winter wonderland our natural world described mathematically

Winter WonderlandWinter Wonderland

Our Natural World Described Mathematically

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Here’s one snowflake…Here’s one snowflake…

Here’s another snowflake…Here’s another snowflake…

Based on geometry which video Based on geometry which video shows a possible real snowflake.shows a possible real snowflake.

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Snowflake A Snowflake B

1. Snowflake A

2. Snowflake B

How do How do snowflakes snowflakes

form?form?

Why do snowflakes have Why do snowflakes have similar shapes?similar shapes?

Are all Are all snowflakes snowflakes identical?identical?

How is snow formed?How is snow formed?

1. Water freezes as it falls to the ground.

2. Water vapor condenses directly into an ice crystal.

3. Ice freezes more in cold weather.

How does the water become snow?How does the water become snow?

When water vapor condenses directly into an ice crystal.

What happens when water freezes as it falls to the ground? SLEET

Let’s look at some Euclidean Let’s look at some Euclidean shapes…shapes…

Some natural shapes to sort…Some natural shapes to sort…

Let’s look at some snowflakes…Let’s look at some snowflakes…

Why do snowflakes have similar Why do snowflakes have similar shapes?shapes?

What shape do these snowflakes What shape do these snowflakes have in common?have in common?

What shape do these snowflakes What shape do these snowflakes have in common?have in common?

Circ

le

Squar

e

Cone

Hex

agon

Pen

tagon

Oct

agon

17% 17% 17%17%17%17%1. Circle

2. Square

3. Cone

4. Hexagon

5. Pentagon

6. Octagon

What is the molecular shape of an What is the molecular shape of an ice crystal?ice crystal?

HO

H

Molecular Structure of IceMolecular Structure of Ice

Are any two snowflakes alike?Are any two snowflakes alike?

Temperature Humidity

Wind

Temperature

Warmer air = smooth ends

Colder air = sharp ends

Humidity

Crystals formed in dry air have simpler shapes than those in higher humidity.

Why do more complex crystals Why do more complex crystals form in more humid air?form in more humid air?

The

humid

ity w

orks

...

More

hum

id a

ir has

...

Dry

air

is w

arm

er.

33% 33%33%1. The humidity works on snowflakes the same way it does hair (frizz).

2. More humid air has more molecules in it.

3. Dry air is warmer.

Wind

Fractal SnowflakesFractal Snowflakes

Materials:Materials:

• Protractor

• Marker/Pencil

• Six-sided die

• Snowflake template

• Air Properties Chart

Draw a diameterDraw a diameter

Congratulations!Congratulations!

Stage Zero

What is the midpoint measurement What is the midpoint measurement of each red line segment?of each red line segment?

What properties determine the What properties determine the growth of a snow crystal?growth of a snow crystal?

Temperature Humidity

Wind

Air Properties ChartClick Here

Air Properties ChartAir Properties Chart

3 T = 0°/32°H = 0.10

30°

T = -7°/20°H = 0.10

30°

T = -12°/10°H = 0.10

45°

T = -18°/0°H = 0.10

45°

2 T = 0°/32°H = 0.05

15°

T = -7°/20°H = 0.05

15°

T = -12°/10°H = 0.05

30°

T = -18°/0°H = 0.05

30°

1 T = 0°C/32°H = 0.0

15°

T = -7°C/20°FH = 0.0

15°

T = -12°C/10°

FH = 0.0

15°

T = -18°C/0°FH = 0.0

15°

1 2 3 4

Air Properties ChartAir Properties Chart

3 T = 0°/32°H = 0.10

30°

T = -7°/20°H = 0.10

30°

T = -12°/10°H = 0.10

45°

T = -18°/0°H = 0.10

45°

2 T = 0°/32°H = 0.05

15°

T = -7°/20°H = 0.05

15°

T = -12°/10°H = 0.05

30°

T = -18°/0°H = 0.05

30°

1 T = 0°C/32°H = 0.0

15°

T = -7°C/20°FH = 0.0

15°

T = -12°C/10°

FH = 0.0

15°

T = -18°C/0°FH = 0.0

15°

1 2 3 4

Air Properties ChartAir Properties Chart

3 T = 0°/32°H = 0.10

30°

T = -7°/20°H = 0.10

30°

T = -12°/10°H = 0.10

45°

T = -18°/0°H = 0.10

45°

2 T = 0°/32°H = 0.05

15°

T = -7°/20°H = 0.05

15°

T = -12°/10°H = 0.05

30°

T = -18°/0°H = 0.05

30°

1 T = 0°C/32°H = 0.0

15°

T = -7°C/20°FH = 0.0

15°

T = -12°C/10°

FH = 0.0

15°

T = -18°C/0°FH = 0.0

15°

1 2 3 4

T = -12°C/10°F

H = 0.05 g/m3

30°

Stage One!Stage One!

Fractal SnowflakeFractal Snowflake

One more timeOne more time

Here’s the other snowflake…Here’s the other snowflake…

Based on geometry which video Based on geometry which video shows a possible real snowflake.shows a possible real snowflake.

50%50%

Snowflake A Snowflake B

1. Snowflake A

2. Snowflake B

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Wonderland today.