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Project Featured On:Digital Art Served — 4/24/12

"256"

by

Daniele De Nigris

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"256"
di De Nigris Daniele
How to create many of regular, crazy and impossible object using only 4 modules. I'm not a mathematician,
so I hope that all arguments are correct. I do not know if these objects were already created, but it was very interesting to create them. However this project will be useful in the future. I hope that this topic interests you. Have a good vision. For any clarification: denigrisdaniel@gmail.com
Create a grid of 15x15 squares
Create this parallelepiped following the grid
Apply two orthogonal lines passing through the center of the figure
Now the figure is composed of 4 different modules
The 4 modules
In this scheme you can see that each module has the same attachment point. All modules can be joined together
Some examples of possible combinations
That is the question
"256" this is the number of the possible combinations (O.M.G.They are too numerous)
When I start to create all the combinations I realize that some are the same (the only difference that they are rotated). So, I decided to study this scheme, to try to eliminate the possibility that repeat themselves.
The 256 combinations are composed of 7 groups:
A: 4 squares with same color
B: 3 squares of the same color + 1 square with a different color
C: 2 squares on the same side with the same color + 2 squares on the same side with the same color
D: 2 squares on the diagonal with the same color + 2 squares on the diagonal with the same color
E: 2 squares on the same side with the same color + 2 squares on the same side with the different color
F: 2 squares on the diagonal with the same color + 2 squares on the diagonal with the different color
G: 4 squares with different color
Group A: There are 4 combinations
All 4 combinations are different
Group B: There are 48 combinations
There are only 12 different combinations. The remaining 36 combinations are the same (they are only rotated)
Group C: There are 24 combinations
There are only 6 different combinations. The remaining 18 combinations are the same (they are only rotated)
Group D: There are 12 combinations
There are only 6 different combinations. The remaining 6 combinations are the same (they are only rotated)
Group E: There are 96 combinations
There are only 24 different combinations. The remaining 72 combinations are the same (they are only rotated)
Group F: There are 48 combinations
There are only 12 different combinations. The remaining 36 combinations are the same (they are only rotated)
Group G: There are 24 combinations
There are only 6 different combinations. The remaining 18 combinations are the same (they are only rotated)
That is the second question
"70" this is the number of the different possible combinations
But now I found another problem: when I start to create this 70 combinations I realize that some combination are specular. So, I decided to study this problem, to try to eliminate the possibility that repeat themselves.
I understand the problem: the red module and the gray module, are specular. So I decide, to reflect all the 70 combinations, to find any similar combinations.
There are only 3 different combinations. The remaining 1 combination is the same (It's specular)
There are only 7 different combinations. The remaining 5 combinations are the same (they are specular)
There are only 4 different combinations. The remaining 2 combinations are the same (they are specular)
There are only 4 different combinations. The remaining 2 combinations are the same (they are specular)
There are only 14 different combinations. The remaining 10 combinations are the same (they are specular)
There are only 7 different combinations. The remaining 5 combinations are the same (they are specular)
There are only 4 different combinations. The remaining 2 combinations are the same (they are specular)
That is the last question
"43" this is the number of the different possible combinations.
From 256 to 43 ...sounds good!
...finally
Take a random combination
We create two areas:
The colored areas: they tell us how to color the picture.
The white areas: for each combination, they will have a different color.
Take the right colors to the "colored areas"
after applying the color to the "Colored areas" we can apply the color the
"white areas"
Remove the excess filets and replace the color combination with the letters.
Now we can created all the 43 combinations.
combinations 1-4
combinations 5-8
combinations 9-12
combinations 13-16
combinations 17-20
combinations 21-24
combinations 25-28
combinations 29-32
combinations 33-36
combinations 37-40
combinations 41-43
One example with a best rendering
One example with a best rendering
THANK YOU FOR WATCHING!

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