Geometry Applications: Three-Dimensional Geometry

[Music]

[Music]

Title: Geometry Applications: Three-Dimensional Geometry

Title: Geometry Applications: Three-Dimensional Geometry

Title: Geometry Applications: Geometry Basics: 3D Geometry

Title: Geometry Applications: Geometry Basics: 3D Geometry

Title: Geometry Applications: Geometry Basics: 3D Geometry

IN ANCIENT GREECE THE PHILOSOPHER PLATO

DESCRIBED A SET OF THREE DIMENSIONAL SHAPES

THAT HAVE SINCE COME TO BEAR HIS NAME:

THE PLATONIC SOLIDS.

LET'S START WITH THEIR TWO DIMENSIONAL COUNTERPARTS

AND BUILD THEIR THREE DIMENSIONAL VERSION.

LOOK AT THIS EQUILATERAL TRIANGLE.

IT IS A REGULAR POLYGON

WHERE ALL SIDES AND ANGLE MEASURES ARE CONGRUENT.

THIS TWO DIMENSIONAL NET WHEN FOLDED THIS WAY

BECOMES THE THREE DIMENSIONAL SOLID

CALLED A TETRAHEDRON,

WHICH IS ONE OF THE PLATONIC SOLIDS.

BECAUSE OF THE UNDERLYING REGULAR POLYGON,

THE TETRAHEDRON HAS CONGRUENT EDGES,

VERTICES AND ANGLES.

NOW LOOK AT THIS SQUARE,

WHICH IS A REGULAR QUADRILATERAL.

WE USE THAT SQUARE TO CONSTRUCT

A TWO DIMENSIONAL NET.

WE FOLD THIS NET TO CONSTRUCT A CUBE,

WHICH IS ANOTHER PLATONIC SOLID

ALSO REFERRED TO AS A HEXAHEDRON.

THE "HEX" IN HEXAHEDRON REFERS TO SIX,

WHICH IS THE NUMBER OF SIDES IN A CUBE.

LIKE THE TETRAHEDRON

THE EDGES, VERTICES AND ANGLES OF THE CUBE

ARE CONGRUENT TO EACH OTHER.

THE OTHER PLATONIC SOLIDS INCLUDE THE

EIGHT-FACED OCTAHEDRON...

THE TWELVE-FACED DODECAHEDRON...

AND THE TWENTY-FACED ICOSAHEDRON.

WITH ALL THESE FIGURES,

THE UNDERLYING REGULAR POLYGON SHAPE

ENSURES CONGRUENT EDGES, ANGLES AND VERTICES.

BEYOND THE PLATONIC SOLIDS

ARE MANY DIFFERENT THREE DIMENSIONAL SHAPES.

IN THIS PROGRAM YOU WILL EXPLORE THE PROPERTIES

OF THREE DIMENSIONAL FIGURES.

UNDERSTANDING THE PROPERTIES OF THESE FIGURES

HELPS US UNDERSTAND CERTAIN NATURAL AND

MAN-MADE STRUCTURES THAT SHARE THESE PROPERTIES.

IN PARTICULAR, THIS PROGRAM WILL COVER

THE FOLLOWING KEY CONCEPTS:

IN THE JUNGLES OF SOUTHERN MEXICO

AN ANCIENT MAYAN CITY ARISES FROM THE TREES AND VINES.

THE CITY OF PALENQUE HAS SOME OF THE BEST

PRESERVED TEMPLES, AND ONE OF THE MOST IMPRESSIVE

IS THE TEMPLE OF THE INSCRIPTIONS

WHERE A FAMOUS MAYAN RULER IS BURIED.

THE TEMPLE OF THE INSCRIPTIONS

HAS ELABORATE MAYAN SYMBOLS,

AND, LIKE SOME OF THE EGYPTIAN PYRAMIDS,

INCLUDES A SECRET CHAMBER WHERE THE KING IS BURIED.

THIS JADE MASK WAS OVER THE KING'S SKULL

WHEN EXPLORERS FIRST FOUND THE SECRET CHAMBER IN 1952.

AT THAT POINT THE TEMPLE WAS OVER 500 YEARS OLD.

THE TEMPLE OF THE INSCRIPTIONS

IS AN EXAMPLE OF A RECTANGULAR PYRAMID.

THIS IS DIFFERENT FROM THE PYRAMIDS IN EGYPT

WHICH ARE SQUARE PYRAMIDS.

THE DESCRIPTIONS "RECTANGULAR" AND "SQUARE"

REFER TO THE BASE OF THE PYRAMID.

IN THE CASE OF THE TEMPLE OF THE INSCRIPTIONS,

THE BASE OF THE PYRAMID IS A RECTANGLE.

SUPPOSE THAT THE BASE OF A RECTANGULAR PYRAMID

HAS SIDE LENGTHS a AND b.

THE VOLUME OF THIS TYPE OF PYRAMID IS ONE-THIRD abh

WHERE h IS THE HEIGHT OF THE PYRAMID.

BUT THIS FORMULA APPLIES TO A PYRAMID THAT IS COMPLETE.

THE TEMPLE OF THE INSCRIPTIONS

HAS A PYRAMID SHAPE UP UNTIL THE LEVEL

WHERE THE MAIN TEMPLE APPEARS.

SO THE TYPE OF PYRAMID SHAPE IS MORE LIKE THIS,

WHICH IS A TRUNCATED PYRAMID.

