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Miguel Ros López . Dpto de Matemáticas. IES Aljada. Puente Tocinos. Murcia

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POLYHEDRA. SPHERE EARTH GLOBE

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POLYHEDRA

Initial activities

a) This is a classroom with teaching

materials. On the left, you can see a

small case with geometric bodies and

a globe.

Separate these 3D shapes in two

groups (polyhedra and bodies with

curved surface) and say the features

of each one. Could you name some

of these 3D shapes?

We have studied prisms and pyramids whose surfaces are linked to cylinders and cones

because they are formed by straight lines (parallel or with one common point, apex).

Anyway, prisms and pyramids are polyhedra as well, but in this theme we are going to

study “other polyhedra”.

Icosahedron development and

view of Peace Camp in Barcelona Forum.

At the bottom, an incinerator.

b) From the icosahedron development, build a tent model with an edge of 4 cm.

If we estimate the real edge at 2 m, find out the scale.

Calculate the areas of the model and tent. Find out the surface scale.

What is the relation between the surface scale and the length scale?

There are 25 tents and the price of the material is € 50 per m2 without VAT. What is the

cost of the material that we need if VAT is 20%?

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Definition and elements

Polyhedron ( in ancient Greek : poly= many, hedron= face) is a part of space bounded

by polygons which are called faces. Its surface is developable (a "surface" that can be

flattened onto a plane without distortion).

The elements of a polyhedron are:

Face: each one of the polygons.

Edge: segment where two faces meet.

Vertex: point where three or more edges meet.

Polyhedron angle: it depends on the number of faces and we have:

- Dihedron angle: part of space bounded by two semiplanes (in polyhedra it is

formed by two faces) and if it is right (90º) we have the basis of a representation

system, Diedric System.

- Thrihedron angle : it is formed where three planes meet in a point (in polyhedra,

three faces which meet in a vertex ) and when the planes, two by two, are

perpendicular we have a three-right thrihedron like the corners of a room where

the floor (or ceiling) and two walls meet.

- Polyhedron angle (in general): it is formed by whatever number of planes meet

in a point (vertex).

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Convex and concave polyhedra

Polyhedra fall into two categories: convex and concave.

A convex polyhedron is defined as follows: no line segment joining two of its points,

contains a point belonging to its exterior (also, every plane which contains one face,

leaves all faces in the same semispace).Figures on the left.

A concave polyhedron will have some line segments that join two of its points with

points lying in its exterior (also, some planes which contain one face, do not leave all

faces in the same semispace).Figures on the right.

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EULER’S POLYHEDRA FORMULA

Leonard Euler (Basel 1707- St.Petersburg 1783) was a mathematician

who worked in different fields of maths. In one of them, Geometry, he

discovered the first topologycal theorem or Euler’s formula. Its more

general expression is V + F = E + K( Euler’s characteristic) and when

K=2 result the most known : Vertices + Faces = Edges + 2 which is

valid for simple polyhedra (those whose surface can be deformed

continuously into the surface of a sphere). For non-simple polyhedra

K≠ 2, for instance, polyhedra with a hole, K=0.

V + F = E + 2

1) Prove Euler´s formula with Deltahedron-6, truncated cube and Corpus

Hypercubus.

2) Demonstrate, using Euler’s theorem, that one of these 3D shapes below is

not a polyhedron (besides, it does not fit with the definition of a polyhedron).

Regular polyhedra. Platonic solids

A regular polyhedron has identical faces, they are regular polygons and the same

number of them meet at each vertex. There are 5 regular polyhedra.

Around Mare Nostrum

Some places on the map:

(1) Syracuse in Sicily island. There,

Archimedes of Syracuse was born.

(2) Crotona in Magna Greece (South of

Italy). The Pythagoreans lived there.

(8) Athens, Plato and his students’ town.

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Regular polyhedra are called “Platonic solids” because of their history. The five regular

polyhedra were discovered by the ancient Greeks. The Pythagoreans knew of the

tetrahedron, the cube, and the dodecahedron, these two last, maybe, from the mineral

forms of pyrite, cube and dodecahedron (iron disulfide, S3 Fe2, "fool's gold") from the

mines of Crotona (2).The mathematician Theaetetus added the octahedron and the

icosahedron. These shapes are also called the Platonic solids, after the ancient Greek

philosopher Plato, who greatly respected Theaetetus' work, speculated that four of these

five solids were the shapes of the fundamental components of the physical universe

(fire, earth, air and water). The fifth element, ether, was named by Aristotle (Plato’s

student) and is linked to the dodecahedron.

