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The mathematics of snowflakes

Ice crystals

Introduction: (Initial Observation)

Every snowflake has an infinite beauty which is enhanced by knowledge that you never find another exactly like it. Nature combines her greatest skill and artistry in the production of snowflakes and generously fashions the most beautiful specimens on a very thin plane. Why do snow crystals form in such intricate, symmetrical shapes? Where is the creative genius that produces these miniature masterpieces of frozen water, quite literally out of thin air?

What are different shapes of snowflakes? How many crystals form each snowflake? Is there any relation between the shapes (sides and angles) of different snowflakes? What is the size of each snowflake? What is the density of snow? Is there any relation between the shape of snowflakes and weather temperature? These are some of the questions that can be studied in relation to the mathematics of snowflakes. Scientists are interested in such information and can use them in solving many scientific problems. For example they can use the shape of a snowflake and determine the temperature in which the snowflake is formed.

Mathematicians however look at snowflakes as fractals and are interested in their formation algorithm and their calculations. They also use computer to draw fractals including snowflake fractal. See mathematics of snowflake fractals click here to go to its own dedicated project guide.


Please note that this project is available in two versions. One is in this page for physical specifications of snowflakes and the other is the mathematics of snowflake fractals accessible via this link. You can view both and choose one for your project.

If you have any questions, click on the help button at the top of this page to send me your questions. I may respond by email, but often I update this page with the information that you need.

Project Advisor

How do I study snow flakes?

It is best if you catch fresh snowflakes and make observations under a magnifier or microscope; however, some studies can be done using existing photographs of authentic snowflakes or you may make your own snowflakes.

Information Gathering:

Find out about snowflakes and their geometry. Read books, magazines or ask professionals who might know in order to learn about the shapes and sizes of snowflake crystals. Keep track of where you got your information from.

Project advisor note: The form of crystals are among the very important physical properties of every substance. Each crystal is a uniformly arranged atomic structure. Shape of crystals can often help us to determine or guess the shape of molecules forming each crystal. Scientists study the crystals by observing their 3D shape and angles between planes and lines. (lines are edges of planes or where two planes cross each other). Other properties are color, refraction index, melting point and density. Some chemicals have more than one form of crystals. Variation in the form of crystals are caused by the conditions in which the crystal is formed. Temperature and pH are among the factors that may affect the form of crystals.

Snowflakes are single crystals or compound/structured crystals of water. By studying snowflakes, you are actually studying water crystals formed in the sky from water vapors.

If you are gathering information from books, note that there are many books about snowflakes. You may also look at books about atmosphere, weather and meteorology. If you prefer searching the Internet, search for compound keywords such as “snowflakes + crystals” or “water crystal + angle” or “snowflakes + size”.

Sample Information:
Following are samples of information that you may find.

The most basic form of an ice crystal is a hexagonal prism. This form occurs because certain surfaces of the crystal, the growth facets, grow very slowly . The reason these facets exist derives from the molecular structure of water, and how water molecules arrange themselves into a crystalline lattice.

* * *

Frozen water, in all its forms ranging from snowflakes to icebergs, has intrigued scientists for many centuries. Already Kepler had speculated on the hexagonal shape of snowflakes, wondering what were the forces that transformed round droplets of water into the beautiful stars that he saw in falling snow. Although he did not resort to an atomic picture, he used a concept with formation by the cold of very small, identical particles, which would grow from an octahedral origin. There would thus be three orthogonal growth directions, leading to the formation of six branches in the star, and a hexagon could then form by a flattering of the star along one of the three-fold axes. Kepler discussed the form of different natural objects, ranging from honeycombs over pomegranates to different crystal forms, and he ended by proposing that different fluids would have in-built abilities that would lead to different forms when frozen.

