|Home Institution:||MORGAN STATE UNIVERSITY|
|Project:||Applications of Dissecting Polygons up to Higher Dimensions|
|Mentor:||Dr Vinay Vaishampayan|
I am working on dissecting rectangles and reassembling them, this can be used to encode and decode bit information and other applications. My mentor has worked on this project before with now PhD in math Antonio Campello, who showed that there exists a dissection in higher dimensions.
The goal of the summer research project is to extend Montucla's dissection into hyperdimensions, find algorithms for coding and maybe even look at other cut algorithms. To accomplish this, I will be using Mathematica, vector analysis techniques, principles in Geometry, and some notes on Campello's work among other resources.
This week I read a draft and paper by Antonio Campello, analyized part of it and Montucla's dissection. I drew up a representaion on Mathematica complete with polygon limits and hash lines representing a brick laying structure for the algorithm to be formed. And of course there was orientation and a presentation of our research plan.
This week, I have algorithms and even extended Montucla's dissection to parallelograms complete with significant relationship and Mathematica drawing. Worked on algorithm for this and use of this Algorithm to write Mathematica reconstuction. Also, worked on candidate relationship for 3-d.
Further relevant reading involving tiling was done with reanalysis for 3-d and higher. My mentor has also introduced me to other source material. I will read for this over the weekend if I have a chance.
This week, I figured out algorithms to systematically cut and translate any parallelogram to another and back in 2-d.
I have adapted it for a working program on mathematica.
And I have begun working with an idea for extension into 3-d.
In 2d, I wrote a progam using region plots, it works but has a tendency to round off corners and lose sections. This is not acceptable. I need to find another way to plot this.
I found a better way for 2d using line plots involving vertices, although more tedious produced better graphs and a visual understanding of the problem.
I presented this to my mentor and got feedback on other interesting things for exploration.
I have prepared drawings in my write up to show Monticula's technique in its alternate forms a key relationship that others may have overlooked, algorithms on movements, observations on expanding vs contracting, efficiency, tesselations and shapes involved for that pesky partial.
I have 3-d for a particular transform of rectangular prisms complete on mathematica. Note I again needed a better technique and in finding one had to trick my way around limitations with this way of plotting.
I figured out how to expand that in 3d on discrete levels.
I Will extract my algorithms and mechanics and write a new ambitious mathematica program next week along with paper outlining details.
I have prepared a mathematica program that does handle discrete levels.
Finding partial levels will allow continuity, and has now been accomplished, including finding critical geometric shapes that were needed.
This gives me a better working knowledge of the problem and allows me to not only come up with algorithms, but formalize an approach for higher dimensions.
I found that my program gave me too many dependent variables. Realizing I had a dimension I could extend into; I found a way that gave me the flexability I needed to take any rectangular prism to another with two predetermined sides,(before I only had one).
I have some reading to do (to relate my findings to past ones) and need to get algorithms and limits for (hyper-)planes for 3d and higher dimensions nailed down.
This is my final week.
I have written a proof utilyzing hyperplanes and vectors to restructure a vector set defining a hyperparallelepiped to one with a set dimension for redimensionalizing. In other words, hypervolume preserving orthogonalizing technique.
I can apply rotational parameters to any dimension level to fix sides and find cuts needed to restructure any hyperparallelepiped to another.
I also have mathematica programs with varying parameters that; does a restructuring of a parallelepiped in 3d, Montucla's recut in 2d and 3d rectangular prism to another rectangular prism
With this Montucala's method is extended to higher dimensions and further applicable to not only to hyper rectangular prisms but also to parallelepipeds I plan to do further work on predetermining rotational vectors to desired final side lengths along with connecting with Campello's work.
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