In 1525, Albert Durero, in 1525, made sheets with the developments of the regular

polyhedra for the first time. We can fold and stick (glue) the development to build each

polyhedron.

Polyhedra Element Vertices Faces Edges

Cubic pyrite (mine in

Crotona, Magna Greece)

Tetrahedron Fire

Hexahedron

(cube)

Earth

Octahedron Air

Dodehedron Ether

Icosahedron Water

3.- Complete the chart. You can use the 3D shapes or the developments.

From the developments, how would you get the number of edges and vertices?

Anyway, you can get the vertices with Euler´s formula when you know faces

and edges.

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4.-Obtain surface area and volume of a octahedron with edge 8 cm. (Hint:

octahedron is the same as two pyramids which join their bases).

5.- Find out the following probabilities when

using each of the polyhedrical dice.

(remember: the probability of an event equals

the number of possible favorable outcomes

divided by the total number of possible

outcomes.)

a) Probability of getting an even number.

b) Probability of getting an odd number.

c) Probability of getting a multiple of 3.

d) Probability of getting a prime number.

Polyhedrical dice from the

regular polyhedra.

Polyhedra can be combined into pairs called duals, where the vertices of one correspond

to the faces (centres of faces) of the other. Tetrahedron is dual with itself (self-dual).

Cube and octahedron are dual. Dodecahedron and icosahedron are dual.

6.-Draw the self-dual

tetrahedron

(approximately)

Your teacher will guide you to

find out area and volume of

tetrahedron of edge 2 m.

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Principal section of regular polyhedra

It is a cross section which is defined by the centre of the polyhedron and an edge. The

principal section contains three very important lengths:

Radius circumscribed sphere = distance from centre to vertex (R).

Radius inscribed sphere = distance centre polyhedron to centre face (r).

Distance from centre to middle point of edge (d).

Principal sections of cube (a rectangle) and

of icosahedron (an irregular hexagon)

Cubes painted by Ibarrola (Basque painter)

7.-The photo shows a lot of cubes.

They are in Llanes, an Asturian

harbour, as defence against waves.

If the edge is 1.2 m, find:

a) Area and volume of a cube.

b) Radii of the inscribed and

circumscribed spheres.

c) Distance centre-edge.

8.-Draw different sections on the

cubes, one is done. Try to find four

cross sections which divide the cube

into two parts with equal area and

volume (when the section is a

rectangle, its area has to be the

biggest).

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9.-Draw the development of half (a) cube

with an edge 6 cm (in order to fit the

development into the A4 sheet ) when the

cross section is a regular hexagon. You will

get a drawing as shown in the picture.

Homework: built from cardboard half a cube

with edge 8 cm and bring it the next lesson.

If you put your half cube in the corner of a

three-right thrihedron with a common vertex

you will be able to see a new polyhedron.

What can you say about it?

Kelvin polyhedron on the beach and mineral “hauerita” (mine in Raddusa near Syracuse,

Archimedes probably knew it before he thought of the 13 semiregular polyhedra).

How many half cubes do you need for building the Kelvin solid?

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Semiregular polyhedra or Archimedean solids

A semiregular polyhedron has different polygons as faces but they are regular polygons

with the same edge and the same number of them meets at each vertex. There are 13

semiregular polyhedra.

Kepler’s drawings to “Harmonices Mundi” 1619

We are going to study: cuboctahedron (8), truncated octahedron or Kelvin solid (5) and

truncated icosahedron (4).

On the beach (Torre de la Horadada, Alicante).

Kelvin solid is made with the ropes and the structure is

a cuboctahedron.

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CUBOCTAHEDRON

10.-Maybe, you made this and calculated its

area the last course (2ndESO). Do you

remember?

Now you are going to do a 2D drawing of

the cuboctahedron from a cube (an

isometric perspective), joining the middle

point of the edges. Calculate its area and

volume if the edge of the cube equals 8cm.

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TRUNCATED OCTAHEDRON OR KELVIN SOLID

We can get this polyhedron by truncation of an octahedron to a third of its edges.

A cube is given in cavalier perspective. You can get the dual octahedron from it.

11.-You will have to get a drawing

like this from the cube on the left.

If the edge is 9 cm, what is the distance between the upper face of Kelvin solid

and the one of the cube? Find out the volume of Kelvin solid taking into account

that you obtained it from your half cube in the corner of the mirrors.

Children playing in a

park (Hagen-Germany).

Electron in crystals.

First zone of Brillouin

according to Bragg´s

law (atomic diffraction).

Glass container for sweets,

almonds... It has been designed

after Kelvin solid.

