Transcript: Grade 7 - Intro & Transformational Geometry | Mar 31, 1999

A title reads “TVO’s Virtual Classroom. Get connected.” The “V” in “Virtual” is a tick, the “A” in “classroom” is an “at sign” with an extended loop that turns into a power cord with a plug at the end, and the first “O” in “classroom” is a spinning globe.

(Lively music plays)

In front of a world Atlas spread on the wall, Mister C., curly-haired and bearded in his forties, with glasses, wearing a white T-shirt with suspenders painted on it, says
GOOD MORNING AND WELCOME
TO THE VIRTUAL CLASSROOM.
MY NAME IS Mr. CRAVEN,
BUT I PREFER JUST Mr. C.
AND I'M REALLY THRILLED
TO BE WITH YOU TO DO
A PROJECT IN MATHEMATICS
THAT I PARTICULARLY ENJOY.
I'VE ALWAYS LOOKED FOR A
CLOSE CONNECTION BETWEEN
ART AND MATHEMATICS
BECAUSE AT ONE TIME
I ACTUALLY PRACTISED AS AN
ARTIST AND I FIND THIS
TO BE A MOST WONDERFUL WAY
TO LOOK AT MATHEMATICS
AT THE SAME TIME.
WE'LL BE LOOKING AT
THE ART OF M.C. ESCHER
PRETTY SOON, BUT
AS I ALWAYS DO,
I LIKE TO OPEN THESE
PROGRAMS WITH A LITTLE BIT
OF A PUZZLE JUST TO GET
THE JUICES FLOWING AND,
AS USUAL, I'M ABOUT
TO SHOW YOU ONE.
WHAT I WILL DO IS, I WILL
READ THROUGH THIS QUESTION -

A sheet of blue paper appears with a question on it titled “The World's Best Train Puzzle” that reads “Out West, there's a single-track railroad that goes through Old Baldy Mountain. Inside the mountain, the tunnel is only wide enough for one train. At exactly 2 o'clock one day, two trains entered the tunnel from opposite directions. 5 minutes later, each train came out of the opposite entrance to the tunnel. Neither train was damaged in any way. How was this possible?” A drawing shows the two trains and the tunnel.

Mister C. continues AND THEN I WILL ASK YOU TO
PHONE IN BY PHONING POUND 9
AND GIVE ME YOUR
ANSWER, YOUR PROPOSAL
AS TO WHAT THE
SOLUTION TO THIS IS.
THIS IS CALLED “THE WORLD'S
BEST TRAIN PUZZLE.”

He then proceeds to read the text of the question.

Mister C. continues READ IT CAREFULLY.
AND ONCE YOU THINK
YOU KNOW THE ANSWER,
PRESS POUND 9 AND
GIVE ME A CALL.
TWO TRAINS ENTERED THE
SAME TUNNEL FROM OPPOSITE
DIRECTIONS, 2 O'CLOCK,
5 MINUTES LATER,
THEY CAME OUT THE OPPOSITE
ENDS, NO DAMAGE DONE.
ONLY A SINGLE TRACK.
HOW IS THIS POSSIBLE?
GIVE ME A CALL AND LET
ME KNOW WHAT YOU THINK.
(PAUSE)
PERHAPS YOU SHOULD LOOK
CLOSELY AT THE TIME
AND THAT MIGHT BE A
BIT OF A CLUE.
THERE IS A SIMPLE
ANSWER TO THIS.
OH, WE HAVE A
CALL FROM COLLEGE.
WE SHOULD BE
CONNECTED SHORTLY.
AND HOPEFULLY WE HAVE A
SOLUTION TO THIS PROBLEM.
IT ALWAYS TAKES
A MOMENT OR TWO
FOR THE TECHNOLOGY TO
CATCH UP TO MY VOICE.
HI! YOUR NAME IS?
AND YOUR NAME IS?

A child's voice says JOEL.

Mister C. says HI, JOEL.
DO YOU KNOW A SOLUTION
TO THIS PROBLEM?
WHAT DO YOU THINK?

Joel says ONE ENTERED AT
2 O'CLOCK A.M.
AND ONE ENTERED
AT P.M.

Mister C. says YOU GOT IT!
VERY, VERY GOOD, JOEL,
THAT'S A GREAT WAY TO START.
A.M. AND P.M.
IT DOESN'T SAY IN THE
PROBLEM A.M. OR P.M.
AND AS YOU KNOW 2 O'CLOCK
HAPPENS TWICE A DAY
WITH THE STANDARD CLOCK, NOT
THE 24-HOUR CLOCK PERHAPS,
BUT THE STANDARD CLOCK.
ANYWAY, I'M GOING TO START
OUR LITTLE SERIES NOW
AND GET US INTO THE
WHOLE BUSINESS OF -

A sheet of paper reads “Investigating Tilings through Escher's Art:”
he reads the text out loud. Another caption reads “M.C. Escher, Dutch Artist, 1898-1972.”

Mister C. continues WHO IS M.C. ESCHER?
WELL, I WON'T TELL
YOU A WHOLE LOT,
BUT I WILL TELL YOU THAT
HE IS A DUTCH ARTIST.
HE'S AN ARTIST OF
THE 20TH CENTURY,
BORN IN 1898 AND
DIED IN 1972,
THIS IS A SELF-PORTRAIT
THAT HE DID
AT THE AGE OF 20.

A black and white lithograph shows a thin young man posing on an armchair in front of some pictures on a wall. Mister C. appears in a small rectangular frame in the bottom right corner.

Mister C. continues NOW, HE DID A WHOLE
BUNCH OF SELF-PORTRAITS
THROUGHOUT HIS LIFE, AND I
WILL TRY TO MAKE A POINT
OF SHOWING YOU ONE NEXT
WEEK WHICH IS DONE
MUCH LATER IN HIS LIFE.
IT'S ONE WHERE HE STARES
INTO HIS REFLECTION
IN A SPHERE.
QUITE A FAMOUS ONE,
BUT THIS WILL GIVE YOU
SOME SENSE OF WHAT HE
LOOKED LIKE THEN.
BUT WHY
ESCHER'S ART?
WHY IS IT PARTICULARLY
INTERESTING
IN TERMS OF
MATHEMATICS?
WELL THIS LITTLE
PUZZLE MIGHT GIVE
A LITTLE WEE
BIT OF A CLUE.

He puts a small square abstract picture showing a flock of birds in blue, white and pink in place of the lithograph. Using a finger, he moves sections of the puzzle.

Mister C. continues THIS PUZZLE IS
A REPRODUCTION
OF ONE OF HIS
GREAT PIECES.
NOW, WE'VE GOT TO
DO A LITTLE BIT
OF MOVING AROUND HERE.
LET'S SEE IF WE CAN GET
THIS BACK INTO POSITION.
I DIDN'T MESS IT UP TOO
MUCH, BECAUSE I DID WANT TO
GET IT BACK INTO
POSITION FAIRLY QUICKLY,
BUT IT IS A
TILING OF BIRDS.
BIRDS FLYING IN
DIFFERENT DIRECTIONS.
ACTUALLY EVEN A
VARIETY OF BIRDS.
BUT EVERYWHERE
THERE'S A GAP,
THERE'S A COMPLETE
IMAGE OF A BIRD,
JUST AS YOU
WOULD EXPECT.
HE IS FAMOUS FOR
THIS TYPE OF IMAGE,
WE'RE GOING TO TAKE A
LOOK AT THE MATHEMATICS
THAT UNDERLIES THIS
KIND OF A PAINTING,
AND YOU'LL FIND THAT
IT'S QUITE EXTENSIVE.
NOW, WHAT I'M
GOING TO DO IS,
I'M GOING TO SPEND A
MOMENT OR TWO AND
I'M GOING TO SHOW
YOU SOME IMAGES FROM
A VERY, VERY GOOD BOOK.

He produces an open book showing on one page quartered sets of geometrical figures. then he moves to the opposite page.

