# Why no two snowflakes are identical

As if to put us in the mood for winter, the Washington Post has a fascinating feature explaining why no two snowflakes are the same:

Newly formed snow crystals with only a handful of molecules would be nearly impossible to distinguish. But that’s not really the issue. We’re talking about real snowflakes, which have something on the order of a quintillion molecules. (That’s the number 1 with 18 zeros.)

Now, it’s not a law of nature that no two snowflakes could be truly identical. So, on a very technical level, it’s possible for two snowflakes to be identical. And it’s entirely possible that two snowflakes have been visibly indistinguishable. But probability dictates that this is incredibly unlikely. Libbrecht draws a helpful visual comparison.

“There are a limited number of ways to arrange a handful of bricks,” he says. “But if you have a lot of bricks, the number of combinations grows very quickly. With enough of them, you can make a driveway, a sidewalk or a house.”

Water molecules in a snowflake are like those bricks. As the number of building blocks increases, the number of possible combinations increases at an incredible rate.

Consider the math, which Libbrecht helps explain using a bookshelf analogy. He points out that, if you have only three books on your bookshelf, there are only six orders in which you can arrange them. (That’s 3 times 2 times 1.) If you have 15 books, there are 1.3 trillion possible arrangements. (Fifteen times 14 times 13, etc.) With 100 books, the number of combinations increases to a number that is far, far greater than the estimated number of atoms in the universe.

An ordinary snowflake has hundreds of branches ribs, and ridges, all arranged in minutely different geometries. To be sure, lots of snowflakes have fallen in the world, but not nearly enough to render two identical snowflakes a reasonable possibility.

If you’re skeptical, you’re more than welcome to undertake your own study. But you might want to block off a pretty big chunk of time. Libbrecht estimates that around a septillion — that’s a 1 with 24 zeros — snowflakes fall every year.

This also makes me want to re-arrange the books on my bookshelves.  I think I’ll put 15 of them on a shelf and make it my life’s work to put them in every possible order!

I am a retired English professor and college administrator. I have written over 20 books on different facets of Christianity & Culture.

• Kirk

The uniqueness of snowflakes is great, but I’ve always been fascinated by their symmetry. Like, why are all the branches exactly the same? What doesn’t one side have all the branches and the other side have none? I just don’t understand how such distinct and obvious patterns can form.

• Kirk

The uniqueness of snowflakes is great, but I’ve always been fascinated by their symmetry. Like, why are all the branches exactly the same? What doesn’t one side have all the branches and the other side have none? I just don’t understand how such distinct and obvious patterns can form.

• Tom Hering

Look at all the hexagons. Why does the number six – six sides, six branches – characterize these structures? How did a hexagon form at the north pole of Saturn?

• Tom Hering

Look at all the hexagons. Why does the number six – six sides, six branches – characterize these structures? How did a hexagon form at the north pole of Saturn?

• Dennis Peskey

Kirk (#1) The symmetry of snowflakes is a consequence of the bonding angle (105degrees) of the two hydrogen atoms to the oxygen atom when the water molecule is formed. As these molecules bond together from their vapor state to a solid state, the resulting structure always aligns in a hexagonal pattern giving the snowflake its’ distinction structural symmetry. In a singular state, the individual snowflake has an inherent beauty; when more than one is involved, the resultant condition is less than beautiful to old northerners. We assault the collective mass with preferred implements of distruction (aka, shovel, scraper or in my case, my utility tractor with front-bucket loader) to remove the offense as quickly and efficiently as possible.

On a related note for Dr. Veith, the process of rearranging your library shelves can be less than productive; indeed, the results can be quite frustrating when a particular literary work is desired and can not be located due to the random filing system proposed above.

Rather, I should think you would find greater satisfaction in a personal one-on-a-few million interaction with these frozen beauties called snowflakes. With this in mind, I invite you to attend my seasonal snowflakes transferal celebration coming soon to my neighborhood. I will provide you with the necessary implements for hoisting thousands of snowflakes into the atmosphere and depositing them into ever increasing piles along my far-too-long driveway. Oh, the sheer joy of launching snowflake after snowflake to the east and to the west as you progress from the county road to my garage. I, of course, would be cheering you onward inside the warm confines of my humble abode contemplating the latest and greatest posts on Cranach; I would even go so far as to prepare a warm cup of coffee or chocolate to celebrate your experience upon completion. I would not consider robbing you of the intimacy of the hands-on experience by utilizing my front-loader on the John Deere – far too mechanical and impersonal. E-Mail your acceptance soon for the flakes have already began their visitation.
Pax,
Dennis

• http://www.geneveith.com Gene Veith

Thanks, Dennis, for your kind offer. But I lived in Wisconsin for 20 years. While digging out my long, long driveway, there were so many snowflakes that two of them matched.

