randommath

The Geometry of a Nation: Unfolding the Nepali Flag

Jan 29, 20254 minEnglish

Nepal's flag is the only national flag that isn't a quadrilateral, its distinctive shape precisely defined by mathematical instructions enshrined within Nepal's constitution. This piece walks through that construction, one decree at a time.

A Flag Unlike Any Other

The Nepali flag is the only national flag that isn't a quadrilateral, its distinctive shape precisely defined by mathematical instructions enshrined within Nepal's constitution.

If you want to see these rules in action, there are tons of resources online. Numberphile beautifully demonstrated hand-drawing the flag on cardboard. It's where I first found out that there is a rule to draw the flag. This GeoGebra project offers a step-by-step visualization using a slider, I took the pictures for this article from that project. And for the coding enthusiasts, Stack Overflow has a thread where some folks generated the flag using JavaScript, Python, SVG, Mathematica, and PostScript.

Here are the extracted flag construction details.

Drawing by Decree: The Geometry of Nepal's Flag

Step 1: The Foundation

It begins with the foundation: a single line, the base.

The images I'm using come from GeoGebra, but if you're making a real flag, you'd traditionally use crimson cloth (Hex: #DC143C, RGB: 220, 20, 60). Let's be honest, though, printing it out is much easier!

But if you were going the traditional route, you'd start at the bottom of your crimson cloth and draw a line, AB, any size. This line is the foundation upon which all other measurements are based, setting the scene for everything that follows.

Step 1: drawing base line AB — On the lower portion of a crimson cloth draw a line AB of the required length from left to right.

Step 2–5: Constructing the Twin Triangles

From point A, a perpendicular line AC is drawn, its length precisely determined as AB plus one-third of AB. This vertical axis becomes the backbone of the upper and lower triangles, setting the stage for their formation.

Step 2: perpendicular AC and point D — 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 BD.

The next step is marking point D, ensuring AD equals AB, followed by connecting BD, the diagonal stroke that gives the flag its first defined shape.

Step 3: mark off E on BD — From BD mark off E making BE equal to AB.

The structure deepens as we bisect BD at E and extend a parallel segment FG, identical in length to AB.

Step 4: parallel line FG — 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.

With CG now established, the foundational framework is complete.

Step 6–18: Constructing the Moon

With the skeleton in place, we turn our attention to the moon and sun.

The moon, positioned on the upper triangle, is a careful assembly of arcs and radii. From AB, we measure a quarter of its length to determine AH, establishing the moon's placement. A vertical line HI extends parallel to AC, guiding the internal construction. Bisecting CF at J and further extending JK, we refine the structure until the central reference points emerge.

Step 6: mark off AH and draw HI — From AB mark off AH making AH equal to one-fourth of line AB and starting from H draw a line HI parallel to line AC touching line CG at point I.

Step 7: bisect CF at J and draw JK — Bisect CF at J and draw a line JK parallel to AB touching CG at point K.

With geometric precision, arcs are drawn from centers L, M, and N, curving into crescents that form the moon's distinct shape. Additional arcs define the eight triangular rays.

Step 8: point L — Let L be the point where lines JK and HI cut one another.

Step 10: point M — Let M be the point where line JG and HI cut one another.

Step 11: mark off N — With center M and with a distance shortest from M to BD mark off N on the lower portion of line HI.

Step 12: draw line through M from O — Touching M and starting from O, a point on AC, draw a line from left to right parallel to AB.

Step 13: semi-circle with center L radius LN — With center L and radius LN draw a semi-circle on the lower portion and let P and Q be the points where it touches the line OM respectively.

Step 14: semi-circle with center M radius MQ — With the center M and radius MQ draw a semi-circle on the lower portion touching P and Q.

Step 15: arc with center N radius NM — With center N and radius NM draw an arc touching PNQ at R and S. Join RS. Let T be the point where RS and HI cut one another.

Step 16: semi-circle with center T radius TS — With center T and radius TS draw a semi-circle on the upper portion of PNQ touching at two points.

Step 17: arc with center T radius TM — With center T and radius TM draw an arc on the upper portion of PNQ touching at two points.

Step 18: the eight triangles of the moon — Eight equal and similar triangles of the moon are to be made in the space lying inside the semi-circle of No (16) and outside the arc of No (17) of this Schedule.

The moon takes shape as a construct of pure geometry, its curves and edges dictated by the same principles that govern the flag's form.

Step 19–22: Constructing the Sun

Below the moon, centered within the lower triangle, shines the sun. The process begins with a bisection of AF at U. A parallel line UV extends to meet BE at V, marking the sun's foundation.

Step 19: bisect AF at U and draw UV — Bisect line AF at U, and draw a line UV parallel to AB line touching line BE at V.

From a central point W, determined by the intersection of UN and HI, two perfect circles are drawn. The outer circle defines the sun's reach, while the inner boundary shapes its core. Within this sphere, twelve equal triangles burst outward, their tips touching the outer edge.

Step 20: circle with center W radius MN — With center W, the point where HI and UN cut one another and radius MN draw a circle.

Step 21: circle with center W radius LN — With center W and radius LN draw a circle.

Step 22: the twelve triangles of the sun — Twelve equal and similar triangles of the sun are to be made in the space enclosed by the circle of No (20) and No (21) with the two apexes of two triangles touching line HI.

Step 23–25: The Final Border

With the sun and the moon in place, the last step in this geometric construction is the deep-blue border, a frame that binds the entire design. The border's width is determined by the segment TN, ensuring that the proportions remain consistent.

Step 23: the deep blue border — The width of the border will be equal to the width of TN. This will be of deep blue color and will be provided on all the sides of the flag. However, on the given angles of the flag the external angles will be equal to the internal angles.

If the flag is to be hoisted, an additional extension may be made on the side along AC, allowing for the necessary fixtures.

A small asymmetry

Because the slope of the upper triangle is less than the bottom triangle, while creating the blue border, the top triangle extends more to the right than the bottom triangle. It is small, but if you look carefully, you will see it.

The aspect ratio of Nepal's flag is unusual, and this weirdness comes from the blue border. Without the blue border, just considering the internal crimson triangle, the aspect ratio is 3:4. But with the blue border, the aspect ratio is:

1 : \frac{6136891429688 - 306253616715\sqrt{2} - (934861968 + 20332617192\sqrt{2})\, \sqrt{118 - 48\sqrt{2}}}{4506606337686}

which is roughly

\approx 1 : 1.219010337829452184570024869930988566\ldots