Mister C. continues THE REASON THAT THIS BOOK
IS PARTICULARLY GOOD
IS NOT ONLY DOES IT SHOW
SOME OF HIS GREAT WORK,
BUT IT SHOWS HOW HE
CREATED SOME OF IT.
NOW, IF YOU TAKE A LOOK
AT THIS DRAWING UP HERE,
YOU SEE THAT THERE'S
PENTAGONS AND ACTUALLY
INSIDE OF HEXAGONS,
SO THAT'S SORT OF
THE INITIAL STAGES
OF ONE DIAGRAM.
IF YOU TAKE A
LOOK AT THIS ONE,
YOU CAN SEE THAT HE'S
GOING TO GO TOWARDS
SOMETHING LIKE CHICKENS OR
SOMETHING ALONG THAT LINE,
BUT THE FUNDAMENTAL
PATTERN COMES FROM
A VERY GEOMETRIC
LOOKING THING.
EVEN IN THIS ONE UP
HERE IN THIS CORNER,
YOU CAN SEE THAT HE'S
DEVELOPING THIS PATTERN
IN A VERY STRONG GRID AND THE
GRID BASICALLY IS MADE UP
OF EQUILATERAL TRIANGLES.
IT'S REALLY QUITE
HARD TO SEE THOSE,
AND ULTIMATELY YOU END
UP WITH SOMETHING
THAT LOOKS LIKE THIS.

He points to a composite drawing of irregular hexagons with spikes at the junctions of three hexagons. The next page shows a yellow and blue colour drawing of equilateral triangles and hexagons in a flower pattern. Another drawing shows three or more different coloured hexagon grids superimposed on each other.

Mister C. continues NOW, LET'S SEE, LET'S
GO TO THE NEXT HERE.
THESE ARE THE KINDS
OF THINGS THAT COME
FROM PURELY
GEOMETRIC PATTERNS.
THIS INVOLVES
TRIANGLES, HEXAGONS,
AND THIS IS THE
INTERPLAY OF HEXAGONS
AND I FIND IT REALLY
QUITE INTERESTING BECAUSE
IT LOOKS KIND OF LIKE
FENCE ON TOP OF FENCE,
BUT IT'S VERY INTRICATE
AND IT'S ALL TIED
AND MESHED TOGETHER.
REALLY A NEAT
PAINTING, BUT AGAIN,
OBVIOUSLY VERY
GEOMETRIC.
BUT HIS WORK
CHANGED OVER TIME.
IT'S STILL VERY,
VERY GEOMETRIC,
BUT HE REALLY GOT INTO
ANIMALS AND ANIMAL SHAPES.
THIS IS THE INTERPLAY OF
FISH WITH BUMBLEBEES
AND YOU NOTICE THAT
THERE'S NOT A GAP.
THERE'S NO SPACE
IN BETWEEN THEM,
AND EVERYTHING FITS
TOGETHER SO PERFECTLY.
AND THIS BEE LOOKS
VERY MUCH LIKE A BEE
AND THE FISH IS A PERFECT
RENDERING OF A FISH.

The pink fish and white bumblebees form a grid pattern.

Mister C. continues LOOK AT THE ONE
ON THIS SIDE.
THIS IS BIRDS GOING
IN BOTH DIRECTIONS.
THEY'RE POINTED IN
SLIGHTLY DIFFERENT
ORIENTATIONS.
THIS BIRD LOOKING
DOWN, THAT ONE ACROSS.
NOTICE, YOU CAN SEE THE
PARALLELOGRAMS AND
THEN YOU CAN SEE THE
BIGGER HEXAGONS HERE.

He points at the visible outlines of parallelograms and hexagons. Below a simpler diagram is based on hexagons only.

Mister C. continues IT IS QUITE CLEAR THE
GEOMETRY BEHIND THIS
AND IF YOU TAKE A
LOOK AT THE BOTTOM,
YOU CAN SEE HOW THIS IS
THE INITIAL DIAGRAM
THAT LEADS TO
SOMETHING LIKE THIS.
ULTIMATELY HE DID PAINTINGS
THAT WERE LIKE THIS.
THIS IS THE LIZARDS
INTERPLAYING WITH LIZARDS.
IT'S THREE DIFFERENT
ORIENTATIONS.

The drawing shows stylized black, brown and white lizards in an unnatural spread-eagled pose occupying all the space in a picture. a similar drawing is based on unnatural looking popeyed blue, brown and yellow fish. Yet another shows a pattern of brown and white Pegasus-type prancing horses. Another one shows an abstract rendering of white and gray parrots.

Mister C. continues PROBABLY BASED
ON A HEXAGON.
I LIKE THIS ONE
BECAUSE OF THE STRONG,
STRONG RICH COLOURS.
AND AGAIN, SEVERAL
DIFFERENT ORIENTATIONS,
VERY INTERESTING PATTERN

RATHER ABSTRACT
FLYING FISH, I WOULD SAY.
AND LAST BUT NOT LEAST
WE'LL SHOW YOU A COUPLE MORE
JUST TO GET
THINGS GOING HERE.
THIS IS ONE WHERE YOU HAVE
FLYING HORSES INTERPLAYING
WITH FLYING HORSES, AND
THIS IS THE KIND OF THING
THAT I MIGHT EXPECT
THAT WE COULD PRODUCE.
IT'S A VERY, VERY SIMPLE
DIAGRAM INVOLVING BIRDS,
AND YOU CAN SEE THAT
THE PATTERN'S
VERY STRAIGHTFORWARD.
IT'S NOT TERRIBLY
COMPLEX, BUT IT IS
QUITE ATTRACTIVE.
SO THIS GIVES YOU SOME
SENSE OF WHAT ESCHER'S
WORK IS, BUT WHAT'S
THE MATHEMATICS?
WELL, FIRST OF ALL I'LL
GO THROUGH BRIEFLY
WHAT OUR LESSON TODAY
IS GOING TO BE ABOUT.

A blue sheet of paper bears the title “Class Number 1 - Monday, March 22, then items A., B. and C.” Mister C. reads from it.

He says SO THERE WE ARE.
WE'LL GET
STRAIGHTENED OUT HERE.
THE FIRST THING I'M GOING
TO DO IS DO A SHORT LESSON
IN TRANSFORMATIONAL GEOMETRY.
THEN NEAR THE
END OF THE CLASS,
I WILL DISCUSS BRIEFLY
THE IN-CLASS ASSIGNMENT,
AND THEN I WILL ASSIGN
THE PROJECT OFFICIALLY.
NOW, THE IN-CLASS ASSIGNMENT,
I REALLY WANT TO MAKE A
POINT NOW - AND I'LL MAKE
THE POINT AGAIN - THAT YOU
HAVE TO DO THAT BETWEEN
NOW AND NEXT MONDAY.
IDEALLY YOU'LL BE
SUBMITTING YOUR WORK
BY FRIDAY OR SO, SO THAT
YOUR TEACHER CAN FAX TO ME
SOME OF THE SOLUTIONS OF THE
WORK THAT YOU'LL BE DOING.
NOW, I'LL WORRY ABOUT
THE FAX NUMBER LATER
IN THE PROGRAM, BUT RIGHT
NOW, I JUST WANTED TO SORT
OF BRING THAT TO
YOUR ATTENTION.
THE ASSIGNMENT
YOU DO RIGHT AWAY,
I'M LOOKING FOR SOME
RESULTS BY FRIDAY
THAT YOU CAN FAX TO ME.

A new sheet of paper reads “Two Key Mathematical Ideas Embedded in Escher's Art - Tessellations and Transformations.”

