Field Guide
Vol. I
SEP 2026
No. 97
Short Science Facts · For Curious Kids, Parents & Teachers
Field Guide Entry 057

the curve that is all corners

In 1872, the German mathematician Karl Weierstrass presented a function to the Prussian Academy of Sciences that changed how mathematicians thought about curves. It could be drawn in one unbroken stroke, so it was continuous everywhere, but no matter where you looked, you could not draw a tangent line to it. There was no point on the curve where the slope settled down. The curve was made by adding infinitely many cosine waves, each one smaller and faster than the last, so it kept wavering at every scale. For much of the 1800s, many mathematicians expected a curve to be smooth at almost every point. Weierstrass showed that this was not always true, and later mathematicians saw his example as one of the first fractal-like objects. It forced people to rethink what a curve, a slope, and smoothness really mean, and it still raises a bigger question about how mathematics describes shapes that never calm down.

Watch the short · 60 sec
02What's Happening

The Mechanism

For most of the 1800s mathematicians assumed a curve you could draw in one unbroken stroke had to be "smooth" almost everywhere — at nearly every point you could draw a tangent line and read off a slope. In 1872 Karl Weierstrass presented the Prussian Academy of Sciences a function that shattered that assumption: a curve that is *continuous everywhere yet differentiable nowhere*. It is built by stacking infinitely many cosine waves, each higher-pitched and shorter than the last, so the curve wiggles at every scale at once. Zoom into any piece of it and you find the same restless jaggedness as the whole — it never settles into a straight tangent anywhere. It was among the first objects we would now call a fractal, decades before the word existed. Mathematicians recoiled: Charles Hermite wrote in 1893 that he turned away "with fright and horror from this lamentable plague of functions with no derivatives," and Poincaré called such things a gallery of monsters. But the monsters were real, and they forced mathematics to rebuild its definitions of continuity, smoothness, and dimension from the ground up.

03Why It Matters

Why It Matters

This curve is remarkable because it breaks a rule that seemed obvious: if a shape is continuous, people expected it to have a tangent line almost everywhere. Weierstrass built an example that is continuous at every point but differentiable at none, so it has no place where a single straight line gives its slope. Even more surprising, it is made from ordinary cosine waves, but stacked in an infinite way that creates roughness at every scale. It helped push mathematics toward modern ideas of fractals and showed that intuition about smooth shapes can fail.

04Common Misconception

Wait — That's Not Quite Right

A common mistake is to think that if a curve can be drawn without lifting your pencil, it must be smooth somewhere. Continuity only means there are no breaks or jumps. It does not guarantee a tangent line or a slope at any point. Weierstrass's function shows that a curve can be perfectly unbroken and still be jagged forever.

05Words to Know

Vocabulary

  • continuity
  • differentiable
  • tangent line
  • slope
  • function
  • cosine wave
  • infinite series
  • fractal
  • Weierstrass
  • smoothness
  • calculus
06Comprehension Check

Quick Quiz

5 questions · For classroom or kitchen table

1
What important property does Weierstrass's curve have at every point?
2
What cannot be drawn flat against the curve at any point?
3
In what year did Weierstrass present this function to the Prussian Academy of Sciences?
4
How was the curve built?
5
Why did mathematicians later care about this curve beyond the puzzle itself?
07Try This at Home

The Experiment

Build a Wiggle Staircase

Take a strip of paper and draw a smooth wave across it with a pencil. Then, on top of that, add a second wave that is smaller and faster. Keep adding even smaller waves if you have room. You will not make Weierstrass's exact function, but you will see the same idea: a shape can keep changing at smaller and smaller scales instead of settling down.

Now fold the paper back and forth a little and look at it from far away and then close up. Ask what changes when the waves get smaller. The real Weierstrass curve uses infinitely many waves, which is why it never becomes smooth enough for a tangent line anywhere. Your drawing shows how adding many layers of wiggles can make a curve look more and more restless.

paper, pencil or pen, ruler optional, adult supervision not required

08Sources

Where this came from

  1. Weierstrass function, Wikipedia (https://en.wikipedia.org/wiki/Weierstrass_function); "Weierstrass's Monster," Tom Rocks Maths / University of Oxford (https://tomrocksmaths.com/wp-content/uploads/2023/07/weierstrasss-monster.pdf).
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