Mathematics

We present a short proof of Cantor's Theorem (circa 1870s): if a n cosnx+ b n sinnx→0 for each x in some (nonempty) open interval, where a n , b n are sequences of complex numbers, then a n and b n converge to 0.

We give an excellent approximation of the average number of spins of a simplified version of a two-player version of the game Dreidel. We also make a conjecture on the average number of spins of the full version of the game.

We show how we found substitution rules for a quasiperiodic tiling with local rotational symmetry and inflation factor 1 + sqrt(3). The base tiles are a square, a rhomb with an acute angle of 30 degrees, and equilateral triangles that are cut in half. These half-triangles follow three different substitution rules and can be recombined into equilateral triangles in nine different ways to make minor variations of the tiling. The tiling contains quasiperiodically repeated 12-fold rosettes. A central rosette can be enlarged to make an arbitrarily large tiling with 12-fold rotational symmetry. An online computer program is provided that allows the user to explore the tiling.

There is little known about the methods used by the ancient Babylonians and Egyptians to arrive at their recorded estimates of the value of Pi. A surprisingly accurate estimate of Pi was recently revealed coded within a verse in the book of 1 Kings, the value of which suggests how it might have been measured. The coded value is 111/106, which is a continued fraction representation of Pi/3. This suggests that the value may have been measured using an iterative measurement of remainders when comparing the two lengths C (circumference of the circle) and 6R (6 times the radius). This article describes a method that could have been used 3000 years ago to make such a measurement, the expected measurement errors, and a computer simulation that assesses the chances of such a method succeeding in obtaining the coded value of 111/106. The result indicate that with the technology available at the time the proposed measurement method would have been possible and would have had about a 75% chance of producing the result 111/106 after a few hundred measurements.

In this article, we give a particular recreational application of the sequence A000533 and A261544 in "The On-line Encyclopedia of Integer Sequences" (OEIS). The recreational application provides a direct extension to "The Repetitious Number" puzzle of Martin Gardner contained in The Second Scientific American Book of Mathematical Puzzles and Diversions published in 1961. We then provide a generalization to the repetitious number puzzle and give a related puzzle as an illustrative example. Finally, as a consequence of the generalization, we define a family of sequence in which the sequences A000533 and A261544 belong.

In this note, we describe an interpretation of the (continuous) Fourier transform from the perspective of the Chinese Remainder Theorem. Some related issues are discussed.

This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the potential to demystify quadratic equations for students worldwide.

We usually construct mathematical objects that are accessible, on which we can put our hands, but a huge part of the mathematical existing is actually wild. Here we explore part of the wild world: its inhabitants are knots that are infinitely knotted, spheres that try to hug themselves, colorful earrings, labyrinths of labyrinths...

In this article, we give another visual proof of π e < e π .

A closed linkage mechanism in three-dimensional space is an object comprising rigid bodies connected with hinges in a circular form like a rosary. Such linkages include Bricard6R and Bennett4R. To design such a closed linkage, it is necessary to solve a high-degree algebraic equation, which is generally difficult. In this lecture, the author proposes a new family of closed linkage mechanisms with an arbitrary number of hinges as an extension of a certain Bricard6R. They have singular properties, such as one-dimensional degree of freedom (1-DOF), and certain energies taking a constant value regardless of the state. These linkage mechanisms can be regarded as discrete Möbius strips and may be of interest in the context of pure mathematics as well. However, many of the properties described here have been confirmed only numerically, with no rigorous mathematical proof, and should be interpreted with caution.