Since then scientists have been equipped with a multitude of tools that allow them to go well beyond visual observation and speculation in the study of nature, but water in all its forms has always been a subject of interest. Methods to reveal the internal atomic arrangement of crystals were established in the first quarter of this century. Ice was among the first compounds to be looked at with X-rays, and when neutron scattering became viable a powder spectrum of ice was among the first to be recorded.
Over the years methods of measurement and interpretation have developed, and at regular intervals ice has been studied, leading to precise and detailed picture of the atomic arrangement in ice Ih. Source…

* * *

Different Looking Snowflakes
No two snowflakes look the same. There are several reasons for this. The temperature changes the size and shape of a snowflake: 30 degrees Fahrenheit (-1 degrees C) – snowflakes usually look like columns. 3 to 10 degrees Fahrenheit (-16 to -12 degrees C) – snowflakes usually shaped like a star. 10 to 18 degrees Fahrenheit (-12 to -7 degrees C)- snowflakes usually shaped like a plate.

  • 18 to 23 degrees Fahrenheit (-7 to -5 degrees C) – columns usually appear again.
  • 23 to 27 degrees Fahrenheit (-5 to -2 degrees C) – snowflakes usually form like a needle.

Dirt also changes the appearance of a snowflake. Falling snow crystals clump together and form a snowflake. If a snow crystal spins like a top as it falls, it will usually be symmetrical when it hits the ground. If a snow crystal falls sideways, then it will end up lopsided. Source…

Snowflake Man

Question/ Purpose:

What do you want to find out? Write a statement that describes what you want to do. Use your observations and questions to write the statement.

The purpose of this project is to identify the general structure of snowflakes. The result of this study can be used to design a realistic paper model of a snowflake or to identify a real snowflake from a false snowflake design.

Studying the shape of snowflakes is not an experimental project; so, you cannot have a hypothesis, results table, and graph that are generally associated with experimental projects.

If you need to study snowflakes as an experimental project, you may study the effect of air temperature on the shape of snow flakes or the size of snowflakes. The problem is that you will have to repeat your observations and measurements in many different snowy days and that may exceed the time you have to complete your project. The other option is to study snow instead of snowflakes. For example you may check the density of snow in different temperatures.

Identify Variables:

When you think you know what variables may be involved, think about ways to change one at a time. If you change more than one at a time, you will not know what variable is causing your observation. Sometimes variables are linked and work together to cause something. At first, try to choose variables that you think act independently of each other.

Independent variable is the snowflake, dependent variables are lines, angles or the shape of individual crystals that form each snowflake. Controlled variables are sampling and observation procedures.


Based on your gathered information, make an educated guess about what types of things affect the system you are working with. Identifying variables is necessary before you can make a hypothesis. Following are two sample hypothesis:

Building blocks of ice crystals and snowflakes are rhombic prism crystals of frozen water.

These crystals have two 60º angles and adjacent sides have different polarities that ultimately affect the order in which they bound to each other. My hypothesis is based on my gathered information and observation of snowflake photos that clearly contain 60º angles and also contain needle like crystals.

To test my hypothesis I will get sample images of different snowflakes and see if I can reconstruct an identical snow flake using parallelograms.

Snowflakes are ice crystals in different forms that bind to each other in a random pattern while falling to the ground. My hypothesis is based on my observation of snow while plowing it.

What do you think? Suggest your hypothesis before further investigation. Do you think that ice crystals bind to each other in random pattern? Do you think that small crystals forming the snowflakes are rectangles or triangles or hexagons?

Experiment Design:

Design an experiment to test each hypothesis. Make a step-by-step list of what you will do to answer each question. This list is called an experimental procedure. For an experiment to give answers you can trust, it must have a “control.” A control is an additional experimental trial or run. It is a separate experiment, done exactly like the others. The only difference is that no experimental variables are changed. A control is a neutral “reference point” for comparison that allows you to see what changing a variable does by comparing it to not changing anything. Dependable controls are sometimes very hard to develop. They can be the hardest part of a project. Without a control you cannot be sure that changing the variable causes your observations. A series of experiments that includes a control is called a “controlled experiment.”