THE VOLUME OF THIS PYRAMID IS EQUAL TO

THE VOLUME OF THE ENTIRE PYRAMID MINUS THE VOLUME

OF THE PYRAMID SHAPED PIECE AT THE TOP.

FOR NOW LET'S ASSUME THIS IS A SQUARE PYRAMID OF BASE b.

THE HEIGHT OF THE FULL PYRAMID IS h1

AND THE HEIGHT OF THE SMALL PYRAMID IS h2.

THE VOLUME OF THE TRUNCATED PYRAMID IS THIS.

BUT THE SMALLER PYRAMID IS PROPORTIONAL

TO THE LARGER ONE.

BOTH HEIGHTS ARE COLLINEAR.

THE SIDES OF THE SMALLER ONE OVERLAP THE LARGER ONE.

MOST IMPORTANT, THE BASES OF THE PYRAMIDS

ARE ON PARALLEL PLANES.

THIS RESULTS IN THE FORMULA

v EQUALS h OVER 3 TIMES THE QUANTITY b1

PLUS b2 PLUS THE SQUARE ROOTS OF b1 TIMES b2.

BUT EVEN THIS FORMULA ASSUMES THAT THE

TRUNCATED PYRAMID HAS A SMOOTH SURFACE.

BUT THAT ISN'T THE CASE WITH THE TEMPLE

OF THE INSCRIPTIONS OR ANY MAYAN PYRAMID.

INSTEAD THEY ARE IN A STAIR-STEP PATTERN.

THIS CAN BE MODELED BY A DIFFERENT TYPE OF

THREE DIMENSIONAL FIGURE.

A RECTANGULAR PRISM HAS SIX RECTANGULAR FACES

AND EIGHT EDGES.

WHEN ALL THE EDGES OF THE RECTANGULAR PRISM

ARE THE SAME LENGTH, THEN THE PRISM IS A CUBE.

IN THE CASE OF THE TEMPLE OF THE INSCRIPTIONS,

THERE IS A STACK OF EIGHT RECTANGULAR PRISMS

THAT DECREASE IN SIZE IN A CONSISTENT MANNER.

IF WE DRAW A RECTANGLE OVER ONE OF THE RECTANGULAR FACES

OF ONE OF THE LEVELS AND SCALE IT DOWN BY TEN PERCENT

WE GET A NEW RECTANGLE THAT FITS IN THE NEXT LAYER.

SUPPOSE THAT THE DIMENSIONS OF THE BOTTOM RECTANGULAR

PRISM AT THE BASE OF THE PYRAMID ARE a, b, AND c.

THE VOLUME OF THIS RECTANGULAR PRISM IS abc.

IF EACH SIDE LENGTH DECREASES BY 10%,

THEN THE VOLUME OF THE SECOND TIER

ISN'T 90% OF TIER ONE,

BUT 72.9% OF TIER ONE AS SHOWN HERE.

THIS IS BECAUSE THE CO-EFFICIENT, 0.9, IS CUBED.

IN THE SAME WAY THE VOLUME OF TIER THREE

IS THE VOLUME OF TIER TWO ALSO MULTIPLIED BY 0.9 CUBED

AND BECOMES TIER ONE TIMES .9 TO THE SIXTH POWER.

THIS TABLE SHOWS THE RELATIONSHIP BETWEEN

THE TIER NUMBER AND THE EXPONENT OF THE .9 TERM

FOR THE FIRST FOUR TIERS.

WE CAN USE THE PATTERN FOUND IN THE TABLE

TO CREATE A GENERAL TERM.

THIS GENERAL TERM IS PART OF A SEQUENCE OF TERMS.

TO FIND THE TOTAL VOLUME OF THE PYRAMID, ADD THE TERMS

IN THE SEQUENCE INCLUDING THE AREA OF THE BOTTOM TIER.

THE SUM OF THE TERMS IN A SEQUENCE IS CALLED A SERIES.

THERE'S A SIMPLE WAY OF WRITING A SERIES

THAT MIGHT HAVE MANY TERMS.

USE THE SUMMATION SYMBOL, SIGMA, AS SHOWN HERE.

THIS IS READ AS THE SUM FROM i EQUALS ZERO TO SEVEN

OF abc TIMES 0.9 TO THE 3i POWER.

AND YOU CAN USE THE TI-NSPIRE TO FIND THE VALUE

OF THE SERIES.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A CALCULATOR WINDOW.

NOW UNLESS YOU ARE USING AN NSPIRE CAS CALCULATOR,

YOUR EXPRESSION SHOULD ONLY HAVE NUMBERS IN IT

SO YOU WILL BE INPUTTING THE FOLLOWING EXPRESSION.

PRESS THE CATALOG BUTTON

WHICH LOOKS LIKE AN OPEN BOOK.

PRESS 4 TO BRING UP THE TAB

THAT INCLUDES THE SUMMATION SYMBOLS.

USE THE NAVIGATION ARROWS

TO ISOLATE THIS SUMMATION SYMBOL.

PRESS ENTER.

THE CURSOR WILL BE AT THE BOTTOM LEFT FIELD.

INPUT THE LETTER i AND THEN PRESS THE RIGHT ARROW KEY.

INPUT THE NUMBER 0 AND PRESS THE RIGHT ARROW KEY AGAIN.

THE CURSER SHOULD NOW BE AT THE MIDDLE PART OF THE

SUMMATION SYMBOL WHERE THE MAIN EXPRESSION GOES.

INPUT 0.9 TO THE 3i AND PRESS THE UP ARROW.

