Title: Geometry Applications: Rotations, Reflections, and Symmetry

Title: Geometry Applications: Rotations, Reflections, and Symmetry

[Music]

[Music]

ON THE SUMMIT OF MAUNA KEA IN HAWAII,

ASTRONOMERS GATHER TO STUDY THE STARS

AND BE DAZZLED BY THE HEAVENS.

A NUMBER OF OBSERVATORIES ARE LOCATED ATOP MAUNA KEA,

AND ONE OF THE MOST RECENT ADDITIONS

IS THE GEMINI OBSERVATORY.

MAUNA KEA PROVIDES SEVERAL ADVANTAGES

OVER OTHER LAND BASED OBSERVATORIES.

THE AIR IS CLEAR AND DRY AND THE ALTITUDE ATOP MAUNA KEA,

OVER 13,000 FEET ABOVE SEA LEVEL,

PROVIDES A CLOUDLESS VIEW OF THE SKY.

AND WHAT A VIEW, AS SEEN IN THIS NIGHT TIME

PHOTOGRAPH OF THE MILKY WAY.

THE GEMINI OBSERVATORY PROVIDES ONE MORE ADVANTAGE.

AS INDICATED BY ITS NAME, GEMINI HAS A TWIN

OBSERVATORY LOCATED IN THE COUNTRY OF CHILE.

ALSO ON A MOUNTAINTOP.

TOGETHER GEMINI NORTH AND SOUTH CAN SEE

ALMOST ALL OF THE STARS IN THE SKY.

AS YOU HAVE SEEN WITH OTHER CIRCULAR STRUCTURES

WE HAVE STUDIED, BUILDINGS IN THESE SHAPES HAVE CERTAIN

ADVANTAGES WHEN IT COMES TO VIEWING THE NIGHT SKY.

THE KIVAS AT CHOCO CANYON PROVIDE AN EXPANSIVE VIEW

OF THE NEW MEXICO SKY.

THE PANTHEON IN ROME PROVIDES A

CELESTIAL MAP OF THE CHANGING SEASONS

THROUGH THE LIGHT SHINING PAST THE OCULUS.

IN BOTH OF THESE CASES

THE ARCHITECTURE OF THE CIRCULAR STRUCTURE

ACCOUNTS FOR THE MOVEMENTS OF THE HEAVENS.

AN OBSERVATORY TAKES THIS IDEA ONE STEP FARTHER.

NOT ONLY DOES THE OBSERVATORY

TAKE THE MOTION OF THE EARTH, THE SOLAR SYSTEM,

AND THE GALAXY INTO ACCOUNT,

THE BUILDING ITSELF MOVES.

BECAUSE OF THE EARTH'S ROTATION, IN ORDER FOR

THE TELESCOPE TO VIEW A STAR, PLANET OR OTHER

CELESTIAL OBJECT FOR AN EXTENDED PERIOD OF TIME,

THE OBSERVATORY NEEDS TO MAINTAIN

A LINE OF SIGHT WITH THE OBJECT.

AN OBSERVATORY DOES THIS BY MEANS OF ROTATION.

AN OBSERVATORY IS BUILT SO THAT ITS BODY,

WHICH IS WHERE THE TELESCOPE IS LOCATED,

CAN ROTATE 360 DEGREES IN A DIRECTION

PARALLEL TO THE EARTH'S SURFACE.

THE TELESCOPE ITSELF CAN ROTATE

IN A VERTICAL DIRECTION.

THE OBSERVATORY'S ABILITY TO ROTATE TAKES ADVANTAGE

OF ANOTHER ASPECT OF A CIRCLE'S GEOMETRY:

ITS SYMMETRY.

THE GEOMETRIC PROPERTY OF SYMMETRY IS ONE THAT

HAS TO DO WITH AN OBJECT'S ABILITY TO MIRROR ITSELF.

FOR EXAMPLE, A SQUARE HAS FOUR LINES OF SYMMETRY.

NOTICE THAT ALONG EACH LINE THE SQUARE

IS SPLIT INTO MIRROR IMAGES OF ITSELF.

A REGULAR HEXAGON HAS SIX LINES OF SYMMETRY.

A REGULAR OCTAGON HAS EIGHT LINES OF SYMMETRY.

AS YOU CAN SEE, THERE IS A PATTERN TO THE NUMBER OF

SIDES AND THE NUMBER OF LINES OF SYMMETRY

FOR AN n SIDED FIGURE.

SO AS n APPROACHES INFINITY,

SO DOES THE NUMBER OF LINES OF SYMMETRY.

AND AS THE NUMBER OF SIDES INCREASES,

THE SHAPE OF A POLYGON

APPROXIMATES THAT OF A CIRCLE.

SO A CIRCLE HAS AN INFINITE NUMBER

OF LINES OF SYMMETRY.

AT ANY ANGLE OF ROTATION THE OBSERVATORY

HAS THE SAME ORIENTATION RELATIVE TO THE SKY.

THE INFINITE LINES OF SYMMETRY ALSO PROVIDE

A PANORAMIC VIEW OF THE SKY.

THE GEMINI OBSERVATORY'S VIEWING TOWER

HAS MOVABLE WALLS THAT EXPAND.

A CIRCULAR BUILDING ALSO HAS ROTATIONAL SYMMETRY.

AS THE NAME SUGGESTS, ROTATIONAL SYMMETRY

HAS TO DO WITH MAINTAINING AN OBJECT'S APPEARANCE

AFTER IT IS ROTATED IN A CERTAIN NUMBER OF DEGREES.

RETURNING TO THE EXAMPLE OF REGULAR POLYGONS,

A SQUARE HAS ROTATIONAL SYMMETRY OF ORDER 4.

