To tessellate: to cover a plane with one or more shapes without overlaps or To tessellate: to cover a plane with one or more shapes without overlaps or gapsgaps
A Regular Tessellation is when only one regular A Regular Tessellation is when only one regular shape is used. A regular shape has congruent shape is used. A regular shape has congruent sides and angles sides and angles
There are exactly 3 Regular Tessellations:There are exactly 3 Regular Tessellations:
Square, Equilateral Triangle & HexagonSquare, Equilateral Triangle & Hexagon
The grid for each of these tessellations was done in Sketchpad.
Then each student colored her creation by hand, except for Mary who chose to do hers in Paint.
A Semi-regular tessellations is when two or more A Semi-regular tessellations is when two or more regular shapes are used. At each vertex the same regular shapes are used. At each vertex the same arrangement of shapes must exist. arrangement of shapes must exist.
There are exactly 8 semi-regular tessellationsThere are exactly 8 semi-regular tessellations
Each of these grids was also done in Sketchpad. Then the students colored them in Paint.
4.8.8.
CaitlinCaitlin
#1
3.4.6.4.
AyannaAyanna#2
3.4.6.3
Jennifer
CatherineCatherine
3.3.4.3.4.#3
3.3.4.3.4
3.3.4.3.4
VanaVana
3.3.3.4.4.
ShannonShannon
#4
3.6.3.6.
LaurieLaurie
#5
#6
Anne Marie
3.3.3.3.6
#7
3.3.3.3.6
Nora
Tiffany
3.12.12
#8
NEON
DRAGONS
by Elizabeth
Laurie
Ayanna’s Ocean
TiffanyTiffany
Tiffany
Finished by Vana
All of the tessellations done in class were by translation only. They are periodic tessellations. For further explanation go to
http://www.geocities.com/dottie4math/slides.ppt
To learn more about other types go to
http://mathforum.org/sum95/suzanne/tess.intro.html
To explore aperiodic tessellations scroll down to Penrose on the homepage
http://www.funmath.org
Continue with art show.
is a fractal created by connecting the midpoints of a triangle to form another triangle. The original triangle is now composed of 4 congruent triangles – each of which is similar to the original. The process continues by connecting the midpoints of the remaining triangles – not the center one. On and on it goes!
Students created Sierpenski triangles after the study of congruent triangles and after deriving the proof that states – when the midpoints of two sides of a triangle are connected, that segment is parallel to the third side and equal in measure to ½ its length.
While they followed the algorithm, we discussed, after each iteration, how the new triangles related in area and perimeter to the original. We created formulas that allowed us to determine each measure after a given number of steps.
For Therapy, each student had the freedom to decorate her triangle in a creative fashion.
SHANNON
Caitlin
Tiffany
Katie
This one was done with Sketchpad.
It’s picture was taken with a digital camera as it appeared on the computer screen. The image was then inserted into the slide show.
Reverse Coloring
Sierpenski on Calculator
Whose idea was this???
HA-HA-HA-HA-HA
HA-HA-HA-HA! HA-HA-HA!
Fractals in the Lab
Who is Fibonacci?
Fibonacci’s sequence of numbers:
1 1 2 3 5 8 13 21 . . .
What is a Palindrome?
Ex. radar
Fibonacci Palindromic Sequence:
8 5 3 2 1 1 1 2 3 5 8
Making a Palindromic Design:
Divide the sides of a polygon into divisions equal in length with the above sequence. Connect corresponding points on adjacent or opposite sides. Then color every other region.
AND
All snowflakes have a hexagonal shape, but we made 7 sided flakes instead. Our hope is that Mother Nature will show us what true snow- flakes look like. (And she sure did!!!)
The Christmas tree has a maze inside. As with a deductive proof, it is sometimes easier if you look at the end first and work backwards.
The star on top was created by wrapping a strip of paper into a regular pentagon. Before the wrapping began, a wish was written on the back side of the strip.
Mother Nature’s Response
There are five Platonic Solids. Each is created with exactly one regular polygon.
One of these solids is called the icosahedron. It is formed with 20 equilateral triangles. At each vertex five triangles meet. If each vertex is truncated, the resultant solid is composed of hexagons and pentagons. The name of this solid is the truncated icosahedron which is one of the Archimedian Solids. There are 13 of these solids which are composed of more than one regular shape.
I have more fun with mine!
Sir David Brewster (1781-1868)
History of Kaleidoscopes
Kaleidoscopes are created by using mirrors to reflect a design.
However, with Sketchpad, the following were made by rotating a triangular design. This is a better technique when the creations are activated. Activation can only be seen in Sketchpad, so in this slide show there is only a pretty image to represent each creation.
If you have Sketchpad, see how to see the action at the end of this slide show.
Anne Marie
Ayanna
Caitlin
Catherine
CatieCatie
Elizabeth
Jennifer
Mary
Nora
Shannon
Laurie
Katie
Vana
Caitlin
Catherine
Ayanna
Catie
Jennifer
Laurie
Katie
Math Art
And for those beautiful kaleidoscopesRemember you must have Sketchpad on your computer!