Mister C. says THERE ARE TWO KEY THINGS
THAT ARE INVOLVED
IN ESCHER'S ART:
ONE THING IS TESSELLATIONS AND
THE OTHER IS TRANSFORMATIONS
NOW, THE FIRST THING I
WANT TO DO IS ASK SOMEBODY
TO PHONE IN AND TELL ME
WHAT A TESSELLATION IS.
WHAT IS A TESSELLATION?
WHERE DOES THIS WORD
COME INTO IN TERMS
OF MATHEMATICS?
AND IT'S VERY SIMPLE
TO PHONE IN, JUST POUND 9.
NOW, ANOTHER WORD
FOR TESSELLATION -
I'M WILLING TO GIVE THIS
ONE AWAY - IS TILING.
SO TILING OR
TESSELLATIONS,
WHAT ARE THEY?
HOW ARE THEY DESCRIBED?
I ENCOURAGE YOU TO PHONE
IN AND LET ME KNOW
WHAT YOU THINK.
AND I THINK I JUST MIGHT
PHONE OUT TO SOMEBODY
AND SEE WHAT ANSWER
THEY MIGHT HAVE.
SO LET'S JUST TRY SOMEBODY
AT RANDOM AND WE'LL GIVE
THEM A CALL AND SEE WHAT
THEY CAN TELL ME ABOUT
TESSELLATIONS OR TILINGS.
SO WE'RE MAKING
A CALL RIGHT NOW.
I DON'T KNOW
WHO IT IS.
WHEN THE PHONE RINGS,
PLEASE PICK IT UP.
AND WE'RE GOING TO COLLEGE
AVENUE AND HOPEFULLY
WE'LL HAVE A QUICK CHAT HERE
AND SEE WHAT YOU KNOW
ABOUT TESSELLATIONS.
HELLO.
HELLO, HAVE YOU
PICKED UP YOUR PHONE?

A voice says YUP.

Mister C. says YOUR NAME IS?

Carolyn says CAROLYN.

Mister C. says CAROLYN, CAN YOU
TELL ME ANYTHING
ABOUT TESSELLATIONS?
DO YOU REMEMBER THEM?

Carolyn says NO.

Mister C. says DO YOU KNOW WHAT A
TILING IS OR WHAT DOES
A TILING BRING
TO MIND?

Carolyn says I DON'T KNOW.

Mister C. says I'M GOING TO GO ONE STEP
FURTHER WITH YOU, CAROLYN.
YOU KNOW WHERE YOU FIND
TILES ON A FLOOR
OR MAYBE ON A
BATHROOM WALL, RIGHT?

Carolyn says UH-HUH.

Mister C. says OKAY, SO IT'S THE
SAME THING, ISN'T IT?

Carolyn says YES.

Mister C. says OKAY.
WHAT CAN YOU TELL ME
ABOUT A FLOOR OR A TILING
PATTERN IN A BATHROOM?
WHAT KINDS OF THINGS
DO YOU NOTE ABOUT IT?

Carolyn says THEY JUST FIT TOGETHER.

Mister C. says THEY FIT TOGETHER, RIGHT,
SO YOU'RE SAYING
THE KINDS OF THINGS I
WANT YOU TO SAY.
TELL ME A LITTLE BIT
ABOUT THE KINDS OF SHAPES
THAT ARE USED.

Carolyn says SQUARES.

Mister C. says MOSTLY SQUARES,
AREN'T THEY,
AND THEY FIT TOGETHER
REALLY WELL.
SO THERE'S NO GAPS AND
THEY FIT TOGETHER, RIGHT?

Carolyn says MM-HMM.

Mister C. says SEE, YOU HELPED ME, AND
THAT'S ALL A TESSELLATION IS.
THANK YOU VERY
MUCH, CAROLYN.
WHAT I'M GOING TO DO IS,
I'M GOING TO ASK
TWO QUESTIONS TO DO WITH
TESSELLATIONS RIGHT NOW,
AND THE FIRST ONE IS
COMING UP SHORTLY.

A blue sheet of paper is titled Question Number 1 and Mister C. reads off it. A gray column on the right contains a blue bar graph that ascends as increasing percentages appear at the top.

He says DO ALL TRAINGLES, REGARDLESS OF SIZE OR SHAPE TILE THE PLANE? PRESS ONE IF YOU THINK IT IS A TRUE THING AND PRESS TWO IF YOU THINK IT IS A FALSE THING. NOW, WE'RE LOOKING
FOR AT LEAST 70 PERCENT,
SO WE'RE GETTING THERE.
AH, WE GOT IT.
MAYBE 80?
OKAY, FANTASTIC.
NOT A WHOLE LOT IN
THE CLASS TODAY
BUT THERE'S OUR GRAPH.

A graph shows a tall green bar and a shorter purple one with the numbers 5 and 3 below them.

Mister C. says THE MAJORITY OF YOU
THOUGHT THAT THIS WAS
A TRUE STATEMENT.
NOW, WHEN I SAY
TILE THE SURFACE,
IF YOU REMEMBER WHAT
I SAID TO CAROLYN,
IT SAID THAT WHEN YOU PUT
THESE TRIANGLES TOGETHER,
THERE ARE NO GAPS.
THE QUESTION IS, IS
THIS TRUE OR NOT?
WELL, AS IT TURNS
OUT WITH TRIANGLES
IT IS ABSOLUTELY TRUE.
YOU CAN PUT THEM TOGETHER
AND IT DOESN'T MATTER
WHAT THE TRIANGLE IS, YOU
CAN FIND A WAY OF PUTTING
THEM TOGETHER SO THAT
THERE ARE NO GAPS.
NOW YOU CAN EXPERIMENT
WITH THAT ON YOUR OWN,
BUT CLEARLY IT WORKS.
I HAVE A SECOND QUESTION
THAT'S PROBABLY A LITTLE BIT
TRICKIER IN A SENSE.

Another question appears with the same question in reference to quadrilaterals.

Mister C. says QUADRILATERALS, REGARDLESS OF SIZE OR SHAPE, DO THEY TILE THE PLANE? PRESS ONE IF YOU THINK IT IS A TRUE STATEMENT AND PRESS TWO IF YOU THINK IT IS A FALSE STATEMENT.
WE GOT 60 PERCENT.
I'D LIKE A COUPLE MORE
PEOPLE TO PUNCH IN HERE.
AH, WE'RE UP TO 80 PERCENT.
THAT'S GREAT.
AND THERE'S OUR GRAPH.

The graph shows the purple bar higher than the green one by 5 to 4.

Mister C. says WELL, IT'S SPLIT DOWN
THE CENTRE PRETTY MUCH,
SLIGHTLY MORE PEOPLE
THINK IT'S FALSE.
WELL, YOU KNOW, IT'S
A TOUGH QUESTION.
HERE ARE A COUPLE OF SHAPES
THAT ARE QUADRILATERALS.
AND YOU KNOW, WHEN I THINK
ABOUT TILING A SURFACE,
THESE ARE THE KINDS
OF SHAPES I MIGHT
HAVE DIFFICULTY WITH.

A drawing shows an irregular quadrilateral with a point on one angle and another shaped like a plane's wings.

Mister C. says THERE'S NOT TOO MUCH DOUBT
IN MY MIND THAT A SQUARE
WILL TILE THE SURFACE.
THAT IS, I CAN PUT THEM
TOGETHER WITHOUT ANY GAPS.
BUT WHEN YOU GET TO
QUADRILATERALS THAT LOOK
LIKE THIS OR THIS, IT'S
NOT QUITE SO OBVIOUS.
NOW, THE ONLY THING I
CAN ASK YOU TO DO
IS ON YOUR OWN OR WITH
YOUR TEACHER IS TO TRY
DIFFERENT SHAPES; CUT OUT
EIGHT OR TEN OF THEM,
SAY OF SOMETHING THAT LOOKS
LIKE THAT OR LIKE THAT,
AND SEE IF YOU CAN GET
THEM TO FIT TOGETHER
TIGHTLY AND SNUGLY.
YOU KNOW WHAT
THE ANSWER IS?
THEY WILL.
YOU JUST HAVE
TO WORK WITH IT.
SO ABSOLUTELY ALL
QUADRILATERALS
WILL TILE THE SURFACE
WITHOUT ANY GAPS.
SO THAT'S NOT A BAD
EXPERIMENT TO WORK WITH
AT SOME POINT.
NOW THIS IS ONE OF THE
IMPORTANT LESSONS
TO DO WITH ESCHER'S ART.
HE PRODUCES IMAGES
WHICH TILE THE SURFACE.
THEY'RE QUITE COMPLEX,
BUT THEY ALWAYS TILE THE
SURFACE AND THERE ARE NO
GAPS BETWEEN HIS SHAPES.
ONCE HE SETS UP A SHAPE,
HE USES IT OVER AND OVER
AND OVER AGAIN, WHICH
LEADS ME TO THE SECOND
PART HERE, WHICH IS,
TRANSFORMATIONS.
AND TRANSFORMATIONS,
THERE WE ARE.
TRANSFORMATIONS IS THE
SECOND PART OF THIS,
AND WHAT I'M GOING TO
DO AT THIS POINT IS,
I'M GOING TO ASK YOU TO
PHONE IN AND I'M GOING TO
USE A COUPLE OF OBJECTS
HERE TO SIMULATE
SOME TRANSFORMATIONS.