Experiment 1: Catch and observe snowflakes

In December , the season of winter begins on the Winter Solstice. Why not take a close look at snowflakes, these beautiful form of water? Here is what to do.

Procedure: Get a cardboard box to store your equipment in and place everything in a sheltered spot at outdoor temperatures. An unheated garden shed or garage works well. Having all your equipment cold will keep the snowflakes from melting too fast while you look at them. In the box, put some black construction paper, a soft paintbrush, some toothpicks, and a magnifying glass. If you would like to try to preserve snowflakes as described in the next experiment, add a can of hair spray or artist’s fixative and some glass microscope slides. If you want to try to preserve snowflakes, you will want to get an adult to help.

When it starts to snow, take your box outside and catch snowflakes on the black paper. If you need to, you can move them around to look at them with the paintbrush or toothpicks. Look at them with the magnifying glass. A magnifying glass works best if you hold it close to your eye and move the paper with the snowflake up close to get it in focus. Try not to breath on the snowflake or it might melt. How many sides does a snowflake have? Do all snowflakes seem to have this same number of sides? Does the size and shape of snowflakes change with the weather? How can you find out?

If you are studying snowflakes in other seasons or where you have no access to fresh snow, try using existing snowflake pictures available at Wilson A. Bentley.

Experiment 2: Preserve snowflakes

In this experiment you will capture some snowflakes and make snowflake impressions that you can keep.

Through this activity you can not only look at the beauty of individual snowflakes but keep one of you own! A fun winter activity for kids of all ages.


  • a large piece of glass – An 8 x 10 picture frame works well.
  • can of hairspray
  • magnifying glass
  • snow


You need to watch the weather for when it is going to snow because both the glass and the hair spray need to be chilled. Placing your equipment outside in advance is a good practice; however you may choose to put them in your freezer for about an hour before you are going to catch your snowflakes.

When the snow is falling, take your glass and hairspray outside. Hold the can about 10 inches from the glass and spray with a quick sweeping motion. Spray from side to side once, then up and down. Use very little spray. Only spray for a second or two.

Collect some snowflakes on the sprayed glass.

Take the glass indoors and let dry for 15 to 20 minutes. Check out your results with a magnifying glass. You should be able to see ice crystals.

Using Microscope:

If you have access to a microscope, you may choose to use microscope slides instead of a larger glass. Procedures are almost the same:

Spray a microscope slide with hairspray or artist’s fixative. Catch a falling snowflake on the sticky surface of the slide. Set the slide somewhere where it will stay cold but where no more snowflakes will fall on it, maybe in your supply box with the lid closed. Leave the slide for a few hours so the hairspray or fixative dries and the water in the snowflake disappears. If you can, look at the finished slide under a microscope. If your microscope is equipped with a camera, you can also print photos of your snowflakes for further observations.

Experiment 3: Study the angles

In this experiment you will determine in what angles to the ice crystals bind to each other to form snowflakes?

This experiment can be performed the best if you use photos of snowflakes. You may take your own photos or you may use variety of real snowflake images found on the Internet.

In either case first print an enlarged image and use it for your observation and measurements.

First identify the center lines and border lines of crystal growth patterns. Then highlight them with color pencil or markers.

When all lines are clearly visible, use a protractor to determine the angles.

Repeat this in multiple snowflakes and compare the results.

Experiment 4: (Testing your hypothesis) Can you make models of snowflakes using the rules suggested in your hypothesis?

Procedure: Cut about 500 pieces of identical small rhombus shapes from construction paper. Make sure that two 60º angles are in two opposite corners. (You may substitute rhombus with any other shape that you think it is the building block crystals of snowflakes as specified in your hypothesis.)

