Solid Geometry

Solid Geometry is the geometry of three-dimensional space,
the kind of space we live in.

Three Dimensions

It is called three-dimensional, or 3D, because there are three dimensions: width, depth and height.

Simple Shapes

Let us start with some of the simplest shapes:

Common 3D Shapes

Properties

Solids have properties (special things about them), such as:

• volume (think of how much water it could hold)
• surface area (think of the area you would have to paint)
• how many vertices (corner points), faces and edges they have

                              ROGER  PENROSE

                    Sir Roger Penrose is widely considered one of the most important mathematical physicists of our time. His work on the properties of black holes, along with Stephen Hawking's, revolutionized our understanding of the nature of the universe.

WORKS OF PENROSE: 

                In the 1960s, working with Stephen Hawking, Penrose was able to prove that all matter within a black hole collapses into a single point, aka a singularity. According to Encyclopedia Britannica, this is "a geometric point in space where mass is compressed to infinite density and zero volume."
               Mass is so concentrated at this point of space that even light photons are unable to escape the black hole's maw.

 PENROSE  TRIANGLE:

                      In the 1950s Penrose, with his father, devised the Penrose triangle or impossible triangle/tribar.  Interestingly, the optical illusions appear to have first been developed by Oscar Reutersvärd in the 1930s. He was a Swedish graphic artist who has come to be known as the "father of the impossible figure". 

               ERIK  CHRISTOPHER  ZEEMAN

Zeeman's research has been in a variety of areas such as topology, in particular PL topology, dynamical systems and mathematical applications to biology and the social sciences. His initial research was in topology and one of his theorems was the unknotting of spheres in five dimensions. Certainly his work in topology would make him one of the leading topologists of all time but he may be known principally for other work.

Perhaps he is best known for his work on catastrophe theory for, although this theory was due initially to René Thom, it was Zeeman who brought it before the general public giving widespread publicity to applications of what was before that time thought of as pure mathematics. In particular Zeeman pioneered the applications of catastrophe theory in the biological and behavioural sciences, as well as the physical sciences.

Among the books which Zeeman has published are the texts Catastrophe theory (1977), Geometry and perspective (1987) and Gyroscopes and boomerangs (1989). One of his many memorable quotes, from his Catastrophe theory text, says much about mathematical philosophy:-

Technical skill is mastery of complexity while creativity is mastery of simplicity.

A shorter introduction to catastrophe theory than his 1977 book was given by Zeeman in his beautifully written survey article Bifurcation and catastrophe theory [Contemp. Math. (1981)]. The article introduces catastrophe theory in a unified way giving both elementary and non-elementary aspects. There is an elementary discussion of the cusp and the pitchfork and a statement of the classification theorem for elementary catastrophes. Asked what were the highlights of his own research he explained:-
 

In 1978, Zeeman gave the Christmas Lectures at the Royal Institution, out of which grew the Mathematics Master classes for 13-year old children that now flourishes in forty centres in the United Kingdom. He was the 63^rd President of the London Mathematical Society in 1986-88 and delivered the Presidential Address to the Society on 18 November 1988 On the classification of dynamical systems.

SYMMETRY OF SHAPES

Activity: Symmetry of Shapes

An Octagon

Let us try the Octagon (the 8-sided shape)

Is this a Line of Symmetry?

Let's try folding it:

Yes! When folded over, the edges match perfectly

So let us draw it on:

I found another way too::

Tried this	It works!

So let us draw it on, too:

In fact I found 8 Lines of Symmetry:

Annulus

An annulus is a flat shape like a ring. Its edges are two circles that have the same center.

The face of a metal washer is an annulus

Something shaped like an annulus is said to be annular.

Saturn has annular rings:

Area

Because it is a circle with a circular hole, you can calculate the area by subtracting the area of the "hole" from the big circle's area:

Area = πR² − πr²
= π( R² − r² )

Example: a steel pipe has an outside diameter (OD) of 100mm and an inside diameter (ID) of 80mm, what is the area of the cross section?

Convert diameter to radius for both outside and inside circles:

• R = 100 mm / 2 = 50 mm
• r = 80 mm / 2 = 40 mm

Now calculate area:

Area = π( R² - r² )

Area = 3.14159... × ( 50² − 40² )

Area = 3.14159... × (2500 − 1600)

Area = 3.14159... × 900

Area = 2827 mm² (to nearest mm²)

BAYES' THEOREM

                      BAYES' THEOREM

       Bayes’ Theorem is a way of finding a probability when we know certain other probabilities.

