*With its 6 Sides, 8 Corners, and 12 Edges, and with colours outside and inside, a 3D BTC Cube gives us a solid look at the exponential power of C x M x Y . . .*

Here are some **fast facts about cubes**, with examples from the colourful building blocks we're getting to know in BreakThroughColour.

## 8 CORNERS

**All cubes have ****8 corners**, no matter how many unit cubes there are along each edge. No matter how big a BTC Cube gets, there will always be **the same 8 Corner Colours**.

In each of these views of a Colour Cube, you can see 7 of the 8 corners. White is front and centre, with the 6 Corner Hues circling around it, and Black is hiding at the back, directly behind White (in the opposite corner of the Cube).

## 12 EDGES

**All cubes have 12 edges** connecting the 8 corners, and all of the edges are the same length. In a BTC Colour Cube, 3 edges connect the element primaries (Cyan, Magenta, and Yellow) directly to White, and 3 edges connect the compound primaries (Red, Green. and Blue) directly to Black on the opposite side. The other 6 edges are the Hue-to-Hue connections around the middle.

## 6 SIDES

**All cubes have 6 sides** (also called faces, facets, or planes). Each side is a perfect square, with 4 corners and 4 edges.

In a BTC Colour Cube, there are **3 Tint sides**; they show the Tints and meet in the White Corner. Spin the cube around and you see the **3 Shade sides**; these show the Shades and meet in the Black Corner. In both views, you can see the circumference of pure Hues.

## OPPOSITE PAIRS

All opposite sides are parallel to each other. In the simplest terms, the 6 sides can be grouped into 3 opposite 'all or nothing' pairs:

- a
**Tint**side with**no Cyan**opposite a**Shade**side with**full Cyan** - a
**Tint**side with**no Magenta**opposite a**Shade**side with**full Magenta** - a
**Tint**side with**no Yellow**opposite a**Shade**side with**full Yellow**

Not only can the sides be paired into opposites, **each of the colours on any side has an opposite complementary colour in the opposite place on the opposite side**, no matter how many unit cubes there are.

## LENGTH, AREA, VOLUME

The dimensions of a cube are the lengths of the three edges which meet at any vertex. In geometry, the dimensions are often described as values of X, Y, and Z. In BreakThroughColour, the dimensions can be described as values of **C, M, and Y**. All three axes start at '0' in the White Corner, and extend in three different directions to the **full values of each** of the three element primary Corner Hues. The **Black** Corner (opposite of White) has **full values of all three**.

The volume of a cube is equal to the product of its dimensions, and since its dimensions are equal, the volume is equal to the third power, or cube, of any one of its dimensions. The volume of a BTC Cube can be described as the product of the unit cube length of **C x M x Y**.

## OUTSIDE and INSIDE

A cube with a unit cube length of 3 or more has an 'outside' and an 'inside.' In a BTC Colour Cube, you can see the colours on the outside, but you can't see the colours on the inside.

The outside colours are the Corner Colours, the Connector Hues (Hue + Hue), the Tints (Hue + White), and the Shades (Hue + Black). The inside colours are the Tones (Hue + White + Black).

In a *Colour Basics* Cube, there are **27 unit cubes** (3 x 3 x 3), and **you can see 26 of them** on the outside. There is **only one unit cube hidden in the middle**: Hue Grey, the midpoint intersection of all 8 Corners (and in fact, the midpoint for all 13 *Colour Basics* opposite pairs.

In the 6 x 6 x 6 *BreakThroughColour* Cube, there are 4 'layers' between each pair of opposite sides, so instead of just one 'middle' Tone (Colour Basics' Hue Grey), **there are 4 x 4 x 4 = 64 different 'Greys' hiding inside the bigger BTC Colour Cube**.

## HALF, GREATER THAN, LESS THAN

The *Colour Basics* Cube has **an uneven unit cube length of 3**, which is why there are 'halfway' colours at the midpoint of each edge, the centre of each face, and at the centre core of the cube itself. These are the 19 Connector Cards we explored in **Who Wants Pizza?**. Any other Colour Cube with an uneven unit cube length will have these same 19 *Colour Basics* Connector Colours (as well as the same constant 8 Corners).

The bigger *BreakThroughColour* Colour Cube has **an even unit cube length of 6**. Because there is no 'halfway' point in a row of 6 (3 are on one side of half and 3 are on the other), there are no exact 'middle' unit cubes along the edges or across the sides of the *BTC *Colour Cube, or any other Colour Cube with an even unit cube length. Pure Hues belong to only one Corner family. Tints are either White with less-than-half Hue, or Hue with less-than-half White; same is true for the Shades. And unlike the *Colour Basics* Cube, which has the amazing **Mr. Hue Grey** at the centre of everything, there is no single central middle Grey in the BTC Cube.

## 8 CORNERS, 8 GREYS

Does that mean the BTC Cube (or any cube with an even unit cube length) has no centre? No. It does have an exact centre, but the point is, it's just a point, and not a unit cube on its own. Instead, there is a core of eight cubes, an 'almost middle Grey' version of each of the eight Corner Colours.

They are connected in the same configuration, with a lighter Hue Grey opposite a darker Hue Grey, and the same six Hues circling between them, far from pure but still maintaining their familial identity.

# COLOUR LAB

There are lots of different ways to connect the geometric and algebraic basics of a cube with the basic and breakthrough aspects of aBreakThroughColourCube. Defining colour in three dimensions is not a new concept, but relative to both the theory and practice of colour throughout history, it's new enough, and for many, it may be the biggest 'Leap' of all.

The time you take now toreview the cube as both a physical object and a mathematical equationwill be time well spent. If you've got Colour Cards, Basics or BTC, have them handy to sort, stack, and connect into lines, squares, grids, and layers.

Get to know theCornersnot just for what they are, but where they are, and why.

Get to know all theConnectors, not just the 19 in the Colour Basics deck, but all the Edge, Face, and Core colours in the whole BTC collection (and that's most of the cards, because if it's not a Corner, it's a Connector).

Get to know thethree different kinds of connections:thestraight'single element' connections that simply shift any colour to another colour right beside it; thediagonal'double duty' connections that shift a colour in two out of three elemental ways; and thetriagonalconnections that take us corner to corner, into and through the hidden Hues, shifting C, M, and Y up and down all at the same time. Get to know (and love!) the exponentialpower of threeinherent inC x M x Y. Shifting 3 to 8, then 27, then 216, and then...?

A colour wheel gets things rolling, but inevitably, it will leave you flat. The square can be a good foundation (or wall, or roof), but it's still just a plain old plane, and we need 3 planes to go 3D. From Point to Line to Shape to Solid, from Corner to Edge to Face to Cube, with CMY as your XYZ, a solid way to understand colour is to **understand colour as a solid**.