## Archive for June 7, 2017

### The Amazing National Flag of Nepal

The National Flag of Nepal is unique.

Here are several trivia questions about the Nepalese flag:

- It is the only non-quadrilateral national flag in the world. What is its shape?
- It is one of only three national flags where the height is not less than the length. What are the other two?
- What is the sum of the three acute interior angles within the flag?

Question 3 may be difficult to answer without knowing more about the exact dimensions of the flag. For help with that, we turn to the Constitution of Nepal, promulgated 20 September 2015, which contains the following geometric description for the construction of the flag:

**SCHEDULE – 1
The method of making the National Flag of Nepal**

- Method of making the shape inside the border
- On the lower portion of a crimson cloth draw a line AB of the required length from left to right.
- From A draw a line AC perpendicular to AB making AC equal to AB plus one third AB. From AC mark off D making the line AD equal to line AB. Join B and D.
- From BD mark off E making BE equal to AB.
- Touching E draw a line FG, starting from the point F on line AC, parallel to AB to the right hand-side. Mark off FG equal to AB.
- Join C and G.

The traditionalist in me wishes that “line segment” were used instead of “line,” or that the overline were used to indicate those segments, and that a few more commas were inserted to make it more readable. Consequently, the math editor in me feels compelled to rewrite the directions as follows:

But the American in me — given how many times someone in the United States has tried to legislate the value of π — well, I’m just excited to see accurate mathematics within a government document.

The description continues for another 19 exhilarating steps, explaining how to construct a crescent moon in the top triangle, a twelve-pointed sun in the bottom triangle, and a border around the shape described above. Those steps are omitted here — because you surely get the gist from what’s above — but the following “explanation” that appears below the method is worthy of examination:

The lines HI, RS, FE, ED, JG, OQ, JK and UV are

imaginary. Similarly, the external and internal circles of the sun and the other arcs except the crescent moon are alsoimaginary. These are not shown on the flag.

The entirety of this construction, as any classical geometrician would hope, can be completed with compass and straightedge. I cheated a bit and used Geometer’s SketchPad, with this being the resultant mess:

The rough part was placing *C* so that *AC* = *AB* + 1/3 *AB*. Geometer’s SketchPad could have easily measured *AB*, calculated 4/3 of its length, and then constructed a “circle by center and radius,” but that felt like cheating. Instead, I…

- located
*Q*, which is halfway between*A*and*D*; - constructed circle
*A*with radius*AQ = AP*; - constructed circle
*P*with radius*PD*; - constructed circle
*D*_{1}with radius*DA*; - located the intersections of circle
*P*and circle*D*_{1}at points*X*and*Y*; - constructed a line through
*X*and*Y*; - located
*R*, which is 1/3 of the way from*D*to*A*; - constructed circle
*D*_{2}with radius*DR*; and, - located
*C*, so that*CD*= 1/3*AB*.

Now, you could use that information to determine *CF* and *FG*, and then use the arctan function to calculate the measures of the two acute angles in the upper pennon. If you were so inclined, you’d find that their measures are 32.06° and 57.94°, respectively.

But the question above asked for the **sum** of the three angle measures. Without any work at all, it’s clear that the sum of those two angles must be 90°, since the construction described above implies that Δ*CFG* is a right triangle.

And because *AB* = *AD* by construction, then Δ*DAB* is an isosceles right triangle, and the measure of the third acute angle must be 45°.

And that brings us to a good point for revealing the answers to the three questions from above.

- Pentagon
- The flags of Switzerland and Vatican City are square, so the height and width are equal.
- 135°

If you’re looking for more flag-related fun, check out the MJ4MF post from Flag Day 2016 about converting each flag to a pie graph.