On a tabletop marked out in a grid of irregular squares and rectangles with a dot at the centre, he lays down two plastic assembly kit toys, each composed of a yellow pentagon, a red square and a blue equilateral triangle.

Mister C. says AND I'M ONLY BASICALLY
GOING TO ASK YOU TO PHONE IN
AND LET ME KNOW WHAT
TRANSFORMATION - WHOOPS,
DON'T WANT TO BREAK
MY LITTLE PIECE OUT.
THERE WE GO.
CAN YOU TELL ME WHAT
TRANSFORMATION THIS ONE IS,
WHAT KIND OF
TRANSFORMATION?
HOW WOULD YOU
DESCRIBE IT?
PHONE IN, POUND 9, AND TELL ME

THIS IS THE OBJECT,
THIS IS THE IMAGE.
WHAT TRANSFORMATION
DID I USE TO GET THIS
FROM HERE TO HERE?
OKAY, WE DO HAVE A CALL.
AND WE'LL BE CONNECTING -
IT TAKES A FEW MOMENTS,
NOT TOO LONG.
FROM HERE TO HERE,
WHAT TRANSFORMATION?
HELLO.

A boy's voice says HELLO.

Mister C. says YOUR NAME IS?

Scott says SCOTT.

Mister C. says HI, SCOTT.
WHAT'S THIS
TRANSFORMATION HERE?

Scott says A PENTAGON?

Mister C. says OKAY, YOU'RE TELLING
ME WHAT THE SHAPE
IS IN THE CENTRE.
IT'S A PENTAGON.
BUT THE TRANSFORMATION
THAT MOVES THIS FROM
HERE TO HERE, WHAT
WOULD YOU CALL THAT?
TRANSFORMATION
MEANS MOVEMENT.
HOW HAS THIS MOVED
FROM HERE TO HERE?

Scott says SLIDE.

Mister C. says A SLIDE.
DO YOU KNOW ANOTHER
NAME OF A SLIDE?
YOU'RE ABSOLUTELY RIGHT.
IT'S CALLED A
TRANSLATION.
A TRANSLATION FROM HERE
TO HERE, OR A SLIDE.
WELL, THANK YOU
VERY MUCH.
WE GOING TO GO TO
ANOTHER ONE NOW AND SEE
IF SOMEBODY CAN
RECOGNIZE THIS ONE.
THIS ONE, I GOT TO DO A
LITTLE BIT OF MOVING STUFF
AROUND BUT IT'S NOT
REALLY A BIG PROBLEM.

He reassembles one of the plastic toys, then places it facing the other.

Mister C. says WHAT'S THIS
ONE CALLED?
WHOOPS, STRAIGHTEN
THAT UP A BIT.
THAT LOOKS BETTER.
WHAT'S THIS
ONE CALLED?
SO IT'S ALSO A
TRANSFORMATION
BUT IT'S DIFFERENT
FROM THE FIRST ONE.
AND WE'VE GOT SOMEBODY ON
THE LINE AND YOUR NAME IS?

A voice says JOEL.

Mister C. says HI, JOEL.
WHAT'S THIS
TRANSFORMATION?

Joel says IS IT A REFLECTION?

Mister C. says IT'S A REFLECTION.
HEY, FANTASTIC, BECAUSE
YOU'VE USED THE BIG WORD
INSTEAD OF FLIP.
OKAY, IT IS A
REFLECTION.
IT'S A HORIZONTAL
REFLECTION,
AND WHAT DO YOU CALL --
IF I WERE TO DRAW
SOMETHING IN HERE,
(He places a pen between the two toys)
WHAT'S THAT CALLED?
(pause)
THE LINE OF?

Joel says SYMMETRY?

Mister C. says THE LINE OF
REFLECTION, ACTUALLY,
ALTHOUGH THESE TWO
THINGS ARE SYMMETRIC.
IT'S THE LINE OF
REFLECTION, OKAY?

JOel says ALL RIGHT.

Mister C. says AND WHEN YOU DO
REFLECTIONS,
THERE'S ALWAYS A LINE
SOMEWHERE ON THE PLANE
THAT WILL SHOW WHERE
THE MIRROR IMAGE IS
OR WHERE THE MIRROR
IS ACTUALLY.
AND THAT WILL
ALWAYS BE THE CASE.
I'M GOING TO TRY
ONE MORE.
SET THIS ONE UP.
WHAT'S THIS
ONE CALLED?

He sets up the two toys pointing toward the dot at the centre of the grid.

Mister C. says AND WE HAVE A CALL
COMING IN FROM COLLEGE,
SO WE SHOULD HAVE
AN ANSWER SHORTLY.
WHAT'S THIS
ONE CALLED?
IT TAKES A
MOMENT OR TWO.
IT ALWAYS TAKES
A BIT OF TIME.
HELLO.

A voice says ROTATION.

Mister C. says IT'S A ROTATION AND
WHAT'S YOUR NAME?

Joel says JOEL.

Mister C. says HI, JOEL, AGAIN.
JOEL, WHAT'S THIS SPOT
CALLED WHEN YOU'RE
DOING A ROTATION? -
IT'S CALLED THE
CENTRE OF ROTATION.
IN THIS CASE IT
IS ANYWAY.
SO THAT'S THE
CENTRE OF ROTATION.
CAN YOU TELL ME
ABOUT WHAT ANGLE
THIS IS ROTATED
THROUGH?

Joel says 90 DEGREES.

Mister C. says PRETTY CLOSE.
I'D SAY THAT'S
REALLY CLOSE.
ONE OTHER THING,
IS IT CLOCKWISE OR
COUNTERCLOCKWISE, IF
THIS IS THE OBJECT
AND THAT'S THE IMAGE.

Joel says COUNTERCLOCKWISE.

Mister C. says RIGHT ON.
GOOD GOING, JOEL.
SO, YOU'VE GOT THREE
DIFFERENT KINDS
OF TRANSFORMATIONS.
ONE'S THE
TRANSLATION.
ANOTHER ONE IS
THE REFLECTION,
AND THE THIRD ONE
IS A ROTATION.
GET ALL MY BITS AND PIECES
BACK TOGETHER AGAIN.

A sheet of paper reads on four lines, from top to bottom - Transformations, Translation - Slide, Reflection - Flip, Rotation - Turn.

Mister C. says THERE ARE THE THREE
KINDS OF TRANSFORMATIONS
WE'RE CONCERNED ABOUT.
NOW, WHAT DOES THIS HAVE
TO DO WITH ESCHER'S ART?
THERE'S ONE OTHER THING
THAT I WANT TO DEAL WITH,
WITH TRANSFORMATIONS
THAT'S REALLY IMPORTANT,
AND I'VE GOT A QUESTION
THAT'S GOING TO LEAD ME
UP TO THAT LAST LITTLE
BIT OF DESCRIPTION.
SO WE'RE GOING TO GO
TO THE LAST QUESTION
AND THERE YOU ARE.

A sheet of paper is titled “Question Number 3.” Mister C. reads off it.
The gray column with the blue bar apears on the right.

He says TWO FIGURES ARE CONGRUENT IF THEY ARE - 1. DIFFERENT IN SIZE AND SHAPE, 2. DIFFERENT IN SIZE BUT THE SAME SHAPE, 3. DIFFERENT IN SHAPE BUT THE SAME SIZE, 4. THE SAME SIZE AND SAME SHAPE.
NOW, THE KEY THING HERE
IS THE WORD CONGRUENT.
SO LET'S SEE IF WE CAN
GET THIS UP TO 70 PERCENT.
DEFINITION OF CONGRUENT.
SEE IF WE CAN GET
A FEW MORE ANSWERS.
OH, WE GOT A
COUPLE MORE THERE.
SO WHAT WE'RE GOING TO DO
IS TAKE A LOOK AT YOUR GRAPH.

