Make a card for which corner A is labeled A on the front and back and
is painted red on the front and back, corner B is labeled and painted green
on the front and back, corner C is labeled and painted black on the front
and back and corner D is labeled and painted blue on the front and back.

An "element card operation" is a ROTATION or a FLIP ACTION on a card. We will use the eight elements; I, R, T, L, H, m, V, and d defined below. Notice that the definition is given several ways. One way is to consider a symbolic lettering pattern. For example the element R, that rotates a card 90 degrees in a clockwise right turn, takes ABCD to BCDA. Notice that the letters are read in a clockwise direction around the card.

Next we look at a permutation notation method that is easier to see if we consider the following four arrow diagram showing how each letter changes position:

A second way to consider an "element card operation" is to look at the
"Geometry Action on Any Card". You can see this Geometry Action
in each "element card operation" defined below where the ACTION is taken on
the starting (IDENTITY) card. Also consider the example below of one
"element card operation", m, followed by a second, R, giving the same
overall "element card operation" achieved by using only one, V.

You may use your own large square card and turn it in a real time way to
have yet another method to work out the ideas in this project.

I is the "element card operation" that takes ABCD to ABCD.

I is the IDENTITY in that it does not change the card position.

R is the "element card operation" that takes ABCD to BCDA.

R is the CLOCKWISE ROTATION BY 90 DEGREES about the CENTER to a new card position. A RIGHT TURN AROUND the CENTER.

T is the "element card operation" that takes ABCD to CDAB.

T is the ROTATION BY 180 DEGREES about the CENTER to a new card position. A TURN AROUND the CENTER. An ABOUT FACE or REVERSED position.

L is the "element card operation" that takes ABCD to DABC.

L is the CLOCKWISE ROTATION BY 270 DEGREES about the CENTER to a new card position. L is the COUNTERCLOCKWISE ROTATION BY 90 DEGREES about the CENTER or a -90 DEGREE ROTATION. L is a LEFT TURN AROUND the CENTER.

H is the "element card operation" that takes ABCD to DCBA.

H is a FLIP over the HORIZONTAL LINE to a new card position. H is a CONJUGATION.

m is the "element card operation" that takes ABCD to ADCB.

m is a FLIP around the MAIN DIAGONAL to a new card position.

V is the "element card operation" that takes ABCD to BADC.

V is a FLIP about the VERTICAL LINE to a new card position.

d is the "element card operation" that takes ABCD to CBAD.

d is a FLIP around the MINOR DIAGONAL to a new card position.

You may call the "element card operations" the elements of a set then you may even define an operation on these elements as follows:

The GEOMETRY ACTION model looks like:

Which translates into the geometry action of first flip the ABCD card around the main diagonal to ADCB then rotate the card 90 degrees clockwise. to get BADC. The overall change from the Identity position to the last position is the same as the element card operation V.

It is nice to have a second method to check answers from a "not often used" operation. The permutation of symbols idea looks like:

Now take the first and the last row to get:

Either way the result is a card position of BADC and that corresponds to a vertical flip of ABCD.

Can you fill in the following table?( 5 points)Extra One Page Table

The operation is: ROW ELEMENT "FOLLOWED BY" COLUMN ELEMENT in the table.

• | I | R | T | L | H | m | V | d | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

I | |||||||||||||||||

R | |||||||||||||||||

T | |||||||||||||||||

L | |||||||||||||||||

H | |||||||||||||||||

m | V | ||||||||||||||||

V | |||||||||||||||||

d |

Some basic algebra properties( such as CLOSURE, ASSOCIATIVE, IDENTITY and INVERSE) seem to be true for the "followed by", • operation. Can you show that the COMMUTATIVE property does not hold for all the combinations of two elements in the table above. State an example from the table that proves the COMMUTATIVE property fails to hold. This method of proof is called a proof by COUNTER EXAMPLE. (3 points if you show how)

How would you prove that one of the other properties holds for the table? (2 points if you show how) Bring your comments to class for a discussion.

Copyright © 1998

with all rights reserved by

William V. Thayer, PedLog