INPUT THE NUMBER 7.

PRESS ENTER TO SEE THE RESULT OF THE CALCULATION.

ROUNDED TO THE NEAREST HUNDRED THE RESULT IS 3.39

SO THE TOTAL VOLUME OF THE TRUNCATED PYRAMID

IS 3.39 TIMES THE VOLUME OF THE FIRST TIER.

IN FACT, LOOK AT THE FUNCTION y=0.9 TO THE 3x.

THE FUNCTION IS BASED ON THE SEQUENCE

YOU JUST INVESTIGATED.

LOOK AT THE GRAPH OF THIS FUNCTION.

CONTINUING WITH THE NSPIRE, CREATE A GRAPH WINDOW.

PRESS HOME AND SELECT THE GRAPHING WINDOW.

PRESS ENTER.

AT THE FUNCTION ENTRY LINE

INPUT THE EXPRESSION 0.9 TO THE 3x.

PRESS ENTER.

YOUR GRAPH SHOULD LOOK LIKE THIS.

RECALL THAT THIS GRAPH REPRESENTS THE TOTAL

AMOUNT OF VOLUME BASED ON THE NUMBER OF TIERS.

SO ALONG THE x AXIS THE NUMBERS REPRESENT THE TIERS

AND ALONG THE y AXIS THE NUMBERS REPRESENT

THE VOLUME AT THE GIVEN TIER.

SO THE TOTAL VOLUME AT ANY GIVEN TIER

IS THE SUM OF THE Y VALUES UP TO THAT POINT.

THE TEMPLE OF THE INSCRIPTIONS

HAS EIGHT TIERS,

SO IF YOU LOOK AT X=8 YOU SEE THAT THIS TIER

MAKES A SMALL CONTRIBUTION TO THE TOTAL VOLUME.

THE TEMPLE OF THE INSCRIPTIONS IS BUILT

IN SUCH A WAY THAT THE GREATEST AMOUNT OF VOLUME

IS CONCENTRATED IN THE SMALLEST NUMBER OF TIERS.

AS YOU CAN SEE FROM THE FUNCTION GRAPH,

THE GRAPH EXTENDS TO INFINITY.

BUT AS X INCREASES IN VALUE,

THE TIERS HAVE A SMALLER AND SMALLER VOLUME.

SO THE TEMPLE SITTING ATOP THE TRUNCATED PYRAMID

REPRESENTS THE INFINITE WHILE THE LOWER TIERS

REPRESENT THE FINITE AND TERRESTRIAL.

THIS IS IMPORTANT SINCE MAYAN TEMPLES

HAD RELIGIOUS SIGNIFICANCE.

WHILE WE DON'T KNOW FOR SURE IF THE MAYANS HAD

DEVELOPED THE MATHEMATICAL CONCEPT OF INFINITY,

THEIR MATHEMATICS CERTAINLY MADE IT POSSIBLE.

FOR EXAMPLE, THE MAYAN NUMERICAL SYSTEM

INCLUDED THE NUMBER 0, A REMARKABLE ACHIEVEMENT

AMONG THE ANCIENT CULTURES.

FURTHERMORE, THE MAYA'S NUMERICAL SYSTEM ALSO

INCLUDED PLACE VALUE IN MUCH THE SAME WAY AS WE USE IT.

WITH PLACE VALUE IT IS EASY TO CREATE

VERY LARGE NUMBERS, AND THE NOTION OF

LARGE NUMBERS IS THE STEPPING STONE TO INFINITY.

THE ONE MAYAN PYRAMID THAT BEST EXEMPLIFIES

THE MATHEMATICAL AND THE SPIRITUAL

IS THE PYRAMID AT CHICHEN ITZA,

WHICH IS SEVERAL HUNDRED MILES FROM PALENQUE.

CHICHEN ITZA IS A SQUARE PYRAMID THAT HAS NINE TIERS

WITH A TEMPLE ATOP THE TRUNCATED PYRAMID.

THE BASE OF THE PYRAMID IS 55.3 METERS ON EACH SIDE.

THE ANGLE THAT THE PYRAMID MAKES WITH THE HORIZON

IS 53.3 DEGREES.

AS WITH THE TEMPLE OF THE INSCRIPTIONS, EACH TIER IS

MADE UP OF A RECTANGULAR PRISM AND THE SIDE LENGTHS

FROM ONE TIER TO ANOTHER DECREASE BY TEN PERCENT.

THE TRUNCATED PYRAMID IS 24 METERS HIGH,

BUT AS YOU KNOW, IT IS A CUT OFF VERSION

OF A TALLER PYRAMID.

WHAT IS THE HEIGHT OF THIS VIRTUAL PYRAMID?

LET'S USE THE NSPIRE TO FIND OUT.

CREATE A GEOMETRY WINDOW.

PRESS HOME AND SELECT GEOMETRY.

PRESS ENTER.

YOU WILL BE CONSTRUCTING A SCALE MODEL

OF A SIDE VIEW OF THE PYRAMID.

THIS WILL BE A 10:1 SCALE DRAWING.

USE THE SEGMENT TOOL TO CONSTRUCT

THE BASE OF THE PYRAMID.

PRESS MENU AND UNDER POINTS & LINES SELECT SEGMENT.

MOVE THE POINTER TO THE MIDDLE

LEFT PART OF THE SCREEN.

PRESS ENTER TO DEFINE THE FIRST ENDPOINT

OF THE SEGMENT.

