A CUBE

Do you know about CUBE ?
Look at the pictures below !

                                       
A box, a rubiks, and a dice above are cuboid. Mention other the cuboid objects! 

Parts of A Cube
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Face :	Also called facets or sides. A cube has six faces which are all squares, so each face has four equal sides and all four interior angles are right angles.
Edge :	A line segment formed where two edges meet. A cube has 12 edges. Because all faces are squares and congruent to each other, all 12 edges are the same length.
Vertex:	A point formed where three edges meet. A cube has 8 vertices

The sides of ABCD.EFGH CUBE is:

1 ) ABCD side

2 ) ABFE side

3 ) ADHE side
4 ) EFGH side
5 ) DCGH side
6 ) BCGF side
So, a cube has six sides.

The edges of ABCD.EFGH CUBE is:
 1 ) AB edge
 2 ) EF edge
 3 ) HG edge
 4 ) DC edge
 5 ) BC edge
 6 ) FG edge

 7 ) EH edge
 8 ) AD edge
 9 ) AE edge
10) BF edge
11) CG edge
12) DH edge
So, a cube has twelve edges.

The vertices of ABCD.EFGH CUBE is:
 1 ) A vertex
 2 ) B vertex
 3 ) C vertex
 4 ) D vertex
 5 ) E vertex
 6 ) F vertex

 7 ) G vertex
 8 ) H vertex
So, a cube has eight vertices.

Base on the description above, we can describe the definition of cube as follows: 

Definition of Cube

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A solid with six congruent square faces. A regular hexahedron.

The Net of A Cube

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A pattern that you can cut and fold to make a model of a solid shape.A pattern from combination of six square shape cuboid. 

This is a net of a cube.

There are eleven nets of a cube.

Volume and Surface Area of A Cube

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To find volume of cube with use this formula:

Volume = (Edge Length)³

To find surface area of cube with use this formula:

Surface Area = 6 × (Edge Length)²

A cube is also called a hexahedron because it is a polyhedron that has 6 (hexa- means 6) sides.

Cubes make nice 6-sided dice, because they are regular in shape, and each face is the same size.
In fact, you can make fair dice out of all of the Platonic Solids.

EXERCISE

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Fibonacci Sequence

The Fibonacci Sequence is the series of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

The next number is found by adding up the two numbers before it.

• The 2 is found by adding the two numbers before it (1+1)
• Similarly, the 3 is found by adding the two numbers before it (1+2),
• And the 5 is (2+3),
• and so on!

Example: the next number in the sequence above is 21+34 = 55

It is that simple!

Here is a longer list:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, ...

Can you figure out the next few numbers?

Makes A Spiral

When we make squares with those widths, we get a nice spiral:

Do you see how the squares fit neatly together?
For example 5 and 8 make 13, 8 and 13 make 21, and so on.

The Rule

The Fibonacci Sequence can be written as a "Rule" (see Sequences and Series).

First, the terms are numbered from 0 onwards like this:

n =	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	...
xn =	0	1	1	2	3	5	8	13	21	34	55	89	144	233	377	...

So term number 6 is called x₆ (which equals 8).

Example: the 8th term is the 7th term plus the 6th term:  x8 = x7 + x6

So we can write the rule:

The Rule is x_n = x_n-1 + x_n-2

where:

• x_n is term number "n"
• x_n-1 is the previous term (n-1)
• x_n-2 is the term before that (n-2)

Example: term 9 is calculated like this:

x9	= x9-1 + x9-2
	= x8 + x7
	= 21 + 13
	= 34

Golden Ratio

And here is a surprise. When we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio "φ" which is approximately 1,618034...

In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:

A	B		B / A
2	3		1,5
3	5		1,666666666...
5	8		1,6
8	13		1,625
...	...		...
144	233		1,618055556...
233	377		1,618025751...
...	...		...

Note: this also works when we pick two random whole numbers to begin the sequence, such as 192 and 16 (we get the sequence 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ...):

A	B		B / A
192	16		0,08333333...
16	208		13
208	224		1,07692308...
224	432		1,92857143...
...	...		...
7408	11984		1,61771058...
11984	19392		1,61815754...
...	...		...

It takes longer to get good values, but it shows that not just the Fibonacci Sequence can do this!

Using The Golden Ratio to Calculate Fibonacci Numbers

And even more surprising is that we can calculate any Fibonacci Number using the Golden Ratio:

The answer always comes out as a whole number, exactly equal to the addition of the previous two terms.

Example:

When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got the answer 8,00000033. A more accurate calculation would be closer to 8.

Try it for yourself!

A Pattern

Here is the Fibonacci sequence again:

n =	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	...
xn =	0	1	1	2	3	5	8	13	21	34	55	89	144	233	377	610	...

There is an interesting pattern:

• Look at the number x₃ = 2. Every 3rd number is a multiple of 2 (2, 8, 34, 144, 610, ...)
• Look at the number x₄ = 3. Every 4th number is a multiple of 3 (3, 21, 144, ...)
• Look at the number x₅ = 5. Every 5th number is a multiple of 5 (5, 55, 610, ...)

And so on (every nth number is a multiple of x_n).

Terms Below Zero

The sequence works below zero also, like this:

n =	...	-6	-5	-4	-3	-2	-1	0	1	2	3	4	5	6	...
xn =	...	-8	5	-3	2	-1	1	0	1	1	2	3	5	8	...

(Prove to yourself that each number is found by adding up the two numbers before it!)

In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a +-+- ... pattern. It can be written like this:

x_−n = (−1)ⁿ⁺¹ x_n

Which says that term "-n" is equal to (−1)ⁿ⁺¹ times term "n", and the value (−1)ⁿ⁺¹ neatly makes the correct 1,-1,1,-1,... pattern.

History

Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before!

About Fibonacci The Man

His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy.

"Fibonacci" was his nickname, which roughly means "Son of Bonacci".

As well as being famous for the Fibonacci Sequence, he helped spread Hindu-Arabic Numerals (like our present numbers 0,1,2,3,4,5,6,7,8,9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble! Thank you Leonardo.

Fibonacci Day

Fibonacci Day is November 23rd, as it has the digits "1, 1, 2, 3" which is part of the sequence. So next Nov 23 let everyone know!

source: https://www.mathsisfun.com/numbers/fibonacci-sequence.html

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