BreakThroughColour

'LEAP' DAY 16: WHAT IS A CUBE?

LEAP!Tracy HolmesComment

In Geometry, a Cube is a solid object with 6 sides, 8 corners, and 12 edges. In Algebra, a Cube is a number multiplied by itself twice. In BreakThroughColour, it's both . . .

 
 

CUBE: THE OBJECT

So, what exactly is a Cube anyway? According to Wikipedia:

In Geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

So, it's like an ice cube, or a dice cube, or, if this is 1980, it's one of these:

 
 

According to official Rubik's Cube trivia, 1 out of 7 people alive have played with a Rubik's Cube, attempting to sort its 54 different colour squares into 6 different colour groups, one on each of its 6 sides.

But a Cube doesn't have to be so multi-coloured. It can be the same one colour on all 6 sides, like 'all' Black, or 'nothing' White

1 x 1 x 1 = 1

A cube with sides that are all one unit long is called a unit cube. In BreakThroughColour, these single-unit solids are all unit cubes:

8cubes.jpg

2 X 2 X 2 = 8

Using the hierarchy of colour, connecting the 'nothing' of White to the three 'single scoop' element primary Corner Hues (Cyan, Magenta, and Yellow), then connecting those with the 'double scoop' Corner Hues (Red, Green, and Blue), and completing the bigger Cube with 'triple scoop' Black in the back, these eight little cubes can connect together to become an all-or-nothing 'nothing-but-corners' composite cube, like this:

 
 

We know from Who Wants Ice Cream? that two versions (all or nothing) of each of the three element primaries gives us 8 different colours. Connecting them to one another in this 'nothing-but-corners' cube, we get a bigger cube with a unit cube value of 2. Another way to say '8' is to say '2 x 2 x 2,' or 'two to the power of three,' or '2 cubed.'

3 x 3 x 3 = 27

We know from Who Wants Pizza? that with three versions of each of the element primaries (none, all, or 'only on half'), we can make 27 different colours. Moving each of the eight Corners out just enough for the Connectors to squeeze in the middle, we increase the size of the cube to 3 unit cubes in each direction. Starting with the White unit cube in the centre, you can count 3 front to back (to the Cyan corner), 3 left to right (to the Magenta corner), and 3 top to bottom (to the Yellow corner). Another way to say '27' is to say '3 x 3 x 3,' or 'three to the power of three,' or '3 cubed.'

Same Corners, more colours...

 
 

6 X 6 X 6 = 216

Doubling the number of steps it takes to get from one corner to the next one doubles the unit cube length, width, and height of the cube. But it more than doubles its volume. Going from o to 5, with nonea littlesomemore than halflots, and all, we can have 6 versions of Cyan x 6 versions of Magenta x 6 versions of Yellow. We end up with '6 x 6 x 6,' or 'six to the power of three,' or '6 cubed.'

 
 

Same corners, way more colours. The more versions we have for Cyan, Magenta, and Yellow, the more ways we can combine them. And with three different primary elements, it's always to the power of three.

CUBE: THE EQUATION

So, what else is a Cube? According to Wikipedia:

In Algebra, the cube of a number 'n' is its third power: the result of the number multiplied by itself twice:

n × n × n

This is also the volume formula for a geometric cube with sides of length 'n,' giving rise to the name.

Which brings us back to Geometry. Scroll to the top, read, and repeat...


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