PRESS THE RIGHT ARROW TO MOVE THE POINTER

IN THAT DIRECTION.

YOU'LL SEE THE SEGMENT TAKE SHAPE.

PRESS ENTER.

MEASURE THE SEGMENT.

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER ABOVE THE SEGMENT YOU JUST CREATED.

PRESS ENTER TO RECORD THE MEASUREMENT.

MOVE THE POINTER BELOW THE SEGMENT AND PRESS ENTER

AGAIN TO PLACE THE MEASUREMENT ONSCREEN.

PRESS ESCAPE AND MOVE THE POINTER

ABOVE THE MEASUREMENT.

PRESS ENTER TWICE TO MAKE THE MEASUREMENT EDITABLE.

PRESS THE "CLEAR" BUTTON AND REPLACE THE VALUE

WITH 5.5 WHICH REPRESENTS ONE-TENTH THE LENGTH

OF THE BASE OF THE PYRAMID AT CHICHEN ITZA.

NOW FIND THE MIDPOINT OF THE SEGMENT.

PRESS MENU AND UNDER CONSTRUCTION

SELECT MIDPOINT.

MOVE THE POINTER ABOVE THE SEGMENT AND PRESS ENTER.

YOU WILL NOW SEE THE MIDPOINT WHERE YOU WILL

CONSTRUCT A LINE PERPENDICULAR TO THE BASE.

PRESS MENU AND UNDER CONSTRUCTION

SELECT PERPENDICULAR.

SINCE THE POINTER IS RIGHT ABOVE THE MIDPOINT,

PRESS ENTER TWICE.

YOU SHOULD NOW SEE A LINE PERPENDICULAR TO THE BASE

AT THE MIDPOINT.

PRESS ESCAPE AND MOVE THE POINTER

TO THE END OF THE LINE.

PRESS AND HOLD THE CLICK KEY

UNTIL THE POINTER CHANGES TO A GRASPING HAND.

USE THE UP ARROW TO EXTEND THE LINE.

EXTEND IT TO THE TOP OF THE SCREEN.

NOW CONSTRUCT AN ANGLE THAT INCLUDES

THE BASE OF THE PYRAMID.

PRESS MENU AND UNDER POINTS & LINES

SELECT SEGMENT.

MOVE THE POINTER TO THE RIGHTHAND ENDPOINT

OF THE BASE OF THE PYRAMID.

MAKE SURE THE POINTER IS RIGHT ABOVE THAT POINT.

PRESS ENTER.

MOVE THE POINTER UP AND TO THE LEFT,

ABOUT HALFWAY UP THE SCREEN.

MAKE SURE THAT THE SEGMENT INTERSECTS

THE PERPENDICULAR LINE YOU PREVIOUSLY CREATED.

MEASURE THE ANGLE.

PRESS MENU AND UNDER MEASUREMENT SELECT ANGLE.

YOU NEED TO IDENTIFY THREE POINTS TO MEASURE AN ANGLE.

SO MOVE THE POINTER TO THE MIDPOINT

OF THE BASE SEGMENT.

PRESS ENTER.

NEXT, MOVE THE POINTER TO THE RIGHTHAND VERTEX

OF THE PYRAMID BASE.

PRESS ENTER AGAIN.

FINALLY, MOVE THE POINTER TO WHERE THE PERPENDICULAR LINE

AND THE SEGMENT YOU JUST CREATED INTERSECT

AND PRESS ENTER ONE MORE TIME.

YOU SHOULD NOW SEE AN ANGLE MEASUREMENT APPEAR.

PRESS ESCAPE AND MOVE THE POINTER TO THE

FARTHEST ENDPOINT OF THE SLANTED SEGMENT.

PRESS AND HOLD THE CLICK KEY TO SELECT THE POINT.

MOVE THE POINT UP OR DOWN

TO CHANGE THE ANGLE MEASURE TO 53.3 DEGREES.

YOU MAY NEED TO MOVE THE ENDPOINT FURTHER UP

AND TO THE LEFT TO GET THE ANGLE RIGHT.

IF YOU CAN'T CHANGE THE ANGLE MEASURE

TO 53.3 DEGREES, MAKE IT AS CLOSE TO THIS VALUE

AS POSSIBLE.

NOW MEASURE THE LENGTH OF THE SEGMENT FROM

THE MIDPOINT AT THE BASE TO THE INTERSECTION POINT

OF THE PERPENDICULAR LINE AND THE SLANTED SEGMENT.

THIS IS THE HEIGHT OF THE COMPLETE PYRAMID.

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER TO THE MIDPOINT AND PRESS ENTER.

MOVE THE POINTER UP TO THE SECOND ENDPOINT

AND PRESS ENTER AGAIN.

YOU SHOULD NOW SEE A MEASUREMENT APPEAR.

MOVE THE POINTER TO THE LEFT OF THE SEGMENT

AND PRESS ENTER ONE MORE TIME

TO RECORD THE MEASUREMENT ONSCREEN.

THE HEIGHT OF THE PYRAMID SHOULD BE 3.7 TO 3.71.

SINCE THIS IS A 10:1 SCALE DRAWING THE ACTUAL

HEIGHT OF THE VIRTUAL PYRAMID IS 37 METERS.

SINCE THE HEIGHT OF THE TRUNCATED PYRAMID

IS 24 METERS, THEN THERE IS 13 METERS OF VIRTUAL PYRAMID.