The graph shows 9 answers in the last column which is red,and two in the penultimate one which is blue.

Mister C. says AND YOUR GRAPH - HEY,
YOU'RE ABSOLUTELY RIGHT.
THERE IS NO DOUBT THAT
IN THIS PARTICULAR CASE,
THE WORD “CONGRUENT” MEANS
THE SAME IN SIZE AND SHAPE.
NOW, WANT TO GO BACK.
THIS IS ONE THING THAT'S
TRUE OF ESCHER'S ART
FOR THE MOST PART, AT LEAST
THE ESCHER'S ART
WE'RE GOING TO DEAL WITH.
THESE TWO OBJECTS ARE
EXACTLY THE SAME SHAPE AND
THE SAME SIZE, AND IN
THE TRANSFORMATIONS
THAT YOU EMPLOY, WHATEVER
SHAPE YOU CREATE,
STAYS THE SAME EXACTLY
IN TERMS OF ITS EXTERNAL
DRAWING AND AT THE SAME
TIME IT MAINTAINS ITS SIZE.
NOW, ESCHER EVENTUALLY DID
EXPERIMENT WITH SHAPES
THAT DECREASED IN SIZE AS
YOU MOVED TO THE EDGE
OR INCREASED, PERHAPS, IN
SIZE AS YOU MOVED TO THE EDGE.
BUT WE'RE NOT GOING
TO DO THAT KIND
OF TRANSFORMATION.
NOW, SO IT'S IMPORTANT TO
UNDERSTAND THE ATTRIBUTES
OF TRANSFORMATIONS, AND
HERE ARE JUST A FEW
REALLY QUICKLY.

A sheet of paper is titled “Attributes, and he reads off it.

Mister C. says THE DIRECTION
OF THE OBJECT.
IN A TRANSLATION, THE
DIRECTION OF THE OBJECT
IS EXACTLY THE SAME, SO THE
ORIENTATION IS THE SAME.
BUT IN A REFLECTION,
THEY POINT IN OPPOSITE
DIRECTIONS, SO THE
ORIENTATION CHANGES.
LIKEWISE WITH A ROTATION.
WHEN YOU DO A REFLECTION,
THERE IS A MIRROR IMAGE
AND THERE IS A LINE OF
REFLECTION BETWEEN
THE TWO OF THEM.
THE CONGRUENCE IDEA THAT
EVERY TIME WE DO ONE
OF THESE TRANSFORMATIONS,
THE SIZE AND
THE SHAPE ARE RETAINED.
THEY ARE THE SAME.
AND WITH A ROTATION, EVERY
SINGLE POINT IN THE OBJECT
IS THE SAME DISTANCE
FROM THE CENTRE POINT
IN THE ROTATION
AS YOU ROTATE,
AND THAT'S A CRITICAL
IDEA AS WELL.
SO THERE'S SOME ATTRIBUTES.
NOW, WHAT I'M GOING TO
DO RIGHT NOW IS GET YOU
TO DO A LITTLE WEE
BIT OF WORK FOR ME
AND SEE WHAT
HAPPENS HERE.
HERE'S WHAT WE'RE
GOING TO DO.
I MADE A REQUEST THAT YOU
BRING SCISSORS AND TAPE
TO THIS PARTICULAR CLASSROOM.
AND I'M JUST PUTTING
MY LITTLE PIECE BACK
TOGETHER AGAIN.
THAT'S PRETTY CLOSE.

He assembles an arrow-shaped black ribbon-end tag on a light blue square of paper.

Mister C. says WHAT I WOULD LIKE YOU TO
DO IF YOU HAVE SCISSORS
AND TAPE THIS MORNING -
I'M ONLY GOING TO GIVE
YOU A MINUTE TO DO
THIS - IS TO CUT OUT
A SQUARE THAT'S
ABOUT 10 CENTIMETRES BY 10 CENTIMETRES.
THEN DRAW SOME KIND
OF OBJECT ON ONE SIDE
AND CUT IT OUT.
ONCE YOU CUT IT OUT,
IN THE FIRST INSTANCE,
ALL I WOULD LIKE YOU TO DO
IS TO SLIDE IT - THERE'S
A TRANSLATION: SLIDE
IT STRAIGHT ACROSS
AND TAPE IT TO
THE FAR SIDE.
SO I'M GOING TO GIVE YOU
ABOUT A MINUTE TO DO THIS,
AND WHAT YOU'RE
DOING IS CREATING
AN INTERESTING NEW SHAPE.
THE QUESTION OF COURSE
IS DOES IT TILE OR TESSELLATE?
SO WHAT WE'RE GOING TO
DO IS CUT A SQUARE OUT,
THEN CUT A
PIECE OUT OF IT,
WHATEVER SHAPE YOU WANT.
SLIDE IT STRAIGHT ACROSS
AND TAPE IT TO THE FAR SIDE.
NOW, WHILE YOU'RE DOING
THAT - I'M GOING TO GIVE
YOU ONE MINUTE TO WORK ON
THAT - I'M GOING TO PUT
A DRAWING, AN ESCHER PIECE,
UNDER THE CAMERA FOR YOU
TO LOOK AT WHILE I'M AWAY.
SO YOU HAVE ONE
MINUTE TO DO THAT.
CUT OUT A SQUARE.
CUT OUT A PIECE AND TAPE
IT TO THE FAR SIDE.
SO ONE MINUTE FROM
NOW, I'LL BE BACK.

The Escher picture shows a tessellation of black, brown and white birds.
(One minute passes)

Mister C. says OK, I'M GOING TO SHOW YOU WHY I
WAS GETTING YOU TO DO THAT.
THE IDEA IS THAT IF
YOU DO ONE OF THESE,
YOU CAN TAKE THE OBJECT
ITSELF ONCE YOU'VE GOT IT,
YOU CAN ACTUALLY TRACE IT
ON A PIECE OF PAPER AND
YOU COULD TRACE
COPIES OF IT.
THE QUESTION IS, CAN YOU
MAKE THESE COPIES FIT
EXACTLY TOGETHER, AND I
THINK IT SHOULD BE PRETTY
SELF EVIDENT THAT IT DOES.
AND WHAT I'M GOING TO DO
TO DEMONSTRATE THAT FACT IS -
I DID THIS IN COMPUTER
WAY SO TO SPEAK -
BUT BASICALLY I
COPIED THOSE IMAGES.
HERE'S THE CUT-OUT PIECE.
PASTED IT OVER THERE
AND, AS YOU CAN SEE,
WHEN YOU HAVE THAT
PIECE CONNECTED,
IT'S GOING TO FIT RIGHT
OVER THAT PIECE EXACTLY.

A sheet of paper shows a grid pattern made with the square and cut out piece. Then a new drawing shows the square rotated and the cut out piece attached on the same side as the cutout mark, with a dot at the mid-point between the piece and the cutout mark.

Mister C. says THERE WON'T BE ANY
GAPS WHATSOEVER.
NOW, WHAT I'D LIKE YOU TO
DO IS TO PHONE IN AND TELL ME
WHAT TRANSFORMATION I
USED TO MAKE THE SHAPE -
THAT IS, I TOOK THIS CUT-OUT
PIECE AND PUT IT HERE.
WHAT WAS MY
TRANSFORMATION?
BECAUSE I'M GOING TO SHOW
YOU THAT ESCHER IN FACT
ONLY USED FOUR DIFFERENT
KINDS OF TRANSFORMATIONS.
FOUR'S GOING TO BE
A BIT OF A TRICK,
BUT IT'S VARIATIONS ON
A THEME, WE SHALL SAY.
SO WHAT I'D LIKE TO
KNOW IN THIS ONE IS,
WHAT TRANSFORMATION DOES
HE USE IN THIS PARTICULAR
KIND OF TESSELLATION?
SO WE GOT A CALL COMING
IN FROM COLLEGE AVE.
AND WE'LL FIND OUT
IN JUST A MOMENT.

