Escher\u27s Problem and Numerical Sequences

Counting problems lead naturally to integer sequences. For example if one asks for the number of subsets of an $n$-set, the answer is $2^n$, or the integer sequence $1,~2,~4,~8,~ldots$. Conversely, given an integer sequence, or part of it, one may ask if there is an associated counting problem. There might be several different counting problems that produce the same integer sequence. To illustrate the nature of mathematical research involving integer sequences, we will consider Escher\u27s counting problem and a generalization, as well as counting problems associated with the Catalan numbers, and the Collatz conjecture. We will also discuss the purpose of the On-Line-Encyclopedia of Integer Sequences

Teaching geometrical principles to design students

We propose a new method of teaching the principles of geometry to design students. The students focus on a field of design in which geometry is the design: tessellation. We review different approaches to geometry and the field of tessellation before we discuss the setup of the course. Instead of employing 2D drawing tools, such as Adobe Illustrator, the students define their tessellation in mathematical formulas, using the Mathematica software. This procedure enables them to understand the mathematical principles on which graphical tools, such as Illustrator are built upon. But we do not stop at a digital representation of their tessellation design we continue to cut their tessellations in Perspex. It moves the abstract concepts of math into the real world, so that the students can experience them directly, which provides a tremendous reward to the students

Burnside\u27s Lemma

Throughout my time at the University of Redlands, I have been drawn towards areas of math that are abstract. I enjoy knowing that an equation made up of completely abstract ideas can can be used to solve real world problems in a variety of subjects other than math. This paper covers Burnside\u27s Lemma including a proof and a variety of examples. It culminates with counting the number of unique Escher paintings that can be made. Also within this paper are discussion and proofs about both Polia Enumeration and Sylow p-subgroups

The Look of Math

If consideration is given to math’s look when it is written out, math can be experienced visually without working with or understanding its logical content. This “meaningless math” then becomes a vessel into which new contextual meaning is injected by a given observer. “The look of math” is a part of ethnomathematics, or the culture of mathematics, which necessarily and anti-Platonically (i.e., taking math to be purely a human endeavor) contains mathematics per se. Formal elements in math-as-art can be categorized as algebraic, geometric, diagrammatic, or organic. The main idea is not “math can be art,” but rather that math has a look, which is not the same thing as math’s inner structure, and which is exploitable in nonmathematical ways. However, math has special properties that amplify its importance as some specific thing that has rarely been isolated and defined as art, thereby granting math potential for contribution to art

The life and work of Major Percy Alexander MacMahon

This thesis describes the life and work of the mathematician Major Percy Alexander MacMahon (1854 - 1929). His early life as a soldier in the Royal Artillery and events which led to him embarking on a career in mathematical research and teaching are dealt with in the first two chapters. Succeeding chapters explain the work in invariant theory and partition theory which brought him to the attention of the British mathematical community and eventually resulted in a Fellowship of the Royal Society, the presidency of the London Mathematical Society, and the award of three prestigious mathematical medals and four honorary doctorates. The development and importance of his recreational mathematical work is traced and discussed. MacMahon's career in the Civil Service as Deputy Warden of the Standards at the Board of Trade is also described. Throughout the thesis, his involvement with the British Association for the Advancement of Science and other scientific organisations is highlighted. The thesis also examines possible reasons why MacMahon's work, held in very high regard at the time, did not lead to the lasting fame accorded to some of his contemporaries. Details of his personal and social life are included to give a picture of MacMahon as a real person working hard to succeed in a difficult context

Impossible Thoughts, Alternative Spaces

In this thesis I explore meaning and possibility in the works of Samuel Beckett and Italo Calvino, with emphasis on how the texts refute coherent meaning and use that refutation to re-inscribe the boundaries of the possible