Estimating Whether No Two Snowflakes are Alike

A recent “Ask the Expert” question was, “How do we actually KNOW that no two snowflakes are alike”?

One Captured Snowflake : Image by hlkljgk (Heather Katsoulis)

One Captured Snowflake : Image by hlkljgk (Heather Katsoulis)

Is every snowflake unique? Unfortunately, the best answers disagree:

  • Yes, by extrapolation from a limited sample.
  • We don’t know but it is simply probable.
  • Depending on exactly what we mean by ‘unique,’ some have been found that are alike.

After reviewing the standard answers, let’s build a math framework to estimate an answer for this “unique snowflake” question.

Review the Best Answers about Unique Snowflakes

By reviewing the best typical answers, we will also learn some background about snowflakes.

Extrapolating Uniqueness from a Sample of Snowflakes

Chris V. Thangham reported that Wilson Alwyn Bentley was the first person to state he had not found any two snowflakes alike. Since then, almost everyone who looked for a counter-example failed.

Probably No Two Snowflakes are Alike

The Koch Snowflack is a Fractal : Image by António Miguel de Campos

The Koch Snowflack is a Fractal : Image by António Miguel de Campos

As the pictures of lacey dendrite snowflakes show, their patterns can be incredibly complex. Whether or not any physical snowflake is a true fractal, the “Koch snowflake” in the GIF image shows how repeated branching leads to a very complicated form.

Since snowflakes can branch differently down to individual water molecules, the number of possibilities is extremely large. Without a restrictive theory that constrains snowflakes to a limited number of shapes, it seems probable that no two snowflakes are alike.

Some Snowflakes Are Alike, Depending on our Definitions

Some snowflakes look more like six-sided plates than lace doilies; other snowflakes are hexagonal prisms. The smallest and simplest plates or prisms may easily have identical twins, since their criteria are nearly limited to the diameter and thickness measured to a reasonable precision.

Thangham also reported that Nancy Knight did indeed find “identical snowflakes of the hollow column type.” Thangham thus supplies the counter-example that disproves his thesis that “no two snowflakes are alike” if we include the simplest shapes.

Approaching Identical Snowflakes with Birthday Mathematics

Snowflakes on Glossy Black : Image by Flare

Snowflakes on Glossy Black : Image by Flare

You may remember the “birthday party” puzzle. “If you want a 50% chance that two people attending a party share the same birthday, how many guests do you randomly invite?

Each person has a 1/365 chance of sharing a birthday with a random stranger. (Readers are encouraged to do the math including leap years).

Eric W. Weisstein provides an estimate that at least two members in a set of size ‘n’ share the same value if there are ‘d’ possible values.

P = 1 – ( 1 – ( n/(2*d) ) )^(n-1), approximately, where…

  • ‘P’ is the probability;
  • ‘n’ is the number of members in the set, or “guests at the party”;
  • ‘d’ is the number of possible values, with d=365 for the birthday problem;
  • ‘^’ is the symbol for exponentiation.

It turns out that “n=23″ gives a probability of about 50% that any two guests at a party share a birthday.

If we knew how many snowflakes are being compared (‘n’), and how many “values” a snowflake might have (‘d’), this formula would estimate the probability that at least two snowflakes were alike.

Click to Read Page Two: What is the Value of a Snowflake?

© Copyright 2012 Mike DeHaan, All rights Reserved. Written For: Decoded Science
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Comments

  1. Mikael says

    First. With the law of large numbers, a perfectly identical snowflake could turn up, even if it is improbable.

    Second. The problem with the everyday statement of saying that “Every snowflake is unique” that every elementary school teacher learns to parrot is that it is a butchered simplified conclusion. And it is so etched into society that when a young child challenges a teacher the credibility of such a statement without proof, the math work, probability, unclear definitions, and the enormous magnitude of time and space such a clearcut statement claims to be valid for, the child will be laughed at and ridiculed for not accepting a “well known fact” without any form of proof.
    Because, you have to take in the moment the Earth was cool enough that snow could fall all the way to the end of the Earth when the Sun dies and swallows the Earth, some estimated 5 billion years from now. And what about snow on exoplanets, do we take that in consideration? And what moment do we measure the snowflake in? Since in can melt, freeze, crack, clump, and break, a snowflake can have many forms from the moment it is formed to the moment it completely melts into water.
    I sated back then as a child that “two identical snowflakes have, are, or will exist, but it is highly unlikely to find one”, and I still believe that statement holds water. You can not have a world where we say that a group of monkey with typewriters and infinite time will recreate the complete works of Shakespeare, and at the same time say that no snowflakes look the same in an absolute term.

  2. hyeonjung says

    Thank you so much! It’s amazing to know there is a great discovering like this behind snowflakes.

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