BUT AS YOU HAVE SEEN FROM THE GRAPH OF THE FUNCTION,

MOST OF THE VOLUME IS INCLUDED IN THE NINE TIERS

THAT ARE THERE.

WHAT PERCENTAGE OF THE VOLUME

IS ENCLOSED IN THE NINE TIERS?

FOR THAT YOU NEED CALCULUS

TO FIND THE AREA OF THE CURVE.

BUT YOU CAN USE THE NSPIRE TO CALCULATE

A CLOSE APPROXIMATION.

CREATE A NEW CALCULATOR WINDOW.

YOU WANT TO FIND THE RATIO OF THE VOLUME

OF THE FIRST NINE TIERS

OVER THE VOLUME OF THE ENTIRE PYRAMID.

USE THE EXPRESSION YOU USED EARLIER

WITH THE TEMPLE OF THE INSCRIPTIONS.

0.9 TO THE 3X.

CREATE A FRACTION TO CALCULATE THE RATIO.

PRESS THE LIBRARY BUTTON

WHICH LOOKS LIKE AN OPEN BOOK.

PRESS 4 TO BRING UP THE TAB

THAT INCLUDES THE FRACTION TEMPLATE.

MOVE THE POINTER TO THE UPPER LEFTHAND ENTRY

AND PRESS ENTER.

IN THE NUMERATOR, INPUT A SUMMATION SIGMA.

PRESS THE LIBRARY BUTTON AGAIN.

SINCE YOU WERE PREVIOUSLY IN TAB 4,

THEN MOVE THE CURSOR TO THE SIGMA TEMPLATE

THAT INCLUDES THE UPPER AND LOWER LIMITS.

PRESS ENTER.

INPUT i IN THE FIRST FIELD AND PRESS THE RIGHT ARROW.

INPUT 0 AND PRESS THE UP ARROW.

INPUT THE EXPRESSION 0.9 TO THE 3i.

PRESS THE UP ARROW AND INPUT 8 IN THE TOP FIELD.

YOU WANT TO COPY THIS ENTIRE EXPRESSION

TO PASTE INTO THE DENOMINATOR.

PRESS THE LEFT ARROW SO THAT IT IS TO THE LEFT OF

THE SUMMATION EXPRESSION IN THE NUMERATOR.

PRESS THE SHIFT KEY FOLLOWED BY THE RIGHT ARROW.

THIS SHOULD HIGHLIGHT THE ENTIRE NUMERATOR.

PRESS CONTROL AND C TO COPY THE EXPRESSION.

PRESS THE DOWN ARROW TO GO TO THE DENOMINATOR.

PRESS CONTROL AND V TO PASTE THE EXPRESSION.

YOU WANT TO CHANGE THE UPPER VALUE FROM 8

TO A MUCH HIGHER VALUE.

IDEALLY THE VALUE SHOULD BE INFINITY.

A GOOD APPROXIMATION WILL RESULT

EVEN WITH THE VALUE OF 100 OR 1,000.

INPUT THE HIGHER VALUE AND PRESS ENTER.

THE TEMPLE AT CHICHEN ITZA WAS ALSO A KIND OF CALENDAR.

THERE ARE 91 STEPS ON EACH OF THE FOUR SIDES

OF THE PYRAMID. THESE ADD UP TO 364.

THE TEMPLE AT THE TOP IS THE 365TH STEP, SO EACH DAY

CAN BE MARKED AND IDENTIFIED ON THE PYRAMID.

THE ORIENTATION OF THE PYRAMID IS SUCH THAT

ON THE FALL AND SPRING EQUINOX

AT A SPECIFIED TIME OF THE AFTERNOON

THE SHADOWS ALONG THE STAIRS FORM SEVEN

ISOSCELES TRIANGLES THAT ARE ALIGNED ALONG THE DIAGONAL.

THEY MEET AT THE BOTTOM OF THE STAIRS

WHERE THE HEAD OF A SERPENT CAN BE SEEN.

TOGETHER, THE LIGHT-FORMED TRIANGLES AND THE SNAKE'S

HEAD ARE MEANT TO EVOKE THE MAYAN GOD, KUKULKAN.

THE FEATHERED SERPENT COMES DOWN FROM THE HEAVENS,

FROM THE INFINITE EXPANSE OF SPACE,

AND APPROACHES THE GROUND WHERE MAN LIVES.

THIS IS ASTONISHINGLY SIMILAR

TO THE BEHAVIOR OF OUR EXPONENTIAL FUNCTION

FOR THE VOLUME OF THE PYRAMID.

ITS GRAPH DESCENDS AS x APPROACHES INFINITY.

NOW THE MAYA WERE NOT AWARE OF CALCULUS,

THE AREA UNDER A CURVE,

THE GRAPHS OF EXPONENTIAL FUNCTIONS,

OR INFINITE SERIES, AND YET THEIR ARCHITECTURE SHOWS

AN AWARENESS OF SOMETHING BEYOND WHAT CAN BE COUNTED.

THROUGH THESE MYSTICAL PYRAMIDS THE MAYA TOOK STEPS

TOWARD THE INFINITE, BEYOND THE DAY TO DAY.

THE GEOMETRY OF THE PYRAMID GIVES US INSIGHT

INTO THEIR VIEW OF THE WORLD AND THE UNIVERSE.

THE CITY OF SHANGHAI IS CHINA'S MAJOR HUB

FOR FINANCE AND BUSINESS,

MUCH THE WAY MANHATTAN IS FOR THE U.S.

