Chords of an ellipse, Lucas polynomials, and cubic equations
Article by Ben Blum-Smith and Japheth Wood
In collections: Easily explained, Fun maths facts, Puzzles
A beautiful result of Thomas Price links the Fibonacci numbers and the Lucas polynomials to the plane geometry of an ellipse. We give a conceptually transparent development of this result that provides a tour of several gems of classical...
URL: arxiv.org/abs/1810.00492v3
PDF: arxiv.org/pdf/1810.00492v3

Catching a mouse on a tree
Article by Vytautas Gruslys and Arès Méroueh
In collections: Animals, Attention-grabbing titles, Combinatorics
In this paper we consider a pursuit-evasion game on a graph. A team of cats, which may choose any vertex of the graph at any turn, tries to catch an invisible mouse, which is constrained to moving along the vertices of the graph. Our main focus shall be on trees. We...
URL: arxiv.org/abs/1502.06591v1
PDF: arxiv.org/pdf/1502.06591v1

Every natural number is the sum of forty-nine palindromes
Article by William D. Banks
In collections: Fun maths facts, Integerology
It is shown that the set of decimal palindromes is an additive basis for the natural numbers. Specifically, we prove that every natural number can be expressed as the sum of forty-nine (possibly zero) decimal palindromes.
URL: arxiv.org/abs/1508.04721v1
PDF: arxiv.org/pdf/1508.04721v1

Beyond Floating Point: Next-Generation Computer Arithmetic
Article by John L. Gustafson
In collections: Basically computer science, Unusual arithmetic
URL: web.stanford.edu/class/ee380/A

Ice cream and orbifold Riemann-Roch
Article by Anita Buckley and Miles Reid and Shengtian Zhou
In collection: Food
We give an orbifold Riemann-Roch formula in closed form for the Hilbert series of a quasismooth polarized n-fold X,D, under the assumption that X is projectively Gorenstein with only isolated orbifold points. Our formula is a sum of parts each of which is integral and Gorenstein symmetric of...
URL: arxiv.org/abs/1208.0457v1
PDF: arxiv.org/pdf/1208.0457v1

Prime Simplicity
Article by Michael Hardy and Catherine Woodgold

Astonishing Numbers
Article by Richard Hoshino
In collections: Attention-grabbing titles, Easily explained, Integerology
We say that an ordered pair of positive integers $$a,b$$ with $$a \lt b$$ is astonishing if the sum of the integers from $$a$$ to $$b$$, inclusive, is equal to the digits of $$a$$ followed by the digits of $$b$$. Determine all astonishing ordered pairs.
URL: cms.math.ca/publications/crux/

Asymptotic statistics of the n-sided planar Poisson–Voronoi cell: I. Exact results
Article by Hilhorst, H.J.
In collections: Probability and statistics, Geometry
URL: iopscience.iop.org/1742-5468/2

No, This is not a Circle
Article by Zoltán Kovács
In collections: Attention-grabbing titles, Easily explained, Drama!, Geometry
A curve, also shown in introductory maths textbooks, seems like a circle. But it is actually a different curve. This paper discusses some easy approaches to classify the result, including a GeoGebra applet construction.
URL: arxiv.org/abs/1704.08483v2
PDF: arxiv.org/pdf/1704.08483v2

Avian egg shape: Form, function, and evolution
Article by Mary Caswell Stoddard and Ee Hou Yong and Derya Akkaynak and Catherine Sheard and Joseph A. Tobias and L. Mahadevan
In collections: Animals, Food, Geometry, Modelling
Avian egg shape is generally explained as an adaptation to life history, yet we currently lack a global synthesis of how egg-shape differences arise and evolve. Here, we apply...
URL: science.sciencemag.org/content
PDF: science.sciencemag.org/content

New entry!
Graphs, friends and acquaintances
Article by C. Dalfó and M. A. Fiol
In collections: Attention-grabbing titles, Easily explained
As is well known, a graph is a mathematical object modeling the existence of a certain relation between pairs of elements of a given set. Therefore, it is not surprising that many of the first results concerning graphs made reference to relationships between people...
URL: arxiv.org/abs/1611.07462v2
PDF: arxiv.org/pdf/1611.07462v2

The Euler spiral: a mathematical history
Article by Levien, Raph
In collections: History, Geometry
The beautiful Euler spiral, deﬁned by the linear relationship between curvature and arclength, was ﬁrst proposed as a problem of elasticity by James Bernoulli, then solved accurately by Leonhard Euler. Since then, it has been independently reinvented twice, ﬁrst by Augustin Fresnel to compute diffraction of light through a slit, and...
URL: raph.levien.com/phd/euler_hist

The Canonical Basis of $$\dot{\mathbf{U}}$$ for Type $$A_{2}$$
Article by Cui, Weideng
The modified quantized enveloping algebra has a remarkable basis, called the canonical basis, which was introduced by Lusztig. In this paper, all these monomial elements of the canonical basis for type $$A_{2}$$ are determined and we also give a conjecture about all polynomial elements of the canonical basis.
URL: arxiv.org/abs/1208.5531
PDF: arxiv.org/pdf/1208.5531v3

Circle Packing for Origami Design Is Hard
Article by Demaine, E.D. and Fekete, S.P. and Lang, R.J.
In collection: Geometry
We show that deciding whether a given set of circles can be packed into a rectangle, an equilateral triangle, or a unit square are NP-hard problems, settling the complexity of these natural packing problems. On the positive side, we show that any set of circles of total area 1 can be...
URL: arxiv.org/abs/1008.1224
PDF: arxiv.org/pdf/1008.1224v2

Irrationality From The Book
Article by Miller, Steven J. and Montague, David
In collections: About proof, Fun maths facts
We generalize Tennenbaum's geometric proof of the irrationality of sqrt(2) to sqrt(n) for n = 3, 5, 6 and 10.
URL: arxiv.org/abs/0909.4913
PDF: arxiv.org/pdf/0909.4913v2

Prime Number Races
Article by Andrew Granville and Greg Martin
In collections: Attention-grabbing titles, Easily explained, Fun maths facts, Integerology
This is a survey article on prime number races. Chebyshev noticed in the first half of the nineteenth century that for any given value of x, there always seem to be more primes of the form 4n+3 less than x then there are of the form 4n+1. Similar...
URL: arxiv.org/abs/math/0408319v1
PDF: arxiv.org/pdf/math/0408319v1

Implications of the Turing Completeness of Reaction-Diffusion Models, informed by GPGPU simulations on an XBox 360: Cardiac Arrythmias, Re-entry and the Halting Problem
In collections: Basically computer science, Basically physics, Unusual computers
URL: ncbi.nlm.nih.gov/pubmed/195775
PDF: research.microsoft.com/pubs/79