And Mikeb, thanks for your spreadsheet. You know, I believe the statistics about there being 1.3 trillion ways to arrange 15 books, but I can’t picture how that can be. Could someone explain that? What is the formula?

(Webmonk, if you are reading, I did check out the time travel formula. Though I didn’t plug in any numbers, I could see how the value of T could be negative. But is that just a trick of the math, equivalent to dividing by zero, something that you just can’t do?)

• Dennis Peskey

Kirk (#1) The symmetry of snowflakes is a consequence of the bonding angle (105degrees) of the two hydrogen atoms to the oxygen atom when the water molecule is formed. As these molecules bond together from their vapor state to a solid state, the resulting structure always aligns in a hexagonal pattern giving the snowflake its’ distinction structural symmetry. In a singular state, the individual snowflake has an inherent beauty; when more than one is involved, the resultant condition is less than beautiful to old northerners. We assault the collective mass with preferred implements of distruction (aka, shovel, scraper or in my case, my utility tractor with front-bucket loader) to remove the offense as quickly and efficiently as possible.

On a related note for Dr. Veith, the process of rearranging your library shelves can be less than productive; indeed, the results can be quite frustrating when a particular literary work is desired and can not be located due to the random filing system proposed above.

Rather, I should think you would find greater satisfaction in a personal one-on-a-few million interaction with these frozen beauties called snowflakes. With this in mind, I invite you to attend my seasonal snowflakes transferal celebration coming soon to my neighborhood. I will provide you with the necessary implements for hoisting thousands of snowflakes into the atmosphere and depositing them into ever increasing piles along my far-too-long driveway. Oh, the sheer joy of launching snowflake after snowflake to the east and to the west as you progress from the county road to my garage. I, of course, would be cheering you onward inside the warm confines of my humble abode contemplating the latest and greatest posts on Cranach; I would even go so far as to prepare a warm cup of coffee or chocolate to celebrate your experience upon completion. I would not consider robbing you of the intimacy of the hands-on experience by utilizing my front-loader on the John Deere – far too mechanical and impersonal. E-Mail your acceptance soon for the flakes have already began their visitation.
Pax,
Dennis

• http://www.geneveith.com Gene Veith

Thanks, Dennis, for your kind offer. But I lived in Wisconsin for 20 years. While digging out my long, long driveway, there were so many snowflakes that two of them matched.

And Mikeb, thanks for your spreadsheet. You know, I believe the statistics about there being 1.3 trillion ways to arrange 15 books, but I can’t picture how that can be. Could someone explain that? What is the formula?

(Webmonk, if you are reading, I did check out the time travel formula. Though I didn’t plug in any numbers, I could see how the value of T could be negative. But is that just a trick of the math, equivalent to dividing by zero, something that you just can’t do?)

• mikeb

Just for fun I opened a spreadsheet to calculate how many ways Dr. Veith could arrange his 15 books. I was shocked that the number was 1.3 Trillion. I tried to figure out how many ways you could arrange 100 books but my Comodore 64 can’t count that high. And neither can I!

• mikeb

Just for fun I opened a spreadsheet to calculate how many ways Dr. Veith could arrange his 15 books. I was shocked that the number was 1.3 Trillion. I tried to figure out how many ways you could arrange 100 books but my Comodore 64 can’t count that high. And neither can I!

• mikeb

Dr. Veith,

To re-hash an old topic, did you see the news today that “2nd test affirms faster-than-light particles”? The director of the lab says more tests are needed but this is still fascinating.

• mikeb

Dr. Veith,

To re-hash an old topic, did you see the news today that “2nd test affirms faster-than-light particles”? The director of the lab says more tests are needed but this is still fascinating.

• mikeb
• mikeb
• http://www.whenisayrunrun.blogspot.com Andrew

When I can’t sleep at night, which is often, instead of waking my wife with my tossing and turning I rearrange my books. It is quite calming. Unless I knock over the huge pile at my night stand.

• http://www.whenisayrunrun.blogspot.com Andrew

When I can’t sleep at night, which is often, instead of waking my wife with my tossing and turning I rearrange my books. It is quite calming. Unless I knock over the huge pile at my night stand.