AND, MUCH LIKE NEW YORK CITY,

SHANGHAI HAS THE LOOK AND FEEL OF HIGH FINANCE.

THE FINANCIAL DISTRICT ALREADY HAS SOME

IMPRESSIVE TOWERS, YET THERE IS A NEW TOWER.

THE SHANGHAI TOWER, UNDER CONSTRUCTION,

WHICH, WHEN COMPLETED IN 2014,

WILL BE THE TALLEST BUILDING IN CHINA

AND THE SECOND TALLEST BUILDING IN THE WORLD.

THIS STREAMLINED TOWER HAS AN IRREGULAR SHAPE,

BUT UNDERNEATH THE FLOWING EXTERIOR

IS A SOLID GEOMETRIC BASE.

THERE IS A STACK OF EIGHT CYLINDRICAL SHAPES.

HERE IS A CUT-AWAY VIEW TO REVEAL THE EIGHT CYLINDERS.

NOTICE THAT EACH SUCCEEDING CYLINDER

IS NARROWER THAN THE ONE RIGHT BELOW IT.

THIS IS BECAUSE THE SIDE OF THE TOWER

IS ALONG A LINE THAT IS TWO DEGREES

FROM THE VERTICAL AXIS OF THE TOWER.

PUT A LITTLE MORE SUBTLY, EACH CYLINDER INCREASES

IN HEIGHT FROM THE ONE RIGHT BELOW IT

AND THE COMBINATION OF BOTH CHANGES IN DIMENSION

MAKES THE TOWER LOOK TALL AND SLEEK.

PART OF THE REASON THAT THE TOWER IS THINNER AT THE TOP

HAS TO DO WITH GEOGRAPHY.

THIS REGION OF CHINA EXPERIENCES TYPHOONS

WHICH HAVE VERY HIGH WINDS.

A BUILDING THAT IS THINNER AT THE TOP

IS LESS SUSCEPTIBLE TO THE FORCE OF THESE WINDS.

IN FACT, THE OUTER LAYER OF THE TOWER

SWIRLS AROUND THE CORE OF THE TOWER IN SUCH A WAY

AS TO DECREASE THE FORCE OF THE WIND ON THE TOWER.

NOT ONLY THAT, THERE ARE WIND TURBINES LOCATED

WITHIN THE TOWER TO USE THE POWER OF WIND

TO GENERATE ELECTRICITY.

THE SHANGHAI TOWER IS AN EXAMPLE OF

GREEN ARCHITECTURE.

THE SWIRLING OUTER LAYER IS ALSO A COLLECTOR

OF RAINWATER THAT IS RECYCLED FOR USE

IN THE AIR CONDITIONING AND HEATING SYSTEMS.

BUT THE HEART OF THE TOWER IS THE STACK OF CYLINDERS

WHICH IS WHERE OFFICE AND LIVING SPACES

WILL BE HOUSED.

AND GOING FROM ONE TIER TO THE NEXT,

THE WIDTH OF THE CYLINDER DECREASES BY 8.2%

AND ITS HEIGHT INCREASES BY 4.1%.

WE CAN USE THESE VALUES TO DETERMINE THE VOLUME

AND THEREFORE THE WEIGHT THAT EACH SECTION CHANGES

FROM ONE LEVEL TO THE NEXT.

OBVIOUSLY THE HEAVIER LEVELS ARE AT THE BOTTOM

BUT HOW MUCH LIGHTER DO THE SUCCEEDING TIERS GET?

A CYLINDER IS A THREE DIMENSIONAL FIGURE

WITH A CIRCULAR BASE AND A RECTANGULAR SIDE.

A NET FOR A CYLINDER CLEARLY SHOWS THE TWO CIRCLES THAT

DEFINE THE BASE AND THE TOP AND THE RECTANGULAR SIDE.

THE SURFACE AREA OF THE CYLINDER IS MADE UP OF

INDIVIDUAL AREAS OF THE CIRCLES AND THE RECTANGLE.

SUPPOSE THE CIRCLE HAS RADIUS r.

SINCE THE TOP SIDE OF THE RECTANGLE WRAPS AROUND

THE CIRCLE, THEN THE RECTANGLE'S SIDE LENGTH

IS THE SAME AS THE CIRCLE'S PERIMETER.

THE OTHER SIDE LENGTH OF THE RECTANGLE

CORRESPONDS TO THE HEIGHT OF THE CYLINDER.

SO THE SURFACE AREA OF THE CYLINDER IS 2 Pi r SQUARED

PLUS 2 Pi rh.

THE VOLUME OF THE CYLINDER IS Pi r SQUARED h.

LET'S USE THE VARIABLES r AND h FOR THE BOTTOM

CYLINDRICAL PORTION OF THE SHANGHAI TOWER.

EACH SUCCEEDING CYLINDER DECREASES ITS WIDTH BY 8.2%

AND INCREASES ITS HEIGHT BY 4.1%.

SINCE THE DIAMETER OF THE CIRCULAR BASE

DECREASES BY 8.2%, THIS MEANS THAT THE RADIUS

DECREASES BY HALF THAT AMOUNT, OR 4.1%.

SO LEVEL TWO IS Pi r SQUARED h TIMES 1.041 TIMES 0.959.

FOR ANY TIER i, THE VOLUME OF THAT CYLINDER

IS FOUND WITH THIS EXPRESSION.

