Symmetry in Math and Science Terri Husted Ithaca City Schools

Symmetry plays an important part in math and science. These are the kinds of symmetry we need to review first... VERTICAL SYMMETRY HORIZONTAL SYMMETRY POINT SYMMETRY

This letter M has vertical symmetry.

This letter E has horizontal symmetry.

What kind of symmetry does this letter have?

These famous polygons have line symmetry. Give the most specific name for each polygon. What kind of symmetry does each one have?

These are the most specific names for these polygons... Rectangle Rhombus Regular Hexagon Square Isosceles Trapezoid Equilateral Triangle

A circle has many lines of symmetry.

Point Symmetry P 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. cience

First, make one 90 degree turn to the right about point P. P

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 symmetry. P

How about…?

After one 90 degree turn...

Then one more 90 degree turn for a total of 180 degrees... It has point symmetry!

Does this figure have point symmetry?

Does this star have point symmetry?

In nature, we can find many examples of symmetry: in flowers and leaves, in our own bodies, and in snowflakes. A snowflake is a single crystal of water. Describe its symmetry.

TRANSFORMATIONS ON A PLANE Now, let’s look at...

There are four kinds of transformations in a plane. 1) Reflection 2) Translation 3) Rotation 4) Dilation

REFLECTION Axis of reflection

Here is another example of a reflection:

TRANSLATION IN A TRANSLATION YOU SLIDE A SHAPE IN ONE DIRECTION LIKE THIS:.

ROTATION 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.

ROTATION A ROTATION IS WHEN YOU ROTATE A SHAPE ABOUT A FIXED POINT.

ROTATION A ROTATION IS WHEN YOU ROTATE A SHAPE ABOUT A FIXED POINT.

ROTATION And so on...

DILATION A DILATION IS WHEN YOU ENLARGE OR SHRINK A SHAPE OR OBJECT IN PROPORTION.

DILATION A DILATION IS WHEN YOU ENLARGE OR SHRINK A SHAPE OR OBJECT IN PROPORTION.

DILATION A DILATION IS WHEN YOU ENLARGE OR SHRINK A SHAPE OR OBJECT IN PROPORTION.

SYMMETRY IN 3D CRYSTALS

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. 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, industrial tools, and communications (fiber optics)? Silicon crystals are used to create microchips. Crystals like quartz keep time in your watch. Diamonds are used in drilling, cutting, and have many uses in 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?  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.  Minerals form crystals when they have room to grow under the right conditions.  Almost all solids are made of crystals.

Crystals have many properties. Among its properties is symmetry. Look at the polyhedra patterns you can often observe in crystals! Cube- Ex: salt, copper, iron, garnet, galena Tetrahedron- Ex: Chalcopyrite (a copper mineral)

Let’s look at the cubic shape... An axis through the center of the top and bottom plane demonstrates 4- fold symmetry. EDGE VERTEX 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... Hexagonal prism and pyramid- Ex: quartz. Octahedron- Ex: gold, platinum, diamond, magnetite.

Pyritohedron - Ex: pyrite Dodecahedron- Ex: gold Let’s build these!

Here are some examples of crystals: magnetite fluorite pyrite

Symmetry of Crystals (Beautiful on the inside as well as the outside.) The polyhedra shapes we’ve seen suggest that crystal molecules might have also have symmetry. Through x- ray 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. 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. Simple cubic Ex: Manganese Body-centered cubic Ex: Sodium, Iron Face-centered cubic Ex: Lead, Gold, Copper, Aluminum Cubic Imaginary lines

Symmetry of metals Most common metals are face centered cubic, body-centered or hexagonal in structure! Some examples: Zinc, beryllium, carbon. What are the angles in a regular hexagon?

Other crystal lattice structures are... Simple tetragonal Simple orthorhombic Ex: Boron, BromineEx: Tin, Chlorine (when crystallized)

Other patterns are: Monoclinic - Ex : Twinned orthoclase Triclinic - Ex: Axinite This drawing is courtesy of Dr. Margret Geselbracht, Reed College, Portland, Oregon.

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.

Here’s a model of a silicon network solid. Notice the cubic unit cell outlined by the yellow. Imaginary lines

How do scientists identify rocks, minerals, crystals and alloys? Theta-Theta X-ray Diffractometer 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. 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!!!

A mathematical formula developed by William H. Braggs ( ) and his son William Lawrence Braggs explains why the faces of crystals reflect x-ray beams at exact angles for each crystal. This formula helps scientist identify which crystal is being studied. n = 2d sin 

Bragg’s Law d   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  ? BEAM n = 2d sin 

Can you see the connections between science and math? Keep taking math and science courses!

Thanks to the Cornell Center for Materials Science and the National Science Foundation for funding the Research for Teachers Experience. Terri Husted Ithaca City Schools