How Long Is Nepal’s Border?

Here is what the widget measured, with four shorter rulers added. The steps were walked on Nepal’s full outline from the map data the site’s 77 Districts game is drawn from, 66,417 points round the border.

Ruler	Steps	Length of the border
160 km	11 (and 40 km left over)	1,800 km
80 km	24	1,953 km
40 km	51	2,068 km
20 km	111	2,238 km
10 km	247	2,474 km
5 km	521	2,609 km
2.5 km	1,121	2,804 km
1.25 km	2,386	2,984 km
625 m	4,992	3,120 km

Read down the last column. Every time the ruler is halved, the border gets longer, and by roughly the same fraction each time: 153 km more, then 115, 170, 236, 136, 195, 179, 137. Those extra kilometers are not shrinking. Each halving adds between 4.6% and 10.5%, about 7% on average.

Where the dividers start matters a little: from eight different starting points the 160 km ruler gives 1,766 to 1,884 km, the 10 km ruler 2,454 to 2,474 km. So the short ruler’s answer isn’t rougher; it is just longer.

Why the shorter ruler finds more

Look at the border close up. Every view in Figure 21.1 uses a ruler suited to its size, and every view shows the same thing: the ruler cuts across bends, and the bends it cuts across have smaller bends of their own.

Each step is a straight line, the shortest way between its ends, so any wiggle between them adds length that only a shorter ruler can see. On a smooth curve, like the rim of a plate, the wiggles shrink away as you look closer. On Nepal’s border they don’t: zoom in on a ridge and you find spurs and gullies; zoom in on a spur and you find rocks.

A circle settles down

Compare a shape whose length does settle. Take a circle about as big as Nepal, radius 300 km, and walk the same dividers round it.

Ruler	Length round the circle	Short of the true length by
160 km	1,862.9 km	22.07 km
80 km	1,879.4 km	5.52 km
40 km	1,883.6 km	1.40 km
20 km	1,884.6 km	0.35 km
10 km	1,884.9 km	0.09 km

The true length is $2\pi \times 300 \approx 1{,}885.0$ km, and the walks close in on it fast. The shortfall shrinks to about a quarter at each halving: 22.07, 5.52, 1.40, 0.35. (Each step falls short of its arc by about step³ ÷ (24 × radius²). Halve the step and each shortfall becomes an eighth as big, but there are twice as many steps: a quarter in all.)

So the circle’s extra lengths, 16.55 km, then 4.12 km, then 1.05 km, and so on, are, very nearly, a geometric series with common ratio $\frac14$, and you know from Grade 10 how to add one. All of them together, for ever:

$$16.55 + 4.12 + 1.05 + \dots = \frac{16.55}{1 - \frac14} \approx 22.07 \text{ km},$$

just the 160 km walk’s shortfall. A sum that goes on for ever can still come to a finite number, when each term is a fixed fraction of the one before and that fraction is less than 1.

Nepal’s extra lengths don’t behave like that. Each one is about 7% of the length so far, so the length is multiplied by about 1.07 at each halving: a geometric sequence with ratio more than 1. If that went on for ever, the sum would never settle; it would pass any number you name.

Others had noticed the puzzle before, among them the Polish mathematician Hugo Steinhaus in 1954, but the one who measured it, ruler after ruler, was the English scientist Lewis Fry Richardson (his story is below). Studying wars, he wanted the lengths of borders and found that the published figures disagreed: the border between Spain and Portugal was given as 987 km and as 1,214 km. Walking dividers along coasts and borders on maps, he found what you found: the shorter the ruler, the longer the border, with no sign of the lengths settling. This is called the Richardson effect, and the puzzle that a coastline or border has no single true length, the coastline paradox.

A curve whose measured length does settle as the ruler shrinks (a circle, the edge of anything smooth) is called rectifiable: its length is the number the walks close in on. Nepal’s border doesn’t seem to be one.

A straight line on the right graph

Richardson plotted his measurements on a special graph, with both scales logarithmic: each step along the bottom is the ruler ten times longer, and each step up is the length multiplied by the same amount. On such a graph, a growth by the same fraction at each halving becomes a straight line. Try it with Nepal’s nine measurements.

A straight line on this graph means the length follows a power law in the ruler. Write L for the length measured with a ruler $\varepsilon$ (the Greek letter epsilon, for a small length). Richardson’s rule is

$$L = F \times \varepsilon^{\,1 - D},$$

where F and D are fixed numbers for each border. With Grade 9’s indices you can see what it does. For Nepal, $D \approx 1.10$, so 1 − D = −0.10 and $L = F \times \varepsilon^{-0.10}$. Halve the ruler and $\varepsilon^{-0.10}$ is multiplied by $(\tfrac12)^{-0.10} = 2^{0.10} \approx 1.07$. With the ruler in kilometers, F is the length a 1 km ruler gives: about 3,050 km for Nepal (the walk gave 3,025).

For a smooth curve, D = 1: then $\varepsilon^{0} = 1$ and L = F whatever the ruler, which is the circle’s flat line. The bigger D is, the faster the length grows as the ruler shrinks, and the more crumpled the border.

Why call D a dimension?