THE TOTAL VOLUME OF ALL THE CYLINDRICAL SECTIONS

IS FOUND USING THIS EXPRESSION.

LET'S USE THE TI-NSPIRE TO CALCULATE THIS VALUE.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A CALCULATOR WINDOW.

UNLESS YOU ARE USING A TI-NSPIRE CAS,

THEN YOU CANNOT USE THE VARIABLES r AND h.

BUT SINCE THESE VARIABLES ARE FOUND IN EVERY TERM,

THEY CAN BE LEFT OUT.

IN FACT, THE EXPRESSION YOU WILL BE INPUTTING IS THIS:

PRESS THE LIBRARY BUTTON

WHICH LOOKS LIKE AN OPEN BOOK.

PRESS THE NUMBER 4 TO BRING UP THE TAB

THAT HAS THE SUMMATION SYMBOL.

MOVE THE POINTER UNTIL THE SIGMA SYMBOL IS HIGHLIGHTED.

PRESS ENTER.

INPUT THE LETTER i IN THE FIRST FIELD

AND PRESS THE RIGHT ARROW KEY.

INPUT THE NUMBER 1.

PRESS THE RIGHT ARROW.

INPUT THE NUMERICAL EXPRESSION AS SHOWN HERE.

THEN PRESS THE UP ARROW AND INPUT THE NUMBER 8.

PRESS ENTER TO CALCULATE THE RESULT.

YOU'LL SEE THAT THE SUM IS ABOUT 6.9.

CONTRAST THIS TO A SITUATION IN WHICH EIGHT CYLINDERS

THE SAME SIZE AS THE BASE CYLINDER ARE STACKED.

THE SUM OF THE VOLUMES OF THE STACKED CYLINDERS

WOULD BE EIGHT TIMES THE VALUE OF THE BASE CYLINDER.

IN OTHER WORDS, THE SHANGHAI TOWER HAS 14% LESS VOLUME

AND CORRESPONDINGLY LESS WEIGHT

WHICH HELPS IN KEEPING THE COST OF MATERIALS DOWN.

HOW IS THE WEIGHT OF THE TOWER DISTRIBUTED?

LET'S ANALYZE THIS IN A SPREADSHEET.

PRESS THE HOME KEY AND SELECT

A LIST AND SPREADSHEET WINDOW.

PRESS ENTER.

MOVE THE CURSOR TO THE VERY TOP OF COLUMN A

AND THE COLUMN HEADING "TIER".

PRESS THE DOWN ARROW.

YOUR CURSOR SHOULD BE ABOVE CELL A1

AND RIGHT BELOW THE COLUMN HEADING.

THIS IS THE FORMULA LINE FOR INSERTING SPREADSHEET

FORMULAS THAT APPLY TO THE ENTIRE COLUMN.

CREATE A CONSECUTIVE SEQUENCE OF NUMBERS

FROM ONE TO EIGHT.

PRESS MENU AND UNDER DATA SELECT GENERATE SEQUENCE.

AT THE DIALOGUE BOX INPUT N+1 AND PRESS TAB.

AT THE NEXT FIELD INPUT 1 AS THE START VALUE

FOR THE SEQUENCE.

IN OTHER WORDS, THE SEQUENCE STARTS AT N=0.

PRESS TAB AND FOR THE MAXIMUM NUMBER OF TERMS

INPUT 7, NOT 8, KEEPING IN MIND THAT THE SEQUENCE

STARTS AT N=0 AND WILL INCLUDE EIGHT TERMS.

PRESS ENTER.

YOU'LL SEE A CONSECUTIVE LIST OF NUMBERS

FROM ONE TO EIGHT.

NEXT, MOVE TO THE FORMULA LINE FOR COLUMN B,

THE CELL BELOW THE COLUMN HEADING AND ABOVE CELL B1.

INPUT THIS EXPRESSION AT THE FORMULA LINE.

PRESS ENTER.

FOR EACH CELL IN COLUMN B, THIS FORMULA

USES THE CORRESPONDING VALUE IN COLUMN A

TO CALCULATE THE VOLUME.

NOTICE THAT THE VALUES OF THE VOLUME

GRADUALLY DECREASE IN GOING FROM

CYLINDRICAL REGION ONE TO EIGHT.

HOW DOES THE CHANGE IN DIMENSIONS

AFFECT THE SURFACE AREA OF THE CYLINDERS?

RECALL THAT THE FORMULA FOR FINDING SURFACE AREA

OF A CYLINDER IS 2 Pi r SQUARED PLUS 2 Pi rh.

FOR NOW LET'S FOCUS ON THE SECOND TERM,

WHICH IS THE SURFACE AREA OF THE SIDE OF THE CYLINDER.

THE AREA OF THE SIDE OF THE CYLINDER

FOR TIER ONE IS 2 Pi rh.

FOR LEVEL 2 THE AREA IS 2 Pi rh TIMES 0.959 TIMES 1.041.

FOR ANY LEVEL i

THE AREA OF THE SIDE OF THE CYLINDER IS THIS.

CREATE A FORMULA IN THE CALCULATOR WINDOW

TO FIND THE SIDE SURFACE AREA.

PRESS THE LIBRARY BUTTON AND SELECT THE SIGMA TEMPLATE.

INPUT THE LETTER i IN THE FIRST FIELD

AND PRESS THE RIGHT ARROW KEY.

INPUT THE NUMBER 1.

PRESS THE RIGHT ARROW.

INPUT THE NUMERICAL EXPRESSION AS SHOWN HERE.

