and behind in sequence. A hexagram is formed the same way but with 6 points. A heptagram is formed with seven points connected either every 2 or 3 points etc.

By this process a tetragram cannot exist because the only options for 4 is 1 or 2 which are equal to 1 and 4/2. Likewise a triagram cannot exist because the only option is 1.

draw 5 points roughly evenly spaced like the points of an imaginary pentagon. As you walk clockwise around the pentagon, label them 1 through 5 in order as you encounter them.

If you draw a line from 1 to 2, 2 to 3, 3 to 4, 4 to 5, and 5 back to 1 you will have drawn a pentagon. Notice how every point is only 1 away from the previous (2-1=1, 3-2=1, 4-3=1 and 5-4=1 I won’t discuss modular arithmetic, but you can clearly see that 5 is also only one point away from 1) to get a pentagram, try connecting dots that are a gap of 2 away from each other. 1 to 3, 3 to 5, 5 to 2 (skipping over 1), 2 to 4, and 4 back to 1 (skipping over 5). Notice every line skips one number, but I didn’t list all of them because it’s obvious for those that aren’t wrapping across the first point. Here, 3-1=2, 5-3=2, and 4-2=2. And the other gaps are also 2 points away, but you’ll have to just count them.

So let’s try 4 points. If you use a gap of 1, you end up with a tetragon (square). If you choose a gap of 2, you draw one line and then become stuck. 1 to 3 (skipping 2) but then 3 can’t go back to 1 (skipping 4) because you already drew that line. So if you also connect 2 to 4, you end up with a plus sign. If you use a gap of 3, it’s the same thing as using a gap of 1 if you counted counterclockwise instead of clockwise when labeling your points. This makes a square again. And a gap of 4 isn’t possible because you skip all the points and draw no lines at all.

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