Count the steps instead of the length. A straight line 100 km long takes 10 steps of a 10 km ruler and 20 steps of a 5 km ruler: halve the ruler, double the steps. Now cover a square with little squares instead: halve their side, and you need four times as many. A line is one-dimensional and a square two-dimensional, and that is the dimension showing: halve the size and the count goes up by $2^1$ for a line, $2^2$ for a square.

Since the length is the number of steps times the ruler, Richardson’s rule says the number of steps grows as $\varepsilon^{-D}$: halve the ruler, and the steps go up by $2^D$. For Nepal’s border that is $2^{1.10} \approx 2.14$: more than a line’s 2, far less than a square’s 4. In this sense the border is a little more than a line, a curve crumpled enough to start filling the space around it. That is why D is called the border’s fractal dimension: a dimension that needn’t be a whole number. The Koch snowflake’s edge in the Grades 6–8 topic Can a Shape Have an Edge That Never Ends? has $D \approx 1.26$: cut the ruler to a third and you need 4 times as many steps, and $3^{1.26} \approx 4$.

Richardson measured several borders and coasts this way. Here are his values of D, with ours for Nepal:

Border or coast	D
A circle (or any smooth curve)	1
The coast of South Africa	1.02
Nepal’s border (our measurement)	1.10
The coast of Australia	1.13
The land border between Spain and Portugal	1.14
The land border of Germany, about 1899	1.15
The west coast of Great Britain	1.25

Back to Rojina’s poster. Here are the numbers she found, side by side.

Who	India border	China border	All round
India’s border police (SSB)	1,751 km
The Kathmandu Post (2018, 2025)	1,880 km
The CIA World Factbook (2025)	1,770 km	1,389 km	3,159 km
Nepali lawmakers (2020, as reported)		1,414.88 km
Our dividers on the national map, 10 km ruler			2,474 km
The same map, its finest detail			3,440 km

Not one of the published figures says which ruler, which map, or which line was measured, and the borders aren’t even the same line: our outline, like the map Nepal published in 2020, includes Kalapani, Lipulekh and Limpiyadhura, which India administers and shows as its own. So the disagreements are no proof that anyone measured badly. They are what the coastline paradox predicts when people measure the same crumpled line in different ways. A length of a border is only half an answer; the other half is the ruler.

A good answer for the poster would be: “About 3,000 km, measured with a 1 km ruler on the national map. Measure with a shorter ruler and it comes out longer.”

Where the rule breaks

Does the border really have no length at all? Richardson’s straight line can’t go on for ever, for three reasons.

The map has a finest detail. Our outline is made of 66,417 straight pieces, half of them shorter than 35 m. A ruler much shorter than the pieces just walks along the straight lines, and the length stops growing at the map’s own 3,440 km. You saw this begin in the widget, with the rulers under 600 m: the dots bend away from the line towards the dashed level. If the 1.07 pattern went on, a ruler of about a meter would give over 6,000 km. It doesn’t, on this map.

The real border is fixed by people. On land, the Nepal–India border is marked by pillars, 8,554 of them by the Department of Survey’s count as of 2025, and the Nepal–China border of 1961 has pillars too. A border agreed as straight lines from pillar to pillar would have a definite length, the sum of its straight pieces. But where the border follows a river, it moves when the river moves: thousands of hectares of land have shifted from one side to the other, and Nepal’s and India’s boundary officials still have to settle where the line runs there. In the high mountains it follows watersheds, the crests of ridges, which are as crumpled as any coast.

Nature has a smallest size too. A stone, a grain of sand, an atom: below some size it makes no sense to speak of the border’s line at all. Mandelbrot’s 1967 paper was careful: a coast is “statistically self-similar”, each piece looking roughly like a smaller copy of the whole, and since “geography is unconcerned with minute details”, the measurer chooses a smallest size worth counting. So a fractal is a model, like a smooth curve is a model. For a border measured with rulers from a few hundred meters to a hundred kilometers, the fractal model is the better one, and the smooth model is simply wrong.

A staircase that never gets shorter

The coastline paradox has an opposite, a trap that catches people who have just learned it.

Draw a staircase across a square of side 1, as in Figure 21.3. However many steps you use, the treads add up to 1 across and the risers to 1 up, so every staircase is 2 long. With a million steps the staircase is so close to the diagonal that no eye could tell them apart, yet its length is still 2, not the diagonal’s $\sqrt 2 \approx 1.41$.

• Can a Shape Have an Edge That Never Ends? (Grades 6–8) The Koch snowflake: an edge that grows for ever round an area that doesn’t, and the Sierpiński triangle.
• Can You Add Halves Forever and Still Get One? (Grades 6–8) When an endless sum settles, as the circle’s did here.
• Why Is a Magnitude 8 Earthquake Not Just a Bit Bigger Than a 7? Log scales, the kind of graph that turned Richardson’s measurements into a straight line.
• Can a Pattern of Tiles Never Repeat? Shapes that fill a floor without ever repeating, another surprise of geometry.
• How Does a Computer Grow a Fern? (Computing, Going Further) A fractal drawn by a short program, copies of copies of a leaf. (Coming soon.)
• Grade 10 Math: Sum of a geometric series The sum this page used to decide whether a length settles.