• –helen

I’d be happy to arrange several more boxes of books once, if I could figure out where to put more bookshelves!

• –helen

I’d be happy to arrange several more boxes of books once, if I could figure out where to put more bookshelves!

• http://www.whenisayrunrun.blogspot.com Andrew

Helen, rotating boxes are part of the fun! Now that I have inherited my dad’s massive library, I will now have more to do on sleepless nights. No box looks alike. Someday I will have a my own office/library with built-in shelves and built-in ladders. I can dream can’t I?

• http://www.whenisayrunrun.blogspot.com Andrew

Helen, rotating boxes are part of the fun! Now that I have inherited my dad’s massive library, I will now have more to do on sleepless nights. No box looks alike. Someday I will have a my own office/library with built-in shelves and built-in ladders. I can dream can’t I?

• –helen

I’ve seen one of those offices, Andrew.
I’ve wondered whether anyone ever climbed those ladders after the books were put in place.
(Such offices are seldom acquired while you are young and athletic!)

• –helen

I’ve seen one of those offices, Andrew.
I’ve wondered whether anyone ever climbed those ladders after the books were put in place.
(Such offices are seldom acquired while you are young and athletic!)

• Booklover

These snowflake photos are stunning. I, too, have been mesmerized by the hexagons and branch patterns of six. I teach at a Montessori school and have been trying to decide how to make these miraculous pictures into a “work” for the children. Matching work? Artwork? Geometric work? It will be an educational and beautiful project either way. Thank you.

Enjoy all of your rearranging of books. I won’t be arranging mine–I lost most of them along with their beautiful oak shelves in a flood. Ah well. I did save many of my homeschool books, which were upstairs, to donate or lend to a lovely young homeschooling mom.

• Booklover

These snowflake photos are stunning. I, too, have been mesmerized by the hexagons and branch patterns of six. I teach at a Montessori school and have been trying to decide how to make these miraculous pictures into a “work” for the children. Matching work? Artwork? Geometric work? It will be an educational and beautiful project either way. Thank you.

Enjoy all of your rearranging of books. I won’t be arranging mine–I lost most of them along with their beautiful oak shelves in a flood. Ah well. I did save many of my homeschool books, which were upstairs, to donate or lend to a lovely young homeschooling mom.

• Dust

Seems like since the snowflakes can be modeled with some kind of geometry and simple mathematics, maybe something similar to fractals? There must be a software application that can create these on a computer and students can watch them as they take form? If so, would suppose students could enter a few constants or parameters and see how that influences the final product? That would be cool, eh?

Or would it be flakey

• Dust

Seems like since the snowflakes can be modeled with some kind of geometry and simple mathematics, maybe something similar to fractals? There must be a software application that can create these on a computer and students can watch them as they take form? If so, would suppose students could enter a few constants or parameters and see how that influences the final product? That would be cool, eh?

Or would it be flakey

Those interested in learning more about snowflake formation should follow the WaPo link to CalTech physics professor Kenneth Libbrecth’s site, which has lots of information, both basic and detailed. (In addition to the primer at that link, check out his FAQ on snowflakes.)

Why are all the branches exactly the same?

That is to say, why are they (hexagonally) symmetrical? Well, one unsatisfying answer is that they aren’t, necessarily. At the technical level, at least. (Actually, if you go to Libbrecth’s site, you’ll see that some snow crystals — his more technical term — aren’t hexagonal flakes at all, but instead form needles and other shapes.) They just mostly appear mostly hexagonally symmetrical.

The reason for that is that the type of crystal growth (sectored plates, dendrites, solid plates, etc.) is apparently dependent on two things, one of which is temperature (the other is supersaturation (?) — see this chart). As snowflakes fall, they make their way through spots of varying temperature — some warmer, some colder — which induce different crystal growths. But the snowflake does this as a whole, so each arm is almost identically affected. Imagine having sextuplets who never leave each others’ sides, who all have the same experiences, and thus grow up to be a lot like each other.

Why does the number six – six sides, six branches – characterize these structures?

The not-quite-satisfying answer here is that is simply characteristic of the bond angle formed by H2O crystals. Different substances form crystals with different angles (if they form crystals at all). Salt forms a cubic lattice of alternating sodium and chloride ions, so even macroscopic salt crystals have a cubic shape to them.