Mark the sides of all pieces for their polarity using some specific colors or letters. For example you may use letters P and N for positive and Negative. Try to construct snowflakes (similar to the photographed snowflakes available at Wilson A. Bentley website) by placing the pieces next to each other. Start by constructing the center part of each snowflake and then expend it to its branches.

Note that when you place the pieces next to each other, one N side should be next to a P side. Note that in real images, some crystals may be incomplete or partially melted. Focus on general direction of growth and all straight lines.

How many different ways you can construct the center part of each new snowflake?

What is polarity? Polarity is a condition in which one side of a molecule is more electronegative than the other side. (In other words the concentration of electrons in one side is more than the other side.). Polarity is a factor that defines how molecules connect to each other. Water is a polar substance. Water’s polarity is responsible for the “stickiness” or cohesion between the molecules.

Cohesion of water causes capillary attraction, the ability of water to move uphill in small spaces. Water will move up the fibers of a plant because of cohesion. This force helps plants get the water they need to survive. In addition, it moves water upwards in soil. Cohesion of water also causes surface tension, water’s invisible skin which allows water striders to walk on water.

Materials and Equipment:

List of material depends on the experiments that you choose to perform. Final list of material can be extracted from the experiments that you perform. Following are some of the materials that you may use:

    1. pictures of snowflakes – there are printable images at the Wilson A. Bentley website.
    2. Ruler
    3. Protractor
    4. Magnifier glass
    5. Black construction paper
    6. Hair spray
    7. Microscope slide
    8. Clear glass about 6″ x 6″

Results of Experiment (Observation):

Experiments are often done in series. A series of experiments can be done by changing one variable a different amount each time. A series of experiments is made up of separate experimental “runs.” During each run you make a measurement of how much the variable affected the system under study. For each run, a different amount of change in the variable is used. This produces a different amount of response in the system. You measure this response, or record data, in a table for this purpose. This is considered “raw data” since it has not been processed or interpreted yet. When raw data gets processed mathematically, for example, it becomes results.


No specific calculation is required for this version of project.

Summary of Results:

Summarize what happened. This can be in the form of a table of processed numerical data, or graphs. It could also be a written statement of what occurred during experiments.

It is from calculations using recorded data that tables and graphs are made. Studying tables and graphs, we can see trends that tell us how different variables cause our observations. Based on these trends, we can draw conclusions about the system under study. These conclusions help us confirm or deny our original hypothesis. Often, mathematical equations can be made from graphs. These equations allow us to predict how a change will affect the system without the need to do additional experiments. Advanced levels of experimental science rely heavily on graphical and mathematical analysis of data. At this level, science becomes even more interesting and powerful.


Using the trends in your experimental data and your experimental observations, try to answer your original questions. Is your hypothesis correct? (Does the results of your experiments support your hypothesis?) Now is the time to pull together what happened, and assess the experiments you did.

Your conclusion can also contain samples and examples like this:

Many artworks such as the one that you see in the right, do not represent real snowflakes. These artworks have 4 or 8 identically repeated patterns while real snowflakes have 3 or 6 identically repeated patterns.

Related Questions & Answers:

What you have learned may allow you to answer other questions. Many questions are related. Several new questions may have occurred to you while doing experiments. You may now be able to understand or verify things that you discovered when gathering information for the project. Questions lead to more questions, which lead to additional hypothesis that need to be tested.

Possible Errors:

If you did not observe anything different than what happened with your control, the variable you changed may not affect the system you are investigating. If you did not observe a consistent, reproducible trend in your series of experimental runs there may be experimental errors affecting your results. The first thing to check is how you are making your measurements. Is the measurement method questionable or unreliable? Maybe you are reading a scale incorrectly, or maybe the measuring instrument is working erratically.

If you determine that experimental errors are influencing your results, carefully rethink the design of your experiments. Review each step of the procedure to find sources of potential errors. If possible, have a scientist review the procedure with you. Sometimes the designer of an experiment can miss the obvious.