A voice says HELLO.

Mister C. says HI, AND YOU'RE NAME?

Scott says SCOTT.

Mister C. says HI, SCOTT, WHAT
TRANSFORMATION DID WE USE
IN THIS PARTICULAR ONE?

Scott says A SLIDE.

Mister C. says IT'S A SLIDE OR
A TRANSLATION.

Scott says YEAH.

Mister C. says THANK YOU VERY MUCH.
NOW, WATCH THIS
ONE CLOSELY.
THE NEXT ONE, YOU CAN
ALSO DO THIS WITH
A PIECE OF PAPER.
WHAT WE DID IN
THIS CASE IS,
WE CUT OUT A
PIECE HERE.
THERE'S A MARK
HERE OF SOME SORT,
AND WE RE-ATTACHED IT IN
THIS ORIENTATION OVER HERE.
SO THE QUESTION IS, WHAT
IS THE TRANSFORMATION
THAT I EMPLOYED
HERE AND WILL THESE
CONNECT TO ONE ANOTHER
IF IT'S DONE CAREFULLY?
SO, SCOTT, ARE
YOU STILL THERE?
ARE YOU STILL
THERE, SCOTT?

Scott says WHAT?

Mister C. says ARE YOU STILL THERE?

Scott says YES.

Mister C. says YOU WANT TO ANSWER
THIS QUESTION AS WELL.

Scott says SURE.

Mister C. says OKAY.
CAN YOU TELL ME WHAT
TRANSFORMATION
I'M PLAYING WITH
ON THIS ONE?

Scott says A ROTATION.

Mister C. says IT'S A ROTATION,
EXCELLENT.
NOW, CAN YOU DESCRIBE
WHERE THE CENTRE
OF THE ROTATION IS?
EXACTLY.
WHERE IS THAT CENTRE?

Scott says WHERE YOUR FINGER IS.

Mister C. says OH, YEAH, OKAY, I'LL
TAKE MY FINGER AWAY.
KEEP GOING.

Scott says HALFWAY IN BETWEEN.

Mister C. says IT'S HALFWAY
IN BETWEEN ON?

Scott says THE LINE.

Mister C. says ON THE EDGE.
EXACTLY, SO THIS IS A
ROTATION ABOUT
THE HALFWAY POINT
ON AN EDGE.
NOW, I'M GOING TO SHOW YOU
THAT THESE THINGS WILL CONNECT
AND HERE'S HOW THEY DO
CONNECT TOGETHER.
NOW, I'M GOING TO ASK
THAT MY PIP BE REMOVED
JUST MOMENTARILY BECAUSE
UNFORTUNATELY -
SORRY FOR THE SLIGHT DELAY
THERE, BUT WHAT YOU FIND IS
WHEN YOU PUT THIS TWO
SHAPES BACK TOGETHER AGAIN,
THAT IN FACT THEY WILL FIT
TOGETHER BUT YOU HAVE
TO FLIP THE WHOLE SHAPE TO
GET THEM TO CONNECT TOGETHER.
AND WHAT YOU END UP WITH
IS A PURE RECTANGLE AND,
OF COURSE, PURE RECTANGLES
WILL TILE THE SURFACE.
SO YOU CAN SEE THAT THIS
KIND OF A TRANSFORMATION
CAN BE USED TO TILE
SURFACES AS WELL,
SO IT'S A ROTATION ABOUT
THE MIDPOINT OF AN EDGE.
LET'S TRY ANOTHER ONE.
LET'S TRY THIS ONE.

A new sheet of paper shows the piece attached laterally - neither on the same side, nor on the opposite side of the cut out, with a dot on the corner between the two sides.

Mister C. says CAN YOU DESCRIBE
THIS TRANSFORMATION?
PHONE IN IF YOU KNOW WHAT
THIS TRANSFORMATION IS.
WE'LL SAVE THE
BEST FOR LAST.
YES, THERE'S A
CALL COMING IN
SO WE'LL CONNECT VERY
SHORTLY AND I THINK
WE'RE LOOKING TO A STUDENT
FROM FLAMBOROUGH. GREAT.
WHAT IS THIS
TRANSFORMATION CALLED?
WOULD THE FACILITATOR AT
FLAMBOROUGH CALL
THE HELP DESK AT
1-888-371-1917?
THERE MAY BE A LITTLE
BIT OF DIFFICULTY HERE
IN OUR MAKING A
CONNECTION.
I'D CERTAINLY LIKE TO
TALK TO A STUDENT
FROM FLAMBOROUGH
AT SOME POINT.
OKAY, I'LL PUT THE QUESTION
OUT TO EVERYBODY AGAIN
AND I THINK WE HAVE
A STUDENT CONNECTING
FROM COLLEGE AT
THIS POINT.
WHAT IS THIS PARTICULAR
TRANSFORMATION?
IT'S SIMILAR TO THE LAST
ONE BUT SLIGHTLY DIFFERENT
AND WHAT'S THE
DIFFERENCE?
THAT'S REALLY WHAT
WE'RE AFTER HERE.
HELLO AND WHO AM
I SPEAKING TO?

A voice says CHRIS.

Mister C. says HI, CHRIS.
WHAT IS THIS
TRANSFORMATION?

Chris says IT'S A 90-DEGREE
COUNTERCLOCKWISE ROTATION.

Mister C. says I REALLY LIKE
YOUR DESCRIPTION.
WHERE IS THE CENTRE?
THE CENTRE
OF ROTATION.

Chris says ON THE CORNER
OF THE SQUARE.

Mister C. says ON THE CORNER
OR THE VERTEX.
EXCELLENT.
YOU DID A BEAUTIFUL
JOB OF THAT.
SO IT'S A 90-DEGREE
COUNTERCLOCKWISE ROTATION
ABOUT THE VERTEX
OF THE SQUARE.
VERY WELL DONE.
THANK YOU VERY MUCH.
OK, I'M GOING TO REALLY
TEST YOU -
OH, DO THOSE CONNECT?
WELL, I GUESS I SHOULD
DO THAT, AS WELL.
HERE WE GO.

A sheet of paper shows a counter-clockwise rotating pattern in which the four squares with the attached pieces fit snugly together.

Mister C. says THEY CONNECT AND WHAT
HAPPENS IS THAT YOU GET
THIS SORT OF - THEY
ROTATE RIGHT AROUND WHEN
THEY CONNECT TO EACH
OTHER AND RE-CREATE
A LARGER SQUARE.
SO THAT'S KIND
OF NEAT, AS WELL,
BUT THEY ALL CONNECT
AND OF COURSE SQUARES
TESSELLATE SO THERE'S
NO PROBLEM WITH THAT.
WELL, LET'S GO TO THE LAST
ONE AND THAT'S THIS ONE.
AND I'M GOING TO PUT IT IN
A DIFFERENT ORIENTATION
HERE BECAUSE IT'S A LITTLE
BIT EASIER TO FIGURE IT OUT.
NOW, WHAT EXACTLY HAPPENED
WITH THIS PARTICULAR SHAPE?

The new drawing shows the piece attached to the side opposite the cut out, but pointing in the opposite direction.

Mister C. says WHAT EXACTLY HAPPENED ON
THIS PARTICULAR SHAPE?
I DO SEE THAT WE
HOPEFULLY HAVE A CALL.
WHAT'S THE TRANSFORMATION
OR TRANSFORMATIONS
THAT MOVE THIS PIECE -
BASICALLY THAT PIECE
FROM HERE TO UP THERE?
SO WE'VE GOT A
CALL COMING IN.
I THINK IT'S
STILL COLLEGE.
HELLO, IS IT JOEL?

Joel says YUP.

Mister C. says OKAY, JOEL, NOW WHAT'S
THIS TRANSFORMATION?

Joel says A SLIDE AND A FLIP.

Mister C. says A SLIDE AND A FLIP, WELL
DONE, EXCELLENT, JOEL.
THERE'S A SPECIAL NAME FOR
IT AND I'M GOING TO PUT
IT ON THERE.
WHAT HAPPENS INITIALLY IS,
THIS IS FLIPPED LIKE THIS,
RIGHT, AND THEN IT'S
SLID UP HERE, RIGHT?