Dennis’ answer (@3) is a little bit off, as it only discusses the bond angle in a single water molecule (104.5 degrees). If you remember your geometry, the (internal) angles formed on the inside of a hexagon should be 120 degrees — and they are, I believe, in a snow crystal. I won’t pretend to explain any further (my wife’s the chemistry teacher; I just eat scraps off the table, as it were), but in forming the more stable snow crystal, the molecules simply (anthropomorphically) “prefer” to form hexagons that don’t “stretch” the molecule too much from its single-molecule state.

Those interested in learning more about snowflake formation should follow the WaPo link to CalTech physics professor Kenneth Libbrecth’s site, which has lots of information, both basic and detailed. (In addition to the primer at that link, check out his FAQ on snowflakes.)

Why are all the branches exactly the same?

That is to say, why are they (hexagonally) symmetrical? Well, one unsatisfying answer is that they aren’t, necessarily. At the technical level, at least. (Actually, if you go to Libbrecth’s site, you’ll see that some snow crystals — his more technical term — aren’t hexagonal flakes at all, but instead form needles and other shapes.) They just mostly appear mostly hexagonally symmetrical.

The reason for that is that the type of crystal growth (sectored plates, dendrites, solid plates, etc.) is apparently dependent on two things, one of which is temperature (the other is supersaturation (?) — see this chart). As snowflakes fall, they make their way through spots of varying temperature — some warmer, some colder — which induce different crystal growths. But the snowflake does this as a whole, so each arm is almost identically affected. Imagine having sextuplets who never leave each others’ sides, who all have the same experiences, and thus grow up to be a lot like each other.

Why does the number six – six sides, six branches – characterize these structures?

The not-quite-satisfying answer here is that is simply characteristic of the bond angle formed by H2O crystals. Different substances form crystals with different angles (if they form crystals at all). Salt forms a cubic lattice of alternating sodium and chloride ions, so even macroscopic salt crystals have a cubic shape to them.

Dennis’ answer (@3) is a little bit off, as it only discusses the bond angle in a single water molecule (104.5 degrees). If you remember your geometry, the (internal) angles formed on the inside of a hexagon should be 120 degrees — and they are, I believe, in a snow crystal. I won’t pretend to explain any further (my wife’s the chemistry teacher; I just eat scraps off the table, as it were), but in forming the more stable snow crystal, the molecules simply (anthropomorphically) “prefer” to form hexagons that don’t “stretch” the molecule too much from its single-molecule state.

• helen

Booklover:
If you want to do a ‘paper snowflake’, there are on line instructions under that Google entry.
(“Aint yo mama’s snowflake” look like mama’s paper doilies, IMO.)
But there are many others, even a 3-D one.

When we came to Texas (real snowflakes once in a dozen years) my daughter would make dozens of paper ones and decorate the front windows with them… about now. [She has since moved back to snow country; somehow I doubt that she does that any more.]

Losing most of your books in a flood is sad! It happened to my best friend, who had a large collection.
Having to halve the collection for a move to smaller quarters is not so much fun either. I done that a couple of times, the last time so quickly that I don’t really know what’s in those storage boxes and what’s gone.

• helen

Booklover:
If you want to do a ‘paper snowflake’, there are on line instructions under that Google entry.
(“Aint yo mama’s snowflake” look like mama’s paper doilies, IMO.)
But there are many others, even a 3-D one.

When we came to Texas (real snowflakes once in a dozen years) my daughter would make dozens of paper ones and decorate the front windows with them… about now. [She has since moved back to snow country; somehow I doubt that she does that any more.]

Losing most of your books in a flood is sad! It happened to my best friend, who had a large collection.
Having to halve the collection for a move to smaller quarters is not so much fun either. I done that a couple of times, the last time so quickly that I don’t really know what’s in those storage boxes and what’s gone.

• Dennis Peskey

Dr. Veith – Your negative RSVP truly disappoints but given twenty years existence in our neighboring State of Wisconsin (Go Pack Go – 16&0!) I can understand. What I can not fathom is how you ascertained a matching pair of the little monsters while enjoying your shoveling experience (did you individually bid each one bon voyage?)

tODD – While I would revel in a thermodynamic examination of the molecular bonding of water molecules, I fear this would become quite boring to most readers at this site (perhaps I err; perhaps not). Simply ask your wife what would be the consequences of a water molecule having the “perfect” symmetry of 120 degrees – I dare say the solid state would be rather difficult to disassemble at the molecular level (goggle nuclear bonding for further study). PS – since Dr. Veith seems to be unable (or unwilling) to attend my annual snowflake tranferal celebration, I graciously extend the invitation to you (and all readers herein) as well.

