# And then there were Three - a Study on Cubes

## One Lump or Two ?

## Cubes

So, having looked at **squares** as numbers multiplied by themselves, or raised to the power of **two**, as in The Very Next Step - Squares and the Power of Two , it makes sense to go on to the next power, which is **three**, the raising of numbers to which gives us what we know as ** cubes**. Why do we call them

**? For pretty much the same sort of reason we give**

*cubes***squares**their name.

Consider for example the box in the illustration above, **three feet** by **three feet** by **three feet**. This is **3³**, or **27 cubic feet**. Here it can be seen that we have added another

**dimension**to what we previously had with

**squares**and

**area**. This time we are dealing with

**, which is**

*volume***x**times

**x**

**²**.

The following is a list of numbers and their ** cubes** :

## Chart of Numbers and their Cubes

We can see for a start how large **cubes **** **do get very quickly, since this time around we are multiplying the same number by itself, and then by itself again.

An interesting thing to note, is that a number can be both a **square** and a ** cube** at the same time. For example, we have

**64**, which is not only

**8²**, but also happens to be

**4³**. As a matter of fact,

**64**also equals

**2**, but we will get to higher powers of numbers in a later Hub, when we look at the

^{6}**Laws of Exponents**.

So, in the way that an ordinary number, such as **five**, which is in fact **5 ^{1}**, gives us the idea of

__length__,

**squares**, like

**5²**=

**25**give us

**area**, and

**, where**

*cubes***5³**=

**125**, furnish us with

**.**

*volume*Incidentally, if you’re curious to know what **5 ^{0}** is, this equals

**one**. As a matter of fact, any number other than

**zero**, which has been raised to the

**power**of

**zero**, will also equal

**one**, and once again, we shall leave this to our discussion on the

**Laws of Exponents**. And by discussion I mean that this is where I speak, and you listen.

One thing to note with **powers**, is that, just because, for example, **2 **×** 3 **=** 3 **×** 2**, this does **NOT** mean that

**2³**=

**3²**, and in fact it doesn’t, since the first equals

**eight**, while the second gives us

**nine**. There are in fact some examples where

**a**=

^{b}**b**, since

^{a}**2**=

^{4}**4²**=

**16**, but otherwise you need to be aware that most of the time the

**power**is

*NOT***interchangeable with its**

**base**.

The simple reason is because we are multiplying different numbers a given number of times. That is, for example, if we have **2³**, we are going **2 **×** 2 **×** 2**, which is divisible by **2**, but not **3**, whereas **3²**, or **3 **×** 3**, ** is** divisible by

**3**, but is not divisible by

**2**.

Now the reason **2 ^{4 }**just

*happens to*equal

**4²**, is because

**4**also divides by

**two**, but most of the time we are dealing with

**bases**not evenly dividable by their

**exponents**, and vice versa. It is only the

**base**which determines what the answer is a product of, while the

**exponent**is simply how many

**the**

*times***base**is to be multiplied by itself, and is not itself normally a factor of the final answer.

There is another important fact to note when dealing with both **areas** and ** volumes**, and that is what happens when you fiddle with the sides of either a

**square**or a

**.**

*cube*Say you have a simple __ length__ of ** five inches**. If I **double** this, my __length__ is simply **doubled**, and it becomes an even **ten inches**. But imagine the **five inches** this time represent the sides of a square, so that its total **area** is equal to **5² inches²**, which is **twenty five square inches**. What happens when I **double** *both* sides, so my square is now **10 **by **10 **?

Well now my **area** is equal to **10² inches², **or **100 square inches**. The **area** has not **doubled**, but rather **quadrupled. **It is now **four times** what it was before. How does this work ? Think about it - you’re dealing with **area**, where things are now **squared**, so the amount by which you multiply your sides is also **squared**, and since you timesed it by **two**, your **area** gets to be multiplied by **2²**, which of course is **4**.

Realise that this only occurs if you **double** both length ** and** width. If you

**double**only one, then you will only

**double**the

**area**. Also, if I

**triple**my length and width from

**five inches**to

**fifteen**, I then end up with

**3²**=

**9**times the

**area**.

Now, if we are dealing with ** volume**, we can see the pattern, so that, if we have a

**of say**

*cube***five inches by five by five**, its

**equals**

*volume***125**. We see that if we

*cubic*inches³**double**all

**three**dimensions to equal

**ten inches**, our

**increases to**

*volume***10³**=

**1000**, which this time is

*cubic*inches**eight**times the volume, simply because

**2³**=

**8**, because we are

**the amount by which we are multiplying the side, also, just as we**

*cubing***squared**before.

So, we ** cube** the factor by which we multiply all

**three**edges. And again, if we do not multiply

*all*these edges of our

*, well, it is best then to take all*

**cube****three**new dimensions, and multiply accordingly to find our new

**. The idea of**

*volume***the amount by which we multiply all**

*cubing***three**edges is to be used as a

*shortcut*to find the new

**, as long as all three edges have been multiplied by the same amount.**

*volume*Now, just as **squares** are sums of successive odd

numbers, there’s a similar pattern to with our ** cubes** :

And as with the **squares**, where there is a way to work out up to which odd number you need to add to find it, we can work out *from* which **odd number**, and *to* which other **odd number** we need to sum, in order to work out our ** cube**.