Joel says YES.

Mister C. says OKAY, THERE'S A VERY
SPECIAL NAME FOR THIS,
JOEL, AND I JUST WANT
TO MAKE A NOTE OF IT.
IT'S CALLED A
“GLIDE REFLECTION.”
THANK YOU VERY MUCH, JOEL,
THAT WAS VERY NICELY DONE.
IT'S NICE TO BE ABLE TO
IDENTIFY MORE THAN ONE.
SO BASICALLY ESCHER
USES FOUR DIFFERENT
TRANSFORMATIONS.
HE USES STRAIGHT
TRANSLATIONS.
HE USES ROTATION ABOUT THE
CENTRE POINT OF AN EDGE.
HE USES A ROTATION
ABOUT A VERTEX.
AND HE ALSO USES
GLIDE REFLECTIONS.
THAT'S BASICALLY IT.
THEY'RE THE ONLY ONES THAT
HE CAN USE AND STILL TILE
THE SURFACE AND HE WORKED
FOR YEARS AND YEARS
TO DETERMINE WHICH ONES WOULD
WORK AND WHICH ONES WOULD NOT.
NOW, WHAT I'M GOING TO DO IS,
I'M GOING TO SPEND A
LITTLE WEE BIT OF TIME NOW
INTRODUCING THE ASSIGNMENT
FOR NEXT WEEK AND A LITTLE
WEE BIT OF TIME DISCUSSING
THE PROJECT THAT YOU HAVE
TO DO FOR THE END OF THIS.
AND ONCE WE'RE DONE
THAT, I'LL CERTAINLY
BE MORE THAN HAPPY TO
TAKE YOUR QUESTIONS.
LET'S START JUST WITH
THE ASSIGNMENT AND
I'M NOT GOING TO GET
INTO TOO MUCH DETAIL.

A sheet of paper appears titled “Discovering the Mathematics in Escher's Art.” Other sheets show dot paper with squares marked on it, and without. then he produces other diagrams showing variants on the cutout theme.

Mister C. says YOU SHOULD HAVE THIS IN
YOUR HANDS BY THE END
OF THE PERIOD TODAY.
SO, THERE ARE A COUPLE OF
ACTIVITIES THAT I START YOU
WITH AND THIS FIRST
ONE INVOLVES JUST USING
TRANSLATIONS AND TILING
THE SURFACE USING
THOSE TRANSLATIONS
WITH SOME DOT PAPER.
YOU WILL BE PROVIDED WITH
DOT PAPER THAT'S GOT SQUARES
ALREADY SET UP
IN ADVANCE.
AND THIS DOT PAPER ALLOWS
YOU TO EXPERIMENT,
TO GO BEYOND THE
SQUARE, PERHAPS TO DO
A PARALLELOGRAMS, MAYBE
EQUILATERAL TRIANGLES.
PERHAPS EVEN
REGULAR HEXAGONS.
AND WHEN YOU COMPLETE,
YOU'LL HAVE SOMETHING
THAT'S COLOURED AND WILL
LOOK SOMETHING LIKE THIS
EVENTUALLY.
NOW, IN ACTIVITY #2,
WE ASK YOU TO DO TWO
SHAPES AT THE SAME
AND EMPLOY
TRANSLATIONS AS WELL.
AND YOU END UP CREATING
A PIECE THAT MIGHT
LOOK SOMETHING LIKE THIS.
REMEMBER, YOUR
CREATIVITY COUNTS,
SO YOU WANT TO DO
SOMETHING DIFFERENT
FROM WHAT I HAVE.
BUT THE IMPORTANT
THING IS,
IS THAT I WANT YOU TO
GO BEYOND AND DO
THE EXTENSIONS AS WELL.
THE FIRST EXTENSION IS
ROTATION ABOUT A MIDPOINT
ON THE SIDE AND YOU'RE
ALREADY AWARE OF THAT.
THE SECOND'S ROTATION
ABOUT A VERTEX.
THE THIRD IS A
GLIDE REFLECTION,
AND IF YOU WANT TO TRY
SOME UNUSUAL SHAPES,
YOU CAN TRY EITHER THE
HEXAGON OR THE TRIANGLE.
NOW, YOUR RESULTS
FROM THIS ASSIGNMENT,
IF YOU CAN HAVE THEM
HANDED IN TO YOUR TEACHER
BEFORE FRIDAY, I WOULD
LOVE TO HAVE YOU
FAX THEM TO ME, AND
WHAT WE'LL DO IS,
WE'LL PUT UP A SLIDE RIGHT
NOW TO SHOW YOU WHERE
WE FAX IT TO.

On a green background a slide titled “Contact Info.” shows phone and fax numbers.

Mister C. says SO HAVE YOUR TEACHER FAX
YOUR COPIES OR COPIES
OF YOUR WORK.
YOU DON'T HAVE TO SEND
THE WHOLE ASSIGNMENT BUT
WHAT'S REALLY GOOD, MAKE
SURE THE STUDENT NAME
IS ON IT, AND THE
FAX NUMBER IS ONE, EIGHT EIGHT EIGHT,
FIVE TWO TWO, SEVEN ONE FOUR ONE.
AND WE'LL ALSO PUT THAT UP
AT THE END OF THE SHOW
SO THAT IF YOU DON'T GET
IT RIGHT THIS INSTANT,
YOU CAN COPY
IT DOWN LATER.
I'LL REPEAT IT
ONE MORE TIME.
(REPEATS)
AND WE'LL SEE THAT RIGHT
AT THE END OF THE PROGRAM.
NOW, I'M GOING TO DESCRIBE
TO YOU THE MAIN PROJECT
AND THERE'S ONLY ONE
LITTLE PIECE I WANT
TO SHOW YOU ON THIS.
YOU'LL RECEIVE THIS
HANDOUT AS WELL.

A drawing within a larger rectangle, captioned “Title,” shows a round-cornered rectangle in the middle with four smaller rectangles around it, one on each side and two at the bottom. Another paper is captioned “Student Project, Due Date - Tuesday April 6.”

Mister C. says AND THE THING WE'RE ASKING
YOU TO DO IS CREATE
A POSTER WHICH WILL
BASICALLY HAVE FIVE THINGS
ON IT AND THE TITLE,
AND I'LL DESCRIBE
IN SOME DETAIL
WHAT I WANT.
THE FIRST THING I
OBVIOUSLY HAVE TO
DEAL WITH IS
THE DUE DATE. IT'S TUESDAY APRIL 6.
NOW, WE'VE ALREADY TALKED
TO YOUR TEACHERS AT LEAST
THROUGH THE IN-SERVICE
THAT WHAT WE'RE GOING TO DO
IS WE'RE GOING TO ASK
THAT PHOTOGRAPHS OF YOU
AND YOUR WORK BE
TAKEN ON THAT DATE,
PREFERABLY WITH
A DIGITAL CAMERA.
AND IF THEY ARE TAKEN
WITH A DIGITAL CAMERA,
THEN IN FACT THEY
CAN BE EMAILED TO US,
AND I THINK WE DO HAVE
AN e-mail ADDRESS
THAT WE CAN PUT UP ON THE
SCENE, SO WE'LL DO THAT.

On a green background, a slide headed “Digital photographs are preferred” reads the e-mail address.

Mister C. reads the e-mail address and says
NOW, IF YOU HAVE
THAT DIGITAL CAMERA,
IT'LL MAKE THINGS
REALLY, REALLY EASY.
BECAUSE WHAT WE'RE GOING
TO DO IS THE FINAL TAKE-UP
IS THAT ONCE WE GET ALL
YOUR PICTURES ARRANGED,
WE WILL SHOW THEM
INDIVIDUALLY AND I WILL
ASK THE STUDENT OR
STUDENTS WHO DID
THE WORK TO TALK
ABOUT WHAT THEY DID.
NOW, I'M GOING TO
REPEAT ONE OTHER THING.
THE FINAL PROJECT CAN
BE DONE INDIVIDUALLY
OR DONE WITH A PARTNER.
I DON'T REALLY CARE WHICH
AND IT'S UP TO YOUR TEACHER
WHICH WAY HE OR SHE WOULD
LIKE TO ARRANGE THAT.
NOW, SO TO GET THE STUFF
ON AIR, ONE MORE TIME,
THE DUE DATE IS TUESDAY APRIL 6,
AND IT HAS TO BE
EMAILED TO US.
IF YOUR SCHOOL DOES NOT
HAVE A DIGITAL CAMERA,
THEN PHONE US AND
WE'LL TRY TO MAKE OTHER
ARRANGEMENTS SO THAT WE
CAN GET YOUR WORK ON AIR.
NOW, THAT POSTER
IN PARTICULAR,
WHAT DOES IT INCLUDE?