As for the mathematical information supplied by Mikeb – when we wrap our brains around the complexity of arranging fifteen books, graduate to the DNA molecule’s alignment and complexity. For a very excellent treatment of the DNA design, I would suggest Steven Meyer’s Signature in the Cell. If your mathematical background can comprehend the data, go directly to chapter’s eight and nine. Unless you are able to discredit the computations, the conclusion renders the evolution’s claim of chance to be impossible. For now, keep playing with the rearrangement of the library and in a few million years, the answer will slowly come to you.
Pax,
Dennis

• Dennis Peskey

Dr. Veith – Your negative RSVP truly disappoints but given twenty years existence in our neighboring State of Wisconsin (Go Pack Go – 16&0!) I can understand. What I can not fathom is how you ascertained a matching pair of the little monsters while enjoying your shoveling experience (did you individually bid each one bon voyage?)

tODD – While I would revel in a thermodynamic examination of the molecular bonding of water molecules, I fear this would become quite boring to most readers at this site (perhaps I err; perhaps not). Simply ask your wife what would be the consequences of a water molecule having the “perfect” symmetry of 120 degrees – I dare say the solid state would be rather difficult to disassemble at the molecular level (goggle nuclear bonding for further study). PS – since Dr. Veith seems to be unable (or unwilling) to attend my annual snowflake tranferal celebration, I graciously extend the invitation to you (and all readers herein) as well.

As for the mathematical information supplied by Mikeb – when we wrap our brains around the complexity of arranging fifteen books, graduate to the DNA molecule’s alignment and complexity. For a very excellent treatment of the DNA design, I would suggest Steven Meyer’s Signature in the Cell. If your mathematical background can comprehend the data, go directly to chapter’s eight and nine. Unless you are able to discredit the computations, the conclusion renders the evolution’s claim of chance to be impossible. For now, keep playing with the rearrangement of the library and in a few million years, the answer will slowly come to you.
Pax,
Dennis

• BS from Texas

He points out that, if you have only three books on your bookshelf, there are only six orders in which you can arrange them. (That’s 3 times 2 times 1).

Andrew,

Obiously, what the article says would be the case if you limit yourself to just placing the books on the shelf in the standing position only. You can stack the books, you can combine laying one book down on the shelf against two standing, or combine laying two books down against one book standing. Many more possibilities to help you through those sleepless nights!

Helen – I’m originally from the part of Texas where it snows every year, usually 20 or more inches. In fact, our last year before we moved to another part of the state we had a total of 48 inches!

• BS from Texas

He points out that, if you have only three books on your bookshelf, there are only six orders in which you can arrange them. (That’s 3 times 2 times 1).

Andrew,

Obiously, what the article says would be the case if you limit yourself to just placing the books on the shelf in the standing position only. You can stack the books, you can combine laying one book down on the shelf against two standing, or combine laying two books down against one book standing. Many more possibilities to help you through those sleepless nights!

Helen – I’m originally from the part of Texas where it snows every year, usually 20 or more inches. In fact, our last year before we moved to another part of the state we had a total of 48 inches!

I believe the statistics about there being 1.3 trillion ways to arrange 15 books, but I can’t picture how that can be. Could someone explain that? What is the formula?

Well, as BS notes (@17), what we’re really talking about is ways to order a unique set (i.e. no duplicates) of 15 books in a linear fashion. That is, we’re not considering anything other than their order from left to right (or vice versa). If we were to consider other arrangement possibilities, there would be way, way, way more possible combinations.

Anyhow, how did they arrive at 1.3 trillion ways to order 15 books? Well, there are 15 options available for what the first book is. Once one is chosen, there are now 14 options available for the second book. Now that we’ve chosen the first two books, there are 13 options available for what the third book might be. And so on, until there is only one possible book left for the last position.

The formula, then, for the number of possible orderings, is just 15! — that’s “fifteen factorial”, not a punctuation mark — which is math shorthand for 15*14*13*12*…3*2*1. (I’m using the asterisk as a multiplication sign).

And 15! = 1,307,674,368,000.

• http://www.geneveith.com Gene Veith

Thanks, Todd. That was just what I needed. Now it makes sense, though my mind is still reeling.

I believe the statistics about there being 1.3 trillion ways to arrange 15 books, but I can’t picture how that can be. Could someone explain that? What is the formula?

