Symmetry
in
Math and Science
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Terri Husted
Ithaca City Schools
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Symmetry plays an important part in math and
science. These are the kinds of symmetry we need to
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1011
first...
• VERTICAL SYMMETRY
• HORIZONTAL SYMMETRY
• POINT SYMMETRY
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This letter M has vertical
symmetry.
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This letter E has horizontal
symmetry.
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What kind of symmetry does this
letter have?
0011 0010 1010 1101 0001 0100 1011
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These famous polygons have line symmetry.
Give the most specific name for each polygon.
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1010 kind
1101 0001
1011
What
of 0100
symmetry
does each one have?
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These are the most specific names for
these
polygons...
0011 0010 1010 1101 0001 0100 1011
Rectangle
Regular
Hexagon
Isosceles
Trapezoid
Rhombus
Square
Equilateral
Triangle
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A circle has many lines of
symmetry.
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Point Symmetry
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P
cience
If a figure can be rotated 180 degrees about a
fixed point (P) and it still looks the same it is
said to have point symmetry.
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First, make one 90 degree turn to the
right
about
point
P.
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P
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Then, another 90 degree turn, for a total
of 180 degrees. If your figure is identical
to
the
original,
it
is
said
to
have
point
0011 0010 1010 1101 0001 0100 1011
symmetry.
P
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How about…?
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After one 90 degree turn...
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Then one more 90 degree turn for a total of 180
degrees...
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It has point symmetry!
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Does this figure have point
symmetry?
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Does this star have point
symmetry?
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In nature, we can find many examples of
symmetry: in flowers and leaves, in our own
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1010 1101
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bodies,
and0001
in0100
snowflakes.
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A snowflake is a single crystal of water.
Describe its symmetry.
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Now, let’s look at ...
TRANSFORMATIONS
ON A PLANE
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There are four kinds of
transformations in a plane.
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•
•
•
•
1) Reflection
2) Translation
3) Rotation
4) Dilation
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REFLECTION
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Axis of reflection
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Here is another example of a
reflection:
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TRANSLATION
0011 0010 1010
0001 0100 1011YOU
IN A1101
TRANSLATION
SLIDE A SHAPE IN
ONE DIRECTION LIKE THIS:
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.
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ROTATION
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A ROTATION IS WHEN YOU ROTATE A SHAPE
ABOUT A FIXED POINT (OR A FIGURE
RETURNS TO ITS ORIGINAL VIEW AFTER
BEING ROTATED A CERTAIN NUMBER OF
DEGREES.
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ROTATION
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A ROTATION IS WHEN YOU ROTATE A
SHAPE ABOUT A FIXED POINT.
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ROTATION
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A ROTATION IS WHEN YOU ROTATE A
SHAPE ABOUT A FIXED POINT.
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ROTATION
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And so on...
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DILATION
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A DILATION IS WHEN YOU ENLARGE OR
SHRINK A SHAPE OR OBJECT IN PROPORTION.
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DILATION
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A DILATION IS WHEN YOU ENLARGE OR
SHRINK A SHAPE OR OBJECT IN PROPORTION.
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DILATION
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A DILATION IS WHEN YOU ENLARGE OR
SHRINK A SHAPE OR OBJECT IN PROPORTION.
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SYMMETRY IN 3D
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CRYSTALS
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Materials scientists are interested in the structure of
materials such as metals, polymers (plastics) and
ceramics. They study how materials behave
under certain conditions in order to create better
products for everyday life.
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An important interest in the field of Materials
Science is the study of crystals, the basic unit
of most solids.
Crystals are used in control circuits, machines, electronics,
0011
0010 1010tools,
1101 and
0001 communications
0100 1011
industrial
(fiber optics)?
Silicon crystals are used to create microchips.
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Crystals like quartz keep time in your watch. Diamonds
are used in drilling, cutting, and have many uses in
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industry and in medicine. Do you know why diamonds are
used for cutting and drilling? Did you know that
surgeons use diamond-bladed scalpels in
delicate eye surgery?
What is a crystal?
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 A crystal is a solid with an orderly arrangement of
molecules which gives it a regular shape.
