The traditional kaleidoscope uses the prism as a basic unit. Through the reflections of the mirrors on the inner surfaces of a prism, we can see many symmetric axles and patterns. However, traditional kaleidoscope only presents two-dimensional images. The three-dimensional kaleidoscope we designed uses the pyramid as a basic unit. We can see three-dimensional images via mirror faces on the inner surfaces of the pyramid. Our examination shows that not all pyramids are able to show three-dimensional images but only pyramids with specific angles do. The kaleidoscope we designed differs from the traditional kaleidoscope in the following ways: ①Our three-dimensional kaleidoscope not only shows three-dimensional space, but also shows it through two-dimensional planes. ②Our three-dimensional kaleidoscope is based on three kinds of pyramid: three-sided pyramid (tetrahedron), four-sided pyramid (square pyramid) and five-sided pyramid (pentagonal pyramid). Each kind of pyramid has its own charm in showing its reflected images. ③Each kind of pyramid is separated from its entire geometry. However, it can present the whole view of the geometry after mirrored. Our exploration shows that two dimensional and three dimensional worlds can change into one another under specific conditions. Perhaps before long we can discover more dimensions and explain the relationships between each kind of dimension. This treatise mainly involves the principles of Euclidean Geometry and Projective Geometry.

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Motivated by an extra credit problem from our Linear Algebra class, we study the invertibility probability of binary matrices (the number of invertible binary matrices divided by the total number of binary matrices). Binary matrices are of interest in combinatorics, information theory, cryptology, and graph theory. It is known that the invertibility probability of n × n binary matrices goes to 1 as n→∞. We conjecture that this probability monotonically increases as the size of the binary matrix increases, and we investigate this by exploring how n×n binary matrices of rank n and rank (n−1) can be enlarged to (n + 1) × (n + 1) invertible binary matrices. Calculating this explicitly for the identity matrix, we obtain a probable bound that would show that, in a sense, our conjecture is asymptotically true. With the use of a computer, we also computed how many (n + 1) × (n + 1) invertible binary matrices can be enlarged from n×n matrices of rank n and rank (n−1) for small n. In addition, we study the invertibility probability of matrices with entries in Z_q.

Tangram is one of the oldest puzzle games in China. It can make up volatile and wonderful patterns. To master the game, it would be ideal if the players can go through several complexity levels from the easiest to the most complicated. However, tangram today does not have the difficulty grade. Only a similar puzzle game—Japanese jigsaw puzzle has the difficulty grade which is not very reasonable. And the grading process is not provided. Therefore, we try to provide the theoretical basis of dividing the difficulty grades by means of probability of the patterns. We hope we can make some complements to this ancient game. Among the complicated influential factors, we sum up three key elements: the number of answers，the length of side, the basic figures. Based on these points, we have devised three plans, the procedural thinking of computer, the length-of-side principle, division principle, to calculate the probability. Then, we made comprehensive analysis on the probability. Finally, we got the difficulty grade of some classical tangram patterns. Moreover, we summarize some skills of piecing together the patterns.

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