LessonThreeNeSGeometricBeautyofSnowflakesPrt3Rvsd121016.pptx

The Geometric Beauty of Snowflakes:
Snow Crystal Geometry

Snowflake Types
illustrated with
Wilson Bentley’s
snowflake
photography in the
public domain unless
otherwise noted.
On the following slides,
please note the connections
between the snowflakes
beauty and their geometric
descriptions and varieties.

Overview of Some of the
Mathematics Found in Snowflakes
Symmetry
• Rotational symmetry – snowflake rotated or turned &
still looks same
• Reflection or mirror symmetry – one half of snowflake
image is reflection of other half
• 6-fold symmetry (Stellar dendrite snowflake types) –
hexagon looks same after being rotated 360° for 6
turns about its axis
• 3-fold symmetry (truncated triangular) - triangle looks
same after being rotated 360° for 3 turns about its axis

Snowflake Lines
Lines
• Perpendicular lines – 2 lines intersecting or
meeting at a right angle or 90°
• Parallel lines – 2 lines always at same distance
from each other but never touch
Left image -
The red &
yellow lines are
perpendicular
to each other.
Right image – The
two yellow lines are
parallel to each other
and the two red lines
are parallel to each
other.

Simple Hexagonal
Prisms
• Basic hexagonal snow
crystal geometry
• Grow into thin plates or
slender hexagonal columns
• Barely seen with eyes
• Facets rarely perfectly flat
• Decorated with beautiful
indents, ridges, or other
features

Hexagonal Plate Geometrics:
Rotational Symmetry
• A regular hexagon
• Rotational symmetry
definition – at every turn,
hexagon looks same at
each turn
• Nature’s snowflakes
mostly asymmetrical
because of atmospheric
conditions & melting as
flakes falls to earth
Point of
intersection of
each symmetry
line of axis (red
lines)
Imagine turning or rotating this hexagon to the right around its center point and
stopping at each new hexagonal point until it rests where you began turning it by
following the yellow arrows. Using the blue ball as a reference marker, imagine
moving it to its next point, one point at a time. How many turns until the blue ball
reaches its original resting position? Answer on next slide.

Hexagonal Plate Geometrics:
Reflection Symmetry
• Reflection symmetry – mirror image of itself or
half of image reflects other side
• Below diagram: observe how yellow square, blue
triangle & red circle are “reflected” like a mirror to
other hexagon side
1
6
1
6
2
5
2
5
3 4
3 4
Answer from slide #5 – “How many turns until the blue ball reaches its original resting position?” 6 turns

Stellar Plates
• Common
• Thin, plate-like crystals
• Star-like shapes
• 6 broad arms
• Elaborate faces with
symmetrical markings
• Temperature Formations
- near -2 C (28° F) or near
-15 C (5° F)
• 6-fold rotation &
reflection symmetries
What are these lines called
again? Answer on next slide.

Sectored Plates
• Stellar plates
• Hexagonal
• Complex
• Prominent ridges on
broad, flat branches
• Geometrics – 6-fold
rotation & reflection
symmetries
Answer to slide #8’s ques2on, “What are these
lines called again?” Symmetry lines or lines of axis

Stellar Dendrites
• Dendritic - "tree-like”
• Plate-like
• With branches and side branches
• Size - usually 2-4 mm in diameter
• Seen with eyes and or magnifying glass
• Most popular snow crystal
• Used in holiday decorations
• Geometrics – 6-fold rotation & reflection
symmetries

Fernlike Stellar Dendrites
• Multi side branches
• Fernlike
• Many fractal or self-similar
• Largest snow crystals
• Diameter – 5 mm or more
• Some snowfalls contain
only stellar dendrites and
fernlike stellar dendrites
• Geometrics – 6-fold
rotation & reflection
symmetries

Hollow Columns
• Small; need magnifier to view
• Symmetrical hexagonal hollow regions
• Air bubbles sometimes found inside
• Geometrics: tall, narrow, hexagonal,
columnar, prisms
• What’s a prism? solid object with two
identical ends and flat sides
What shape is
this found on
both ends?
Image in the Public Domain

• Slender, columnar ice
crystals
• Temperature growth: -5 C
(23 F)
• Resemble white hair bits
• Mystery: growth change
from thin flat plates to
long needles with
temperature changes
• Geometrics – needles are
straight lines
Needles

Capped Columns
• 2 thin, plate-like crystals
growing on ice column ends
• Blow into cloud region
where growth becomes
plate-like
• Grow into stubby columns
• Geometrics – 6-fold
symmetrical hexagonal
plates (both capped ends)
with rotation & reflection
symmetries
Rime on both ends of a
columnar snow crystal.

Double
Plates
• Double capped column
• Short central column
• One hexagonal plate grows out faster and
shields other from water vapor source
• Geometrics – columnar with two hexagonal
rotation & reflection symmetrical plates on
both ends

Triangular Crystals
• Rare
• Temperature growth - near -2 C (28° F)
• Can grow as truncated (cut tips; see examples
below) triangles
• Sometimes plate corners sprout branch-like arms
with unusual appearance
• Geometrics – truncated or non-truncated
triangles; 3-fold rotation & reflection symmetries

Snowflake Fractals
• Fractal – a
never
ending self
similar
pattern
across
different
size scales
• Crystalline
arms branch
into
increasingly
smaller sizes
• Branching
consistent
repeatable
pattern at
smaller and
smaller scales
• Common with
fern-like
stellar
dendrites
• Other fractal
examples:
rivers, trees,
coastlines,
mountains, &
abstract
through
computer
generation

Visit Natureglo’s eScience MathArt Virtual
Library’s Snowflake resource page at:
http://hascmathart.weebly.com/snowflakes.html
Background image –
“Heavy snow”. By
Steve Webster

References
1. Kenneth Libbrecht (Winter 2004/2005). "Snowflake Science". American
Educator. Retrieved 2009-07-14.
2. Alaska Air Flight Service Station (2007-04-10). "SA-METAR". Federal
Aviation Administration via the Internet Wayback Machine. Archived from
the original on 2008-05-01. Retrieved 2009-08-29.
3. Nelson, Jon. "Branch Growth and Sidebranching in Snow Crystals".
4. Bohannon, John. "ScienceShot: The True Shape of Snowflakes".
ScienceNOW. American Association for the Advancement of Science.
Retrieved 12 April 2013.
5. Jon Nelson (2008-09-26). "Origin of diversity in falling snow".
Atmospheric Chemistry and Physics. Retrieved 2011-08-30.
6. Nelson, Jon. "The Six-fold Nature of Snow”.

Thank you for watching!
Most of the black and white images, save for the ones otherwise noted, are
Wilson Bentley’s, the “Snowflake man” and are in the public domain.