This polyhedron appears in nature (mineral and atomic world) and it has been used by

human beings in some applications.

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TRUNCATED ICOSAHEDRON

It is the most famous semiregular polyhedron in the world because it is the ball which

soccer is played with, a football. Also this polyhedron is got from another polyehedron,

the icosahedron, by truncation to one third of its edges.

12.-How many faces does this polyhedron have? What polygons are they?

When you blow up (inflate) this polyhedron, you get a ball, a sphere. It’s cool.

Isn´t it?

13.-Connecting polyhedra with algebra.

Demonstrate that in a convex polyhedron with faces, pentagons and hexagons,

the number of pentagons is an invariant (it is always the same number, which

one?).

Hint: let’s P, number of pentagons, and H, number of hexagons, V vertices and

E edges; you should take into account that: each edge is shared by two faces

and three faces meet in each vertex, then use Euler´s theorem to get the first-

degree equation that solves it.

Here we have a microscopic diatom algae. Its

surface is formed by hexagons and

pentagons… but heptagons appear as well.

You might investigate about this in :

Radiolarian Aulonia Hexagona

Symmetry: cultural-historical and ontological

aspects of ... - Google

From viruses to fullerene molecules

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READING

Harry Kroto (on the right) hold a model of fullerene. On the left the model with 60-

carbon atoms (the famous ball). The Nobel prize in Chemestry was shared by Curl,

Kroto and Smalley in 1996.

Graphite (in your pencil) and diamond are minerals, both of them pure carbon, but with

different positions of the atoms. Graphite is very soft and cheap but diamond is

very,very hard and expensive. Fullerene is another possibility.

Concerning the question of what kind of 60-carbon atom structure might give rise to a

superstable species, we suggest a truncated icosahedron, a polygon with 60 vertices and

32 faces, 12 of which are pentagonal and 20 hexagonal”.

Truncated icosahedron: technical term for a soccer ball in the United States, a

football everywhere else.

The scientists who vaporized the graphite to produce C60 named the new allotrope

buckminsterfullerene (shortened to fullerenes or buckyballs) because the geodesic

domes designed by inventor and architect Buckminster Fuller provided a clue to the

molecule’s structure (front page of this theme).

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SPHERE AND SPHERICAL SURFACE

Sphere is the part of greatest space bounded by a given surface. This spherical surface,

can be defined in different ways and so we have:

Do you believe the coin is spinning?

a) Locus (set of points with the same

property): set of points whose

distance to one point called centre is

the same, its radius.

b) Surface of revolution which is

created by a semicircumference

spinning around the diameter.

c) Limit polyhedron with infinity

faces at the same distance of its

centre.

Spherical surface is not developable (it

can´t be flattened onto a plane without

distortion). For example, skin of an

orange.

Emma Castelnuovo is an Italian

Maths teacher. She wrote a

great didactic book “Geometry”.

In this book, volume and surface of sphere are

deduced from the Galileo’s bowl.

Equal area in every section at different heights and

then we have the same volume for cone and bowl.

V cone = V bowl = V cylinder – V semisphere

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When we know the volume of a sphere we can suppose that it is compound by infinity

of pyramids with common vertex in the centre of the sphere and height equals the radius

and base just a little bit of spherical surface then Σ pyramids = Sphere.

Volume V = (4 π r3 )/3 Surface area A = 4 π r2

Max Bill’s sculpture in grey granite.

“ Semisphere centred on two axes”

14.-Built this sculpture with Plasticine

from half an orange skin.

If the real sculpture has a radius of 2m,

find out the spherical surface and the

circles surface.

Calculate its volume and its mass if the

density of granite is 2.7 g/cm3.

15.-A water melon with a diameter of 20 cm goes into a cubic box of edge 20

cm.

a) What percentage of the box is occupied by the melon? What percentage is

empty?

b) If we cut a slice by two planes which meet on the diameter with any angle

we get a spherical slice (volume) with its partial-spherical surface (area). Find

the volume of a slice and area of its skin when the angle is 30º. (Spherical

wedge and spherical lune)

16.-In the refinery of Escombreras there are cylindrical and spherical tanks to

store oil and gas. The spherical tank has radius 40 m, find out its capacity and

its surface.

There are more cylindrical tanks because of they are easier to build, the radius

is the same and takes up equal space on the ground. What should the height be

in order to have equal capacity? What is its surface? Could you come to any

conclusion?

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17.-A hemisphere over a cylinder with

equal radius. In this case, radius is 1.1 m

and total height 180 cm.

a) Draw plant and front elevation to scale

1:20

b) Find out the area which is taken on

the ground and the space occupied by

the oven.

c) Calculate the area that is whitewashed

(suppose the hole like a rectangle 60 cm

x 30 cm plus a semicircumference on it).