IN OTHER WORDS, A SQUARE CAN BE ROTATED

90 DEGREES FOUR TIMES.

EACH TIME THE SQUARE'S APPEARANCE IS INTACT.

A REGULAR HEXAGON HAS ROTATIONAL SYMMETRY

OF ORDER 6.

AS WITH LINE SYMMETRY,

AS THE NUMBER OF SIDES INCREASES,

THE ORDER OF ROTATIONAL SYMMETRY INCREASES.

AS n APPROACHES INFINITY,

SO DOES THE ORDER OF ROTATIONAL SYMMETRY.

SO A CIRCLE HAS INFINITE ROTATIONAL SYMMETRY.

FINALLY, A CIRCULAR BUILDING THAT ROTATES

ABOUT ITS CENTER HAS POINT SYMMETRY.

WITH POINT SYMMETRY A MIRROR IMAGE OF AN OBJECT

IS CREATED ABOUT THE POINT AFTER A 180 DEGREE ROTATION.

SO WITH POINT, LINE, AND ROTATIONAL SYMMETRY,

AN OBSERVATORY OFFERS A GREAT DEAL OF FLEXIBILITY

FOR VIEWING THE NIGHT SKY.

THE NIGHT SKY GOES THROUGH PERIODIC CHANGES.

MOST PEOPLE ARE FAMILIAR WITH THE PHASES OF THE MOON,

BUT THERE ARE CONTINUAL CHANGES

OCCURRING WITH THE STARS.

AS THE EARTH ORBITS AROUND THE SUN,

THE PANORAMA OF STARS

CHANGE THROUGHOUT THE YEAR.

FOR STARS AND CONSTELLATIONS

THAT ARE CLOSER TO THE NORTH STAR,

RATHER THAN SLIDING ACROSS THE SKY

THEY WILL ROTATE AROUND THE NORTH STAR.

THE BIG DIPPER IS AN EXAMPLE OF THIS.

THIS CONSTELLATION IS MADE UP OF SEVEN STARS

THAT ARE CLOSELY ARRAYED NEAR THE NORTH STAR.

THROUGHOUT THE YEAR THE CLUSTER OF STARS

MAINTAINS ITS SHAPE,

BUT IT ALSO ROTATES AROUND THE NORTH STAR.

YOU CAN THINK OF THE NORTH STAR

AS THE AXIS OF ROTATION.

WE CAN USE THE NSPIRE TO CREATE A

MODEL OF ROTATION OF THE BIG DIPPER.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GEOMETRY WINDOW.

PLACE A POINT IN THE CENTER OF THE SCREEN

TO REPRESENT THE NORTH STAR.

PRESS MENU AND UNDER POINTS AND LINES SELECT POINT.

MOVE THE POINTER TO THE MIDDLE OF THE SCREEN

AND PRESS ENTER.

NOW USE THE POLYGON TOOL

TO CONSTRUCT THE CONSTELLATION.

PRESS MENU AND UNDER SHAPES SELECT POLYGON.

MOVE THE POINTER TO THE LOWER LEFT

QUADRANT OF THE SCREEN AND PRESS ENTER.

THIS DEFINES THE FIRST POINT OF THE POLYGON.

MOVE THE POINTER DOWN TO DEFINE ONE OF THE

VERTICAL SIDES OF THE BIG DIPPER.

PRESS ENTER.

MOVE THE POINTER TO THE LEFT

TO DEFINE THE LOWER PART OF THE DIPPER.

PRESS ENTER.

MOVE THE POINTER UP

TO COMPLETE A QUADRILATERAL SHAPE.

PRESS ENTER.

NOW MOVE THE POINTER DIRECTLY TO THE LEFT

OF THE POINT YOU JUST CREATED

TO SIMULATE THE HANDLE OF THE DIPPER.

PRESS ENTER.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

YOU NOW HAVE A SIMULATION OF THE BIG DIPPER

NEAR THE NORTH STAR.

TO ROTATE THIS CONSTELLATION AROUND

THE POINT REPRESENTING THE NORTH STAR,

PRESS MENU AND UNDER TRANSFORMATION

SELECT ROTATION.

MOVE THE POINTER ABOVE THE BIG DIPPER SHAPE

UNTIL YOU SEE THE ONSCREEN LABEL "POLYGON".

PRESS ENTER.

NOTICE THAT THE POINTER CHANGES TO A DIFFERENT ICON,

ONE THAT LOOKS LIKE A POINT

WITH TWO CURVED ARROWS AROUND IT.

THIS ICON IS FOR DEFINING THE CENTER OF ROTATION.

MOVE THE POINTER ABOVE THE SOLITARY POINT YOU CREATED.

PRESS ENTER.

THE POINTER NOW CHANGES TO A PENCIL.

USE THIS PENCIL TO DEFINE A LINE.

PRESS ENTER.

MOVE THE POINTER AND PRESS ENTER AGAIN.

THIS LINE CONTROLS THE ROTATION.

MOVE THE POINTER ACROSS THE SCREEN AND YOU WILL SEE

THE BIG DIPPER ROTATE ABOUT THE NORTH STAR.

NOTICE THAT AS THE DIPPER ROTATES,

THE RELATIVE POSITIONS OF THE STARS REMAIN THE SAME.

IN OTHER WORDS, THE BIG DIPPER RETAINS ITS SHAPE.

SO THE GEMINI TELESCOPE ON MAUNA KEA RELIES ON

THE PROPERTIES OF ROTATION, SYMMETRY AND REFLECTION

TO DO THE IMPORTANT WORK OF EXPLORING THE HEAVENS.