I’m sure most of you are familiar with the line by line algebra showing how each cube or prism in the Trinomial Cube is constructed mathematically, turning lengths a, b and c into the 27 three dimensional pieces needed to create it. We would like to offer a newer solution to this problem. One that organizes the algebra slightly differently and includes geometry at every step. For example, what would happen if we chose not to collect like terms and instead viewed each and every newly formed part of the solution as having it’s own special geometry and orientation.
We have discovered that this method, when done correctly, not only places the cubes and prisms into their correct locations, it also orients the pieces as they are found in the geometric solid solution you are familiar with from the shelf. Basically, this method constructs the cube in three dimensions, piece by piece.
Let’s take a look at the equation…
There are 3 parts. The first part, let’s call height. The second part, let’s call length and the third part, let’s call width. Now let’s imagine a prism of height = h, length = l and width = w, as depicted in the diagram below. Viewing the equation this way, it shouldn’t be difficult to understand that the solution will form a cube since it’s components (a + b + c) are all equal. Which is the same as saying h = l = w.
When we multiply out the first two parts of the equation, we are in reality multiplying the first two components of height x length or (a + b + c) (a +b + c). We know from experience that height multiplied by length produces an Area. Since (a + b + c) is equal to (a + b + c) the area produced is a square. Then, If we divide the square into 3 x 3 locations we have a spot for each product that ensues. We have locations for the solution on a 3 x 3 square grid, instead of somewhere on a line. (see diagram below) It then becomes more apparent that we are multiplying height (a + b + c) times length (a + b + c) producing 9 areas.
The above diagram is the solution for (a + b + c) multiplied by (a + b + c), height times length. The first row is height a x length (a + b + c) , the second row is height b x length (a + b + c), the third row is height c x length (a + b + c).
If you take a look at each product in our square you will see that the products make either a square, or a rectangle. For example (a x a) is a square, or (a x b = ab) is a rectangle. At this point we are working with 9 areas. However, we are not yet finished. We still need to multiply by a third component, width. The width is again (a + b + c), so we need to multiply each part of our solution by this amount. In order to do that we will first multiply each of the nine areas in our square grid by width a, then by width b, then by width c, resulting in three new squares each having a different width. (we are now working with 9 areas x 3 different widths which equals 27 volumes)
(It should be noted at this point that only one product abc actually occurs. When we choose to include location and orientation it is just not possible to collect like terms because they are no longer alike. Every piece has a unique position and a unique orientation)
As a next step we will simplify to reveal the locations of the three cubes and leave the rest as they are.
We now need to make a simple adjustment for height. For example, we suppose that each layer should have the same height so that each layer fits together layer by layer to form the cube. If you remember? height is the first letter of each element. Re-arranging the rows by height, we take the top row of each layer to form the first, bottom a – layer. We take the second row (in order) from each layer to form the second, middle b – layer. And we take the third row (in order) from each layer to form the third c – layer at the top. This will result in the Trinomial Cube as commonly known.
Here we have it. The Trinomial Cube, layer by layer, with each cube or prism in it’s correct location and having it’s correct orientation. Basically, by choosing not to collect like terms we end up with a geometric solution, enabling one to build the 3D cube quite easily. (I can do it myself!) This is how each piece of the puzzle appears as a solid. The first layer has height a, the second layer has height b, and the third layer has height c.
It should also be noted that when understood this becomes a 3 step mechanical process that will work on any number of terms, cubed. The first step is to create the square grid and solve for the areas, h x l. The number of elements in the grid will depend on the number of terms, squared. For example, the decanomial will have 10 x 10 = 100 elements, just the same as the decanomial square. The second step is to multiply this square times each of the widths, creating volumes h x l x w. This step results in the geometric solution with the cubes correctly located, however a third step is required to organize the pieces into layers. The third step is to rearrange the rows. The first row in each layer, is the bottom layer. The second row in each layer becomes the second layer and so on like this until you have re-created every layer. And this is it. One would now have the position of each piece in the cube and the orientation, as well as the volumes written in the format h x l x w. (no painful process of collecting a mess of like terms and a logical place to put things)