Typical oven from Murcia.

Shapes 3D and sphere : formulas (Source www.vitutor *)

Spherical wedge and spherical lune Spherical cap

( f . i : melon wedge) ( f . i : lens into the human eye)

* *

We can get area and volume by R = (r2 + h2) /2h Area= 2πRh

proport ional i ty (angle n) . Volume= πh2(3R – h) /3

*

Spherical segment and spherical zone

R = (r 2 + h2) /2h Area= 2πRh

V= πh(h2 + 3R2 +3r2) /6

( f . i : par t of the Earth between

equator and a t ropic)

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TORUS (SPHERICAL RING)

In geometry a torus (pl. tori) is a surface of revolution generated by revolving a circle in three dimensional space around an axis coplanar with the circle. In most contexts it is assumed that the axis does not touch the circle - in this case the surface has a ring shape and is called a ring torus or simply torus if the ring shape is implicit or spherical ring because a sphere can run inside it.

Surface area A= 4π2 R r

Volume V = 2π2 R r2

These formulas are the same as for a cylinder of length 2πR and radius r, created by cutting the tube and unrolling it by straightening out the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side.

A torus is the product of two circles, in this case the small circle is swept around the axis defining the big circle. R is the radius of the big circle, r is the radius of the small one. 18.-Draw an outline of the cylinder created by cutting the ring and stretching it. Put the data, letters, on the outline. Deduce those formulas.

Internal structure of PLT (Princeton Large

Torus). Model of nuclear fusion.

19.-The measurements of this plasmatic

torus are R = 130 cm and r = 45 cm .

Work out its area and volume.

Nuclear fusion and nuclear fission

are two different types of energy-

releasing reactions in which energy

is released from high-powered

atomic bonds between the particles

within the nucleus. The main

difference between these two

processes is that fission is the

splitting of an atom into two or more

smaller ones while fusion is the

fusing of two or more smaller atoms

into a larger one . Fusion happens

inside the Sun H2 + H2 → He2

(from hydrogen to helium) and it is

suggested nuclear fusion will be the

source of energy for the future.

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Sydney Opera House Coordinates: 33º 51’ 31’’ S , 151º 12’ 50’’ E

READING

The Sidney Opera House is a multi-venue performing arts centre in the Australian city of Sydney. It was conceived and largely built by Danish architect Jørn Utzon, finally opening in 1973 after a long gestation starting with his competition-winning design in 1957. Utzon received the Pritzker Prize, architecture's highest honour, in 2003. The Pritzker Prize citation stated:

“ There is no doubt that the Sydney Opera House is his masterpiece. It is one of the great iconic buildings of the 20th century, an image of great beauty that has become known throughout the world – a symbol for not only a city, but a whole country and continent. ”

The Sydney Opera House was made a UNESCO World Heritage Site on 28 June 2007. It is one of the 20th century's most distinctive buildings and one of the most famous performing arts centres in the world.

The design work on the shells involved one of the earliest uses of computers in structural analysis, in order to understand the complex forces to which the shells would be subjected. In mid-1961, the design team found a solution to the problem: the shells all being created as sections from a sphere. This solution allows arches of varying length to be cast in a common mould, and a number of arch segments of common length to be placed adjacent to one another, to form a spherical section.

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EARTH GLOBE

The shape of the Earth approximates a

sphere flattened along the axis from pole to

pole such that there is a bulge around the

equator. This bulge results from the

rotation of the Earth, and causes the

diameter at the equator to be 43 km larger

than the pole-to-pole diameter.

20.- The diameter of this plastic globe

is 60 cm and the real radius of the

Earth is 6,370 km

a) What is the scale?

b) Calculate the plastic surface and the

volume of the model.

Erathostenes knew the Earth was round

and he was the first man in getting its

radius with an excellent aproximation,

with shadows, sticks and mind.

Working in Syene and Alexandria, which

Eratosthenes assumed were on the same

meridian, he estimated the distance

between the cities to be about 5,000 stades

(a stade is believed to be approximately

one-tenth of a mile and a mile is 1,609 m).

At summer solstice, at noon, the Sun cast

no shadow in Syene, but in Alexandria a

shadow was visible. Using a gnomon (a

vertical stick), Eratosthenes measured the

shadow's angle (in the figureA = B) to be

about one-fiftieth of a circle.

21.- Calculate the radius of the Earth

with Erathostenes’ data.