A new sheet of paper reads “Create a poster on a piece of Bristol Board wbich includes all of the following - 1. A sample of a tiling which employs a “translation.” 2. A sample of a tiling which employs a “rotation about the midpoint of a side.” 3. A sample of a tiling which employs a “rotation about a vertex.” 4. A sample of a tiling which employs a “glide reflection.” 5. A sample of a complex tiling of shapes. These shapes could be birds, lizards, fish, buildings, etc. Each tile must resemble an animal or object and cannot be “purely” abstract. Use appropriate detail, inclujding effective use of colour, to enhance your tiling. This is your opportunity to create your own “Escher-like” art piece.”

Mister C. says THERE ARE FIVE SPOTS
ON THE BOARD ITSELF.

He reads points one through four off the slide.

Mister C. says SO IN OTHER WORDS, ALL
THAT WORK YOU'RE DOING
IN THIS ASSIGNMENT COMING
UP FOR NEXT MONDAY,
YOU'RE GOING TO TAKE THAT
WORK AND REFINE IT AND PUT IT
RIGHT ONTO YOUR POSTER;
THEY ARE REALLY TWO
DIFFERENT THINGS, BUT
YOU'RE PREPARING FOR IT
BY DOING THE ASSIGNMENT
THIS WEEK.
THE IMPORTANT PART THOUGH,
THE PART THAT YOU'RE
NOT DOING IN THE
ASSIGNMENT IS THIS PART,
WHICH SHOULD
DOMINATE YOUR POSTER.

He reads point five off the slide.

Mister C. says SO IN OTHER WORDS, WHAT
I'M ASKING YOU TO DO
IS TO ACTUALLY “BE ESCHER.”
BE AN ARTIST, AND CREATE A
TILING WITH CONGRUENT SHAPES,
BUT YOU CAN ADD AS MUCH
DETAIL TO THOSE SHAPES
AS YOU WANT TO
CREATE THE IMAGES OF FISH,
LIZARDS, INSECTS, BIRDS,
BUILDINGS, WHATEVER.
NOW, I'VE SAID THAT EACH
OBJECT MUST BE RECOGNIZABLE AND NOT PURELY ABSTRACT.
THE ESCHER PRINTS THAT I
LIKE THE BEST ARE USUALLY
THE ONES THAT ARE
MOST COLOURFUL. AND THIS IS YOUR CHANCE
TO DO ART, SO I HOPE THAT YOU REALLY
PUT A LOT OF EFFORT INTO THIS.
I AM GOING TO INDICATE
TO YOU LEVEL 4.

A new sheet of paper reads “You can evaluate the project using this rubric (only Level 4 shown). He reads off it.

LEVEL 4 SIMPLY SAYS,
IF I WERE TO GIVE YOU
THE HIGHEST MARK I
COULD GIVE YOU,
WHAT WOULD THIS PIECE
OF WORK LOOK LIKE?
WELL, THE POSTER HAS TO BE WELL ORGANIZED AND EYE-CATCHING.
SO I'LL BE ABLE TO TELL THAT
FROM YOUR PHOTOGRAPHS.
EACH ONE OF THE FIRST FOUR TILINGS IS CORRECT IN ITS
USE OF THE TRANSFORMATIONS AND HIGHLY PRECISE.
THAT WORD “HIGHLY” IS IMPORTANT.
SO YOU'VE DONE A REALLY
CAREFUL JOB AND YOU'VE USED
THE APPROPRIATE
TRANSFORMATION IN EACH ONE
OF THE FIRST FOUR.
THE ESCHER-LIKE RTILING IS MADE UP OF
OBJECTS WHICH ARE EASILY IDENTIFIABLE.
THE ESCHER-LIKE TILING IS ESTHETICALLY PLEASING,
BOTH IN THE USE OF COLOUR AND THE QUALITY OF THE DESIGN.
SO I'M REALLY LOOKING FOR
SOMETHING THAT IS AN ART PIECE.
AND LAST, BUT NOT
LEAST, I'M LOOKING FOR
ORIGINALITY, THAT IS,
IN ALL OF THE TILINGS.
ORIGINALITY REALLY DOES
COUNT FOR SOMETHING.
IT SHOWS THAT YOU PUT
SOMETHING OF YOURSELF
INTO YOUR WORK.
NOW, I'LL INVITE YOU FOR
THE LAST COUPLE OF MINUTES
IF YOU HAVE ANY QUESTIONS
ABOUT THE ASSIGNMENT
OR THE PROJECT, NOW'S
YOUR CHANCE TO PHONE IN.
SO IF YOU GOT A
QUESTION, BY ALL MEANS
PHONE IN RIGHT NOW.
(PAUSE)
FLAMBOROUGH, THIS
MAY BE A CHANCE.
I UNDERSTAND THAT YOU
ARE ABLE TO NOW CONNECT,
SO MAYBE YOU
HAVE A QUESTION,
ALTHOUGH I SEE SOMEBODY
FROM COLLEGE PHONING IN.
SO IF YOU HAVE
ANY QUESTIONS,
NOW'S THE TIME TO ASK
ME AND HOPEFULLY
I HAVE AN APPROPRIATE RESPONSE.
AND I THINK WE'LL BE
CONNECTED IN JUST A MOMENT.
HELLO.

A voice says WHAT EXACTLY DO YOU
WANT FOR THIS FRIDAY?

Mister C. says FOR THIS FRIDAY, YOU'VE
GOT A HANDOUT ASSIGNMENT
WHICH HAS TWO
ACTIVITIES PLUS I THINK
IT'S THREE EXTENSIONS.
SO IT ASKS FOR
SOME TILINGS,
SOME PICTURES OF TILINGS.
IF YOU WERE AN ARTIST - LET'S
PUT IT TO YOU THIS WAY -
YOU DO SOME
PRELIMINARY DRAWINGS JUST
TO GET A SENSE OF
HOW IT ALL WORKS.
WELL, THAT'S WHAT THE
ASSIGNMENT'S ABOUT.
YOU'RE DOING SOME OF THOSE
PRELIMINARY DRAWINGS.
YOU'RE NOT DOING THE FINAL
REFINED PIECE OF WORK.
BUT THE ASSIGNMENT ALLOWS
ME TO MAKE SURE
THAT NEXT WEEK YOU UNDERSTAND
WHAT A TRANSLATION IS,
WHAT A ROTATION
ABOUT THE VERTEX IS,
THAT KIND OF THING.
SO IN FACT THIS ASSIGNMENT
ASKS YOU STEP BY STEP
TO DO SOME OF
THOSE TILINGS.
IS THAT OKAY?
DOES THAT MAKE
SENSE TO YOU?

The voice says YEAH.

Mister C. says YOU SHOULD HAVE THE FULL
HANDOUT AND IF YOU DO HAVE
ANY QUESTIONS, THEN YOU
CAN ACTUALLY HAVE
YOUR TEACHER CONTACT ME.

The voice says SURE.

Mister C. says OKAY?
ARE THERE ANY
OTHER QUESTIONS?
WELL, I DON'T SEE ANY SO
I'M LOOKING FORWARD
TO SEEING YOU SAME TIME,
SOME PLACE, NEXT WEEK.
SO LONG.

A green slate Reads “Contact Info - Phone - Fax, with the corresponding numbers.

A green slate appears on screen. It shows a text that reads “Please remember to log off! Pick up handset. Press pound then seven. Press 1 to confirm. Hang up handset. See you next time!”

Watch: Grade 7 - Intro & Transformational Geometry