We see for a start that there are as many odds to add as the number you are ** cubing**. That is,

**5³**is the sum of

**five**odds, and so forth.

But the way to find both **odd numbers** that are the first and last of the summation sequence to be added with all the odds in between, is to find your value of **x ** to be ** cubed**, multiply it by

**( x**-

**1 )**then add

**one**, and this is your initial number, then to find the final one, you multiply

**x**by

**( x**+

**1 )**, then subtract

**one**, so that your first and last numbers in the sequence are given by

**x²**-

**x**+

**1**, and

**x²**+

**x**-

**1**, and that these become the

**bounds**of your sum.

For example, if you wish to work out ** from** which initial

**odd**number

**which last**

*to***odd**number to sum in order to determine

**9³**, we go :

**9**×

**8**+

**1 (**=

**73 )**, and then

**9**×

**10**-

**1 (**=

**89 )**, so that the

**nine**

**odd**

**numbers**you sum to work out

**9³**are :

**73**,

**75**,

**77**,

**79**,

**81**,

**83**,

**85**,

**87**and

**89**, all of which, like

**9**

**³**, equals

**729**.

The formulas **x²** -** x **+** 1** and **x² **+** x **-** 1** were worked out by looking at the list of sums of odds for each ** cube**, and finding the pattern.

Another way to find the ** cube** of a given number is to get the sums of the

**odd numbers**used to find the

**square**of that particular number, and then multiply it by that number, since after all,

**3³**is just the same as saying

**3²**×

**3**.

For Example :

## Patterns amongst Cubes

The pattern we get from these is that when we have at least **three** of them, we can see that each of the numbers added in parentheses **( )**, which will give us the sum equal to the ** cube**, are the same distance apart. For example, where we add

**(3**+

**9**+

**15)**, each of these has a difference of

**6**between itself and the one after it. In this case, for the sum that is equal to

**3³**, this alternative gap of

**6**equals

**3**×

**2**, where

**x**=

**3**, and

**2**is the normal gap we get between the

**odd numbers**we sum in order to get the

**square**of whatever

**x**is.

Thus we should be able to conclude that in each case if we want to find the numbers which sum to equal **x³**, the first number to which we add the next ones will be **x** itself, and then the **gap** between this and the next number we add will be **2x**, which as noted, is the **gap** between all successive numbers after that in the sum to **x****³**.

As another example :** 2² **=** (1** +** 3) **; ×** 2 **=** (2 **+** 6)**, and** 2 **here in parentheses equals **x**, since **2** is the number we are trying to find the ** cube** of. The gap from the

**2**to the

**6**that are in the parentheses is

**4**, which is

**2**times

**2**, or if you like,

**2x**.

If we analyse these numbers in parentheses, these ** multiples **of the **odd numbers** summed to equal a **square**, which now add to be a ** cube**, we see a pattern. They become sums of successive

**odd numbers**, each multiplied by

**x**, which makes sense, because this is in effect what we did to get them. If you think about it, it stands to reason, that if a certain sum of numbers adds to a

**square**, then multiplying that whole sum, or each member of that sum individually by the number you have

**squared**, will then give you its

*, since as noted before,*

**cube****x³**=

**x²**×

**x**.

## Cubic Gnomons

## Cubic Gnomons

*The above illustration shows how the *** cubes** of successive numbers can be arranged into

*, where each colour represents a sum adding to the*

**Gnomons****of the lowest number in the sum, known as the**

*cube***Base**. This is owing to the fact that each

**is a multiple of its own base for a start.**

*cube*Now each sum has a different formula. Take the fact that **27 **=** 3 **+** 6 **+** 9 **+** 6 **+** 3**. If we take **x **as equal to **3** then this ends up being ** x **+** 2x **+** 3x **+** 2x **+** x**, so it is that we get a kind of triangular sum of **x’s**, but we can also note that it could be seen in this specific case as **x³** =** x² **+** 6x**.

What this means graphically is that when we draw that function, then **x³** =** x² **+** 6x** implies also that **x³** -** x² **-** 6x **=** 0**, and on the graph this would occur when ** x **=** 3 **and **y** =** 0**

## Final Word

There shall indeed be more of this to come, with relevant explanations of these value specific functions.

We continue this journey in the next Hub, Moving on to Higher Powers - a First look at Exponents, and if You are curious, take a look at the other Hubs, The Maths They Never Taught Us - Part One, The Maths They Never Taught Us - Part Two , The Maths They Never Taught Us - Part Three, The Very Next Step - Squares and the Power of Two , The Power of Many More - more on the Use of Exponents, Mathematics - the Science of Patterns , More on the Patterns of Maths, Mathematics of Cricket , The Shape of Things to Come , Trigonometry to begin with, Pythagorean Theorem and Triplets, Things to do with Shapes, Pyramids - How to find their Height and Volume, How to find the Area of Regular Polygons, The Wonder and Amusement of Triangles - Part One, The Wonder and Amusement of Triangles - Part Two, the Law of Missing Lengths, The Wonder and Amusement of Triangles – Part Three : the Sine Rule, and The Wonder and Amusement of Triangles - Part Four : the Cosine Rule.

Also, feel free to check out my non Maths Hubs :

Bartholomew Webb , They Came and The Great New Zealand Flag

## Disclaimer

Most of the material here are my own discoveries, so anything that is Copyright and or a Registered Trademark will be acknowledged as such. Thus, any reference to any Copyright or Registered Trademark is credited as such. Some discoveries are my own, but may also have been found independently by others as Mathematics is a living language, and it was **Ralph Waldo Emerson** who described the demise of any language that does not keep growing. Some information has been referenced in a number of publications, most in the public domain, as well as on Wikipedia ( copyright 2013 Wikimedia Foundation ).