Find out the percentage of error if we

consider the radius 6,370 km.

You can read about this in COSMOS by Carl Sagan, pages 14-17, (the book is in our library, in Spanish) or on Earth radius - Wikipedia, the free encyclopedia

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Geographic coordinate system and geographical coordinates

A geographic coordinate system is a

coordinate system that enables every location on

the Earth to be specified by a set of numbers. A

common choice of coordinates is latitude,

longitude and elevation but we are not going to

consider the elevation, so we have the

geographical coordinates (latitude, longitude).

Latitude (L) of the point P is the angle which forms the radius to P with the equatorial

plane and it runs from 0º to 90º, North or South . It is defined by the parallel.

Longitude (l) of the point P is the angle which forms the planes of meridians 0º

(Greenwich) and of P and it runs from 0º to 180º, East or West.

22.- One metre is defined as one

tenmillionth part of a quarter of

meridian.

From the length of a meridian deduce

the radius of the Earth.

Draw on the graph paper a quarter of

meridian with the convenient scale

and from it obtain the radius and

lenghth of the parallel where Murcia

lies (38º N) and the same for the

Tropic of Capricorn (23° 26′ 16″S)

Use a protractor for the angles.

Buy a political world map without names (dumb) to do the following activities

23.- Put these points on the map

London(50º N, 0º); Quito(0º, 80º W); Sidney(34º S, 151º E); Oslo(60º N, 10º E)

24.- Estimate the coordinates of these places

Murcia ; La Coruña ; Galápagos Islands ; the most eastern and western points

of Spain ; the more southern point of Africa ; one point on the Arctic Circle.

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25.- You know that between two points on different meridians there is a time

difference. What geographical coordinate influences that difference ? Why?

What is the angle for an hour difference on the time zones map?

Give the time difference (h, m, s) between the most eastern and western points

of Spain.

26.- Two points that are antipodal to one another are connected by a straight

line running through the centre of the Earth.

What is the relation between their latitudes? And between their longitudes?

Put on the map the antipodal point of La Coruña. Where is it?

27.- Christophorus Colombus departed from Palos de Moguer (Huelva)(37º N,7º

W) in the direction of La Gomera (Canary Islands)(28º N, 18º W), where he

took in supplies for crossing the Atlantic Ocean toward La Española (Santo

Domingo)(20º N, 70º W) in tha Caribean Sea. Put these three points on the map

and supposing straight lines, find the real distance Colombus travelled.

Mercator (Netherlands) drew in 1569 the

first rigorous representation of the Earth.

He used the circumscribed cylinder as

surface projection. This map conserves

distances.

Peters (Germany) introduced in 1976 a

map which conserves areas and was

drawn in 1856 by James Gall . He used

as surface projection a cylinder which

goes through parallels 45º N and 45º S.

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Because the sphere is not developable the flat maps have a deformation. They are

projections from the spherical surface to other (plane or developable). On one hand,

Mercator’s map is useful in navigation (by plane or ship) but it is not valid to areas, on

the other hand, Peters’ map conserves areas but not distances (useful, for example, for

studying natural resources).

Gall-Peters map (Gall drew it in 1856, and Peters popularized it in 1976)

WRITING

Write a short composition (50-80 words) about the picture below these lines. It has to

start with these words: “ The designer of this advertisement…

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EVALUATION POLYHEDRA SPHERE THE GLOBE

1.-When you cut a cube with a plane it is possible to get four sections which divide it

into two parts with equal area and volume.

a) Draw these four sections on the cubes below this line (0.5 points)

b) Draw the development of half a cube when that section is a rectangle (the biggest

one) and the edge of the cube is 5 cm. (1 point)

c) Find out area and volume of that half a cube. (1 point)

2.-Reason to find out area and volume of the regular

octahedron of the figure when its edge equals 2 m.

Work with ABC triangle and take into account

octahedron equals two pyramids.

(2.5 points)

3.-The Titanic was a huge ship which sank on its first journey in 1912.

It left Southampton (51º N, 1ºW) towards New York (41º N, 72º W) and it sank at the

position (42º N, 50º W).

a) Put the three points on the world map. (0.75 points)

b) Find the distance ran by the Titanic before sinking (use graphic scale) (0.75

points)

c) What is the difference of time (h, m, s) between Southampton and NY (1 point)

4.-A cylinder whose height is 21 cm contains three tennis balls which are tangents to the

cylinder and its bases.

a) Draw an outline with the measurements. (0.5 points)

b) Find the radius, area and volume of one ball. (0.75 points)

c) Calculate the percentage of the cylinder which is empty. (1.25 points)