Well, as BS notes (@17), what we’re really talking about is ways to order a unique set (i.e. no duplicates) of 15 books in a linear fashion. That is, we’re not considering anything other than their order from left to right (or vice versa). If we were to consider other arrangement possibilities, there would be way, way, way more possible combinations.

Anyhow, how did they arrive at 1.3 trillion ways to order 15 books? Well, there are 15 options available for what the first book is. Once one is chosen, there are now 14 options available for the second book. Now that we’ve chosen the first two books, there are 13 options available for what the third book might be. And so on, until there is only one possible book left for the last position.

The formula, then, for the number of possible orderings, is just 15! — that’s “fifteen factorial”, not a punctuation mark — which is math shorthand for 15*14*13*12*…3*2*1. (I’m using the asterisk as a multiplication sign).

And 15! = 1,307,674,368,000.

• http://www.geneveith.com Gene Veith

Thanks, Todd. That was just what I needed. Now it makes sense, though my mind is still reeling.

• helen

BS from Texas @17
You’re right, I should have specified the Texas coast. Here in Austin we are not unlikely to be grounded by ice, usually about MLK day. The daughter spent one winter in San Antonio, the one when SA got 15 inches of snow overnite. Her complaint: she had no boots or mittens to enjoy it!

• helen

BS from Texas @17
You’re right, I should have specified the Texas coast. Here in Austin we are not unlikely to be grounded by ice, usually about MLK day. The daughter spent one winter in San Antonio, the one when SA got 15 inches of snow overnite. Her complaint: she had no boots or mittens to enjoy it!

• Susan

Infinite variety, fixed order, benevolence and loveliness on a sublime level. Even the smallest things shout His glory, love and care for His creation!

• Susan

Infinite variety, fixed order, benevolence and loveliness on a sublime level. Even the smallest things shout His glory, love and care for His creation!

• helen

Thanks to Susan! The first snowfall evokes all that.
In January, the pre dawn buildup of frost on every twig is at least as astonishing, till the lightest of sunrise breezes brings the whole thing crashing down.

• helen

Thanks to Susan! The first snowfall evokes all that.
In January, the pre dawn buildup of frost on every twig is at least as astonishing, till the lightest of sunrise breezes brings the whole thing crashing down.

• Dust

Susan at 20 and helen at 22…..hope you enjoy:

http://www.poetryfoundation.org/poem/174747

cheers!

• Dust

Susan at 20 and helen at 22…..hope you enjoy:

http://www.poetryfoundation.org/poem/174747

cheers!

• George A. Marquart

From a scientific point of view, it is not correct to say, “no two snowflakes are the same.” There may be astronomical odds against two snowflakes being identical, but that simply describes likelihood. The odds in the posting describe the extreme condition, but they should be adjusted for some of the factors that narrow the odds: the law that mandates a hexagonal structure, and whatever other laws that require all molecules to act in a certain way, making identical snow flakes more likely. We constantly hear about things that happen against overwhelming odds: lottery winners, achievements in sports, and events in the cosmos.

There is no law in physics or chemistry that prohibits two snow flakes from being identical.

We can only insist on the uniqueness of every snowflake if we can examine every snowflake, also taking into account all snow flakes that ever fell since the first snow flake (notice how I cleverly avoided a time span which might set creationists or evolutionists against me). The fact is that, in spite of the huge odds against it, the second snow flake could have been identical to the first. The law underlying this statement is used by every engineer who has ever designed a mechanical part for a certain life span.

Keep crunching those numbers.
George A. Marquart

• George A. Marquart

From a scientific point of view, it is not correct to say, “no two snowflakes are the same.” There may be astronomical odds against two snowflakes being identical, but that simply describes likelihood. The odds in the posting describe the extreme condition, but they should be adjusted for some of the factors that narrow the odds: the law that mandates a hexagonal structure, and whatever other laws that require all molecules to act in a certain way, making identical snow flakes more likely. We constantly hear about things that happen against overwhelming odds: lottery winners, achievements in sports, and events in the cosmos.

There is no law in physics or chemistry that prohibits two snow flakes from being identical.

We can only insist on the uniqueness of every snowflake if we can examine every snowflake, also taking into account all snow flakes that ever fell since the first snow flake (notice how I cleverly avoided a time span which might set creationists or evolutionists against me). The fact is that, in spite of the huge odds against it, the second snow flake could have been identical to the first. The law underlying this statement is used by every engineer who has ever designed a mechanical part for a certain life span.

Keep crunching those numbers.
George A. Marquart