=
Using a t-number of 38 I’m going to work it out mentally and then using the formula to see if they are the same.
Formula= (5x38=190)-63=127
Adding= (28+29+21+20+19) = 127
That is my proven formula.
I then tried the same technique on a 10x10 grid
And an 8x8 grid
And then came up with the formulae:
8x8 = 5n-56
9x9 = 5n-63
10x10 = 5n-70
I saw that they all had 5n at the start. I then observed that the difference between the minus parts of each was 7 x the width of the grid. Making the formula 5n-7g to work out the t-total for any size grid (g stands for the grid width).
Another thing I am able to work out is – the bigger the t-number, the bigger the t-total in every single case.
Translations on a Vector
In this section I am going to work out the formula for translating on a vector. Translating on a vector is basically moving the shape from one position to another using a ‘code’ to determine how far. A vector looks like (and will be referred to as) (a/b). ‘a’ being the distance the shape should be moved right (if a negative number is there, the shape moves left). The same occurs for the second number, but in a vertical manner (a positive means upwards, negative downwards).
I am going to firstly create a formula for just translations finding the t-number, not the t-total. I will then combine the formula I found with my earlier formula to develop one that calculates the t-total after translations.
To begin with, I drew out the t-shapes on a grid
And formed a table with the results I found
.
From examining how the results were affected by the vectors, I was able to determine that every positive ‘a’ you move it, it adds on ‘a’ to every part of the ‘t’, like pictured in red in the table below. Also, every positive ‘b’ it was moved, it added ‘–gb’ to each square, like pictured in blue below. After adding up these new additions to my t-formula-square, I developed the formula ‘n+a-gb’ for finding the t-number.
In finding the previous formula I simultaneously discovered the formula for the t-total after translation. I developed the formula ‘5(n-g x b)-7g’. This formula can be used to discover the t-total after translation by a vector
The limitations of translation were few but there is one very large one. The highest number ‘a’ or ‘b’ could be was (g-3) this was because the t shape was unable to go out of the grid and the formula I developed came from the fact the t-shape was three squares across and three downwards.
Rotation
I beginning by drawing the t-shapes I am going to use and then rotating them by 90 degrees in a clockwise direction. I have also realized that a 90 degree clockwise rotation is the exact same as a 270 degree anti-clockwise rotation. My centre of rotation to begin with will be the t-number. I’ll begin on a 7x7 grid.
I noticed the 5n would be there again, similarly to earlier. So I just worked out the rest using simple mathematics.
16x5=80 87-80=7 → so ((16x5) +7)
17x5=85 92-85=7 → so ((17x5) +7)
18x5=90 97-90=7 → so ((18x5) +7)
So therefore the formula is 5n+7 for a 7x7 grid, I saw again a correlation between the added amount and the grid width. I guessed that a formula of 5n+g would work. I’m now going to prove it.
The displayed formulae t-shape proves my formula for 90 degrees clockwise rotation or a 270 degrees anti- clockwise rotation on a 7x7 grid. (Highlighted red)
I now have to find a formula for any size grid and a 90 degree clockwise/270 degree anti-clockwise rotation. I investigated the correlation between the +7 part of the formula and the grid width; I tested this theory and proved it using this t-formula-shape (in yellow).
I came up with the formula of 5n+7. This seven can be related to the grid width and changed to a g. I will try to prove this below.
I was surprised to see that the formula actually turned out to be 5n-7 as all of the others had involved a g. I incorrectly guessed that but prevented making a mistake with my testing.
I now have to move onto 180 degree rotations. I have realised that 180 degrees in either direction, clockwise or anti-clockwise, will result on the same location so have saved myself some work.
I used a 7x7 number square again and made a t-formula-shape again (in green).
I saw the same thing as before where the 7 in the t-formula-shape is the same as the grid width so came up with this new one (in turquoise).
From this I developed the formula of 5n+7g for the rotation of a t-shape by 180 degrees.
I now have to create a formula for rotation by 270 degrees around the t-number. To accommodate the rotation I had to translate my t-shape (1/0) on my grid.
I developed the formula of 5n-7. I noticed this was the opposite of my 90 degree rotation. I also noticed that each of the formulas for the different rotations contained seven. This furthered my suspicions that 7 was in fact ‘g’ but further testing disproved that theory.
5n-7g 0*
5n+7 90*
5n+7g 180*
5n-7 270*
I noticed that a rotation of 180 degrees from all points of view resulted in the positive being changed into a negative or vice-versa. This can relate to the vectors explained earlier, where the squares translating left result in a negative outcome as well as the squares travelling down.
Conclusion
In conclusion, during the project I discovered a series of formula so I was able to determine the values of t-shapes in any size grids, after various transformations. I found the project steadily became harder as it developed. The formulae were harder to piece together as the transformations changed. In the end I was unable to discover a formula for rotation from a point outside of the t-shape.
I did enjoy the coursework and found it adequately challenging but not so difficult that I was unable to complete it.
