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Browse High School Symmetry/Tessellations

Stars indicate particularly interesting answers or good places to begin browsing.

Cutting Shapes into N Congruent Parts [03/23/2017]
Given a dozen different shapes to cut into congruent halves, a teen seeks a general method. Identifying the source of the puzzles, Doctor Peterson goes on to share his insights as well as those of the original author, emphasizing patience and pattern- matching.

2^4 = 16 AND 4^2 = 16 [10/29/2001]
Can you think of any other pair of unequal numbers that share the same relation as 2 and 4 in the above example? What was your strategy?

Axes of Symmetry [2/3/1996]
Can you help a year 5 student understand axes of symmetry of a triangle?

Center of Mass [02/22/1999]
Find the center of mass of a thin plate of constant density covering the region bounded by the parabolas y = 2x^2-4x and y = 2x-x^2.

Circle and Polygons: Lines of Symmetry [04/14/1997]
How many lines of symmetry are there in a circle?

Colour Combinations on a Cube [02/08/2002]
If each side of a cube is painted red or yellow or blue, how many distinct colour patterns are possible?

Cone Symmetry [7/17/1996]
Why does a circular cone have infinite lines of symmetry?

Counting Squares in Bigger Squares [02/29/2000]
How many edge 2 squares (2x2 squares) can be found in an edge 4 square (a 4x4 square)?

Diagonals and Axes of Symmetry [03/31/1998]
Could you explain the concepts behind the diagonals and axes of symmetry in a regular octagon?

Diagonals and Symmetry in Polyhedra [09/15/2002]
I would like a formula for finding the number of diagonals in a Platonic solid.

Ellipse Area and Circumference [04/19/2001]
How can I draw an ellipse and find the area and circumference?

Etymology of the Word Tessellation [11/05/2001]
Are tessellations related to the tesseract in Madeleine L'Engle's _Wrinkle in Time_ series?

Explanation of Orientation in Transformations [05/24/2007]
We've been learning about transformations and my teacher says that a translation, rotation, and dilation all preserve the orientation. I'm confused as to what orientation is. Could you explain it?

Group theory [11/22/1994]
The four rotational symmetries of the square satisfy the four requirements for a group, and so they are called a subgroup of the full symmetry group. (Notice that the identity is one of these rotational symmetries and that the product of two rotations is another rotation in the subgroup.) a. Do the four line symmetries of the square form a subgroup? b. Does the symmetry group of the equilateral triangle have a subgroup?

History of Tessellations [12/10/1996]
A students asks questions about the history of tessellations.

Horizontal and Vertical Symmetry [10/27/2001]
I have to find an example of horizontal symmetry, and I have no idea what that is.

How Do You Do Translations? [12/09/2009]
If (4,7) translates to (-3,9) what is the translated x-coordinate for the point (10,-5)?

Inversions [01/09/2002]
Is there a name for words such as SWIMS that look the same after a rotation of 180 degrees (flipped and upside-down to my grade 3 students)?

KaleidoTile [11/15/1995]
I would appreciate it if you would tell me a bit about KaleidoTile.

Lines of Symmetry [01/29/2001]
How do you find the number of lines of symmetry there are in a polygon?

Lines of Symmetry in Regular Polygons [03/13/2001]
Is there a formula for finding all the lines of reflectional symmetry in regular polygons?

Mapping Points [6/25/1996]
How do you map points inside a 4-point convex polygon onto another 4 point convex polygon?

Math in Card Games [10/06/2000]
How is math involved in creating and playing card games?

Numbering the Faces of Dice [02/27/2001]
How many ways are there to make dice out of the Platonic solids (i.e. 4, 6, 8, 12, and 20 sides)? How many of those ways have opposite face sums equal? What would the opposing face sums be for each type?

Number of Lines of Symmetry in a Regular Polygon [03/12/1998]
In a regular polygon, is the number of lines of symmetry the same as the number of lines or angles of that polygon?

Point and Line Symmetry in the Alphabet [12/05/1997]
What letters of the alphabet have point symmetry, line symmetry, or both? How many have neither form of symmetry?

Point Symmetry [08/19/2001]
What exactly is point symmetry? How can one tell if point symmetry is present?

Product of Isometries [10/05/1997]
Use three isometries (translation, rotation, and reflection) in composition with each other and deduce the net result of the two transformations.

Reflection and Rotation [02/20/2003]
Can rotation of a figure and reflection of that same figure yield the same result at times?

Reflective Properties of a Semicircular Mirror [05/16/2000]
What are the reflective properties of a semicircular mirror? Will a ray exit a semicircular mirror parallel to its entry line?

Rotating a Point [04/08/2003]
Find the image of a triangle with vertices A(0,1),B(-2,0),and C(-4,-5) under a rotation of 90 degrees counterclockwise about the origin. Is there a formula I can use instead of drawing a picture?

Rotational Symmetry [11/05/1997]
I am looking for a precise definition of rotational symmetry of a figure in a two-dimensional plane.

Rotation, Translation, and Reflection in the Plane [12/11/2005]
Can you explain what the words "rotate", "translate", and "reflect" mean?

Shortest Distance between Points [01/17/1998]
I am doing a project on the shortest distance between two points via another plane. I need help with my theorems.

Symmetries of a Cube [10/09/2003]
Prove that the group of symmetries of a cube is isomorphic to S_4.

Symmetry [04/29/2001]
What is symmetry?

Symmetry in Platonic Solids [01/24/1997]
How many planes of symmetry does each of the platonic solids have?

Symmetry Lines [03/30/2000]
How many symmetry lines are in a circle?

Symmetry Proof [09/27/2001]
Given an angle with vertex O and a point P inside the angle, drop perpendiculars PA, PB to the two sides of the angle, draw AB, and drop perpendiculars OC, PD to line AB. Then show that AC=BD.

Tessellation [02/26/1998]
Are there any non-regular convex polygons with more than four sides that can tessellate?

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