THEN PRESS THE UP ARROW AND INPUT THE NUMBER 8.

PRESS ENTER TO CALCULATE THE RESULT.

YOU'LL SEE THAT THE SURFACE AREA IS ALMOST 8.

SO WHILE THE TOWER HAS 14% LESS VOLUME AND WEIGHT,

IT HAS ALMOST THE SAME FLOOR SPACE

AS IF THE BUILDING WERE NOT TAPERED.

THIS IS AN EFFICIENT WAY TO CREATE A LIGHTER WEIGHT

BUILDING WITHOUT SACRIFICING SPACE.

BY CAREFULLY MANAGING THE CHANGE IN DIMENSIONS

OF A BUILDING FROM ONE LEVEL TO ANOTHER,

AN ARCHITECT COULD MAKE SUBSTANTIAL CHANGES

TO THE SURFACE AREA OR VOLUME OF A BUILDING.

THIS IS CLEARLY SEEN IN COMPARING THE GRAPHS

OF THE VOLUME AND SURFACE AREA EQUATIONS.

PRESS THE HOME KEY AND SELECT THE GRAPH WINDOW.

PRESS ENTER.

AT FUNCTION ENTRY LINE F1 INPUT THIS FUNCTION

WHICH REPRESENTS THE CHANGE IN VOLUME.

PRESS THE DOWN ARROW.

AT THE F2 FUNCTION ENTRY LINE INPUT THIS FUNCTION

WHICH REPRESENTS THE CHANGE IN THE SIDE SURFACE AREA.

PRESS ENTER.

TO GET A BETTER VIEW OF THE GRAPH, PRESS MENU

AND UNDER WINDOW/ZOOM SELECT ZOOM-FIT.

YOUR GRAPH SHOULD LOOK LIKE THIS.

BOTH GRAPHS ARE DECREASING FUNCTIONS

WHICH MEANS THAT FOR HIGHER VALUES OF X

THE GRAPH IS AT A LOWER Y COORDINATE.

BUT NOTICE THAT THE GRAPH OF THE VOLUME

DECREASES AT A MUCH FASTER RATE

THAN THE SIDE SURFACE AREA GRAPH.

THIS MEANS THE SHANGHAI SHEDS VOLUME AND THEREFORE

WEIGHT AT A FASTER RATE THAN IT SHEDS SURFACE AREA.

THIS GRAPH CONFIRMS THAT THE ARCHITECTS

ARE ABLE TO GET MORE OFFICE AND LIVING SPACE

WITHOUT INCREASING THE WEIGHT OF THE BUILDING.

THE SLOWER DECREASE IN THE SURFACE AREA

ALSO HELPS WITH THE OVERALL HEIGHT OF THE TOWER.

AT THE TIME THIS BUILDING WAS BEING PLANNED,

ITS HEIGHT WOULD HAVE MADE IT THE TALLEST BUILDING

IN THE WORLD, AND PART OF THIS WOULD HAVE BEEN DUE TO

THE INCREASING HEIGHT OF EACH CYLINDRICAL SECTION

WHICH WOULD HAVE ADDED JUST ENOUGH HEIGHT

TO HAVE PUSHED IT OVER THE TOP.

HOWEVER, SINCE THE ORIGINAL PLANNING OF THIS BUILDING

THE BURJ KHALIFA TOWER IN DUBAI

STANDS AS THE TALLEST BUILDING.

NOTICE THAT THIS TOWER, TOO,

HAS SOME OF THE FEATURES OF THE SHANGHAI TOWER:

THE CYLINDRICAL SECTIONS IN GOING FROM TOP TO BOTTOM

INCREASE IN HEIGHT AND DECREASE IN WIDTH.

THIS IS AN EFFECTIVE COMBINATION

FOR CREATING A SUPER TALL SKYSCRAPER.

BUILDINGS LIKE THE SHANGHAI TOWER AND THE BURJ KHALIFA

ARE SO MASSIVE THAT THEY ARE MINIATURE CITIES

AND SO THE ARCHITECTURE OF BUILDINGS LIKE THESE

RELY ON GREEN SUSTAINABLE TECHNIQUES

TO MAKE THEM MORE LIVABLE AND MANAGEABLE.

ONE OF THE LIKELY CHARACTERISTICS

OF FUTURE SUPER TALL SKYSCRAPERS

IS THE TAPERED LOOK - WIDER AT THE BOTTOM

AND COMING TO A POINT NEAR THE TOP.

THE USE OF CYLINDRICAL FORMS IN COMBINATION

WITH A TAPERING DESIGN TENDS TO

GIVE THESE BUILDINGS THE LOOK OF A CONE.

COMPARING THE VOLUME AND SURFACE AREA

OF A CONE AND CYLINDER SHOWS WHY.

THE CONE HAS A THIRD OF THE VOLUME OF THE CYLINDER

AND SIGNIFICANTLY LESS SURFACE AREA.

AND OF COURSE AS A BUILDING

APPROXIMATES THE SHAPE OF A CONE

IT ALSO BEGINS TO EVOKE THE SHAPE OF A PYRAMID.

SHANGHAI IS A CITY IN TRANSITION

AND CHINA IS A COUNTRY IN TRANSITION.

AS SKYSCRAPERS REACH FOR THE HEAVENS,

ARCHITECTS TAP INTO TECHNIQUES

THAT GO BACK TO THE AWE-INSPIRING WORKS

OF THE ANCIENT WORLD.