The formula is:

P(A|B) = P(A) P(B|A)P(B)

Which tells us:		how often A happens given that B happens, written P(A|B),
When we know:		how often B happens given that A happens, written P(B|A)
		and how likely A is on its own, written P(A)
		and how likely B is on its own, written P(B)

Let us say P(Fire) means how often there is fire, and P(Smoke) means how often we see smoke, then:

P(Fire|Smoke) means how often there is fire when we can see smoke
P(Smoke|Fire) means how often we can see smoke when there is fire

So the formula kind of tells us "forwards" P(Fire|Smoke) when we know "backwards" P(Smoke|Fire)

Example:

• dangerous fires are rare (1%)
• but smoke is fairly common (10%) due to barbecues,
• and 90% of dangerous fires make smoke

We can then discover the probability of dangerous Fire when there is Smoke:

P(Fire|Smoke) =P(Fire) P(Smoke|Fire)P(Smoke)

=1% x 90%10%

=9%

WHAT IS TREE DIAGRAM?

Probability Tree Diagrams

                Calculating probabilities can be hard, sometimes we add them, sometimes we multiply them, and often it is hard to figure out what to do ... tree diagrams to the rescue!

Here is a tree diagram for the toss of a coin:

There are two "branches" (Heads and Tails) The probability of each branch is written on the branch The outcome is written at the end of the branch

We can extend the tree diagram to two tosses of a coin:

How do we calculate the overall probabilities?

• We multiply probabilities along the branches
• We add probabilities down columns

EXPERIMENT WITH A DIE

Activity: An Experiment with a Die

You will need: A single die

Interesting point

Many people think that one of these cubes is called "a dice". But no!

The plural is dice, but the singular is die. (i.e. 1 die, 2 dice.)

The common die has six faces:

We usually call the faces 1, 2, 3, 4, 5 and 6.

High, Low, and Most Likely

Before we start, let's think about what might happen.

Question: If you roll a die:

• 1. What is the least possible score?
• 2. What is the greatest possible score?
• 3. What do you think is the most likely score?

The first two questions are quite easy to answer:

• 1. The least possible score must be 1
• 2. The greatest possible score must be 6
• 3. The most likely score is ... ???

MEAN FROM FREQUENCY TABLE

The Mean from a Frequency Table

It is easy to calculate the Mean:

Add up all the numbers,
then divide by how many numbers there are.

         But sometimes we don't have a simple list of numbers, it might be a frequency table like this (the "frequency" says how often they occur):

Score	Frequency
1	2
2	5
3	4
4	2
5	1

(it says that score 1 occurred 2 times, score 2 occurred 5 times, etc)

We could list all the numbers like this:

Mean = 1+1 + 2+2+2+2+2 + 3+3+3+3 + 4+4 + 5(how many numbers)

But rather than do lots of adds (like 3+3+3+3) it is easier to use multiplication:

Mean = 2×1 + 5×2 + 4×3 + 2×4 + 1×5(how many numbers)

And rather than count how many numbers there are, we can add up the frequencies:

Mean = 2×1 + 5×2 + 4×3 + 2×4 + 1×52 + 5 + 4 + 2 + 1

And now we calculate:

Mean = 2 + 10 + 12 + 8 + 514
=  3714  =  2.64...

Geometric Mean

                The Geometric Mean is a special type of average where we multiply the numbers together and then take a square root (for two numbers), cube root (for three numbers) etc.

Example: What is the Geometric Mean of 2 and 18?

• First we multiply them: 2 × 18 = 36
• Then (as there are two numbers) take the square root: √36 = 6

In one line:

Geometric Mean of 2 and 18 = √(2 × 18) = 6

It is like the area is the same!

Example: What is the Geometric Mean of 10, 51.2 and 8?

• First we multiply them: 10 × 51.2 × 8 = 4096
• Then (as there are three numbers) take the cube root: ³√4096 = 16

In one line:

Geometric Mean = ³√(10 × 51.2 × 8) = 16

It is like the volume is the same:

MEAN MACHINE

The Mean Machine

See how the Arithmetic Mean (or Average) is calculated:

To find the Mean (or Average) divide the
Sum of the numbers by the Count of the numbers

 Numbers: 

7

7

21

7

= 3

 Count

 Sum

 Count

Divide

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How to Find the Mean

The mean is the average of the numbers.

It is easy to calculate: add up all the numbers, then divide by how many numbers there are.

In other words it is the sum divided by the count.

Example 1: What is the Mean of these numbers?

6, 11, 7

• Add the numbers: 6 + 11 + 7 = 24
• Divide by how many numbers (there are 3 numbers): 24 / 3 = 8

The Mean is 8