 In minerals the atoms are arranged in patterns which
are very specific for each mineral.
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 Minerals form crystals when they have room to
grow under the right conditions.
 Almost all solids are made of crystals.
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Crystals have many properties. Among its properties is
symmetry. Look at the polyhedra patterns you can
observe
in crystals!
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0010 1010
1101 0001
0100 1011
Cube- Ex: salt, copper,
iron, garnet, galena
Tetrahedron- Ex:
Chalcopyrite (a
copper mineral)
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Let’s look at the cubic shape...
VERTEX
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An axis through
the center of the
top and bottom
plane
demonstrates 4fold symmetry.
EDGE
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Four-fold symmetry means that in one full
turn around one axis the figure will look the
same four times. The cube has 3 such axes. We
will be exploring Euler’s Theorem.
Other polyhedra patterns found in crystals
are...
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Hexagonal prism and pyramidEx: quartz.
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Octahedron- Ex: gold, platinum,
diamond, magnetite.
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Dodecahedron0011 0010 1010 1101 0001 0100
1011
Ex: gold
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Pyritohedron - Ex: pyrite
Let’s build these!
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Here are some examples of crystals:
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pyrite
fluorite
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magnetite
Symmetry of Crystals
(Beautiful on the inside as well as the outside.)
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• The polyhedra shapes we’ve seen suggest that crystal
molecules might have also have symmetry. Through xray diffraction instruments scientists know that
metals crystallize into one of seven types of lattice
structures. Lattice structures are the imaginary lines
that connect the centers of atoms in a pattern.
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• A material’s physical and mechanical properties
depend on the crystal structure of that material which
is why scientists are interested in these structures.
Crystal Lattice Structures
Every lattice structure has its own unit cell.
Cubic
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Simple cubic
Body-centered cubic Ex:
Sodium, Iron
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Ex: Manganese
Imaginary
lines
Face-centered cubic
Ex: Lead, Gold, Copper, Aluminum
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Symmetry of metals
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Most common metals are face centered cubic,
body-centered or hexagonal in structure!
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Some examples: Zinc, beryllium, carbon.
What are the angles in a regular hexagon?
Other crystal lattice structures are...
0011 0010 1010 1101 0001 0100 1011Simple
Simple tetragonal
Ex: Tin, Chlorine
(when crystallized)
orthorhombic
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Ex: Boron, Bromine
Other patterns are:
Monoclinic - Ex : Twinned orthoclase
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Triclinic - Ex: Axinite
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This drawing is courtesy of
Dr. Margret Geselbracht,
Reed College, Portland, Oregon.
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There are a total of 7 crystal structures
but what is most important for you to
know is that crystal shape and size
depends on the particular metal,
temperature, pressure and cooling rate.
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Here’s a model of a silicon network solid.
Notice the cubic unit cell outlined by the yellow .
Imaginary
lines
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How do scientists identify rocks, minerals, crystals and alloys?
Theta-Theta X-ray Diffractometer
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An x-ray beam hits a crystal which is usually powdered and
the lattice planes of the crystals create unique patterns that
can be interpreted and used to identify the composition and
structure of the material being studied.
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Courtesy of Cornell University -Maura Weathers,
Director of X-Ray Facility.
The way the x-rays are diffracted from the crystal are interpreted by
a computer and the material being studied is identified. The peaks
you see are unique for that particular crystal!!!
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A mathematical formula developed by William H. Braggs
(1862-1942) and his son William Lawrence Braggs
explains why the faces of crystals reflect x-ray beams at
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exact angles for each crystal. This formula helps scientist
identify which crystal is being studied.
n = 2d sin 
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BEAM
Bragg’s Law
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d
n = 2d sin


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The distance between the atomic layers in a crystal is called d
(the hypotenuse). Lambda  is the wavelength of the x-ray
beam. Can you see that Lambda is the opposite side from
angle ?
Can you see the connections between science and math?
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Keep taking math
and science
courses!
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Thanks to the Cornell Center for Materials Science and the
National Science Foundation for funding the Research for
Teachers
Experience.
0011 0010
1010 1101
0001 0100 1011
Terri Husted
Ithaca City Schools
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