Related papers: A square from similar rectangles
In this paper, we construct certain infinite families of imaginary quadratic fields whose class number is divisible by a given positive integer.
We present and discuss different algorithms for converting rectangular imagery into elliptical regions. We mainly focus on methods that use mathematical mappings with explicit and invertible equations. The key idea is to start with…
We consider the problem of finding 4 rational squares, such that the product of any two plus the sum of the same two always gives a square. We give some historical background and exhibit one such quadruple.
Standard approaches to probabilistic reasoning require that one possesses an explicit model of the distribution in question. But, the empirical learning of models of probability distributions from partial observations is a problem for which…
We prove that two polygons $A$ and $B$ have a reversible hinged dissection (a chain hinged dissection that reverses inside and outside boundaries when folding between $A$ and $B$) if and only if $A$ and $B$ are two noncrossing nets of a…
A new method is proposed to divide a spherical surface into equal-area cells. The method is based on dividing a sphere into several latitudinal bands of near-constant span with further division of each band into equal-area cells. It is…
We prove an integral formula for continuous paths of rectangles inscribed in a piecewise smooth loop. We then use this integral formula to show that (with a very mild genericity hypothesis) the number of rectangle coincidences, informally…
Let T be a tile in the Cartesian plane made up of finitely many rectangles whose corners have rational coordinates and whose sides are parallel to the coordinate axes. This paper gives necessary and sufficient conditions for a square to be…
Magic squares have been an enthralling topic in mathematics for centuries. They are formed by filling in all the cells of a square matrix with the numbers starting from one so that the sum of all rows, columns, and diagonals is the same.…
We present a conceptually simple and intuitive method to calculate and to measure the dissimilarities among 2D shapes. Several methods to interpret and to visualize the resulting dissimilarity matrix are presented and compared.
In this paper, we show that if we decompose a polygon into two smaller polygons, then by comparing the number of extremal vertices in the original polygon versus the sum of the two smaller polygons, we can gain at most two globally extremal…
We study rationality constructions for smooth complete intersections of two quadrics over nonclosed fields. Over the real numbers, we establish a criterion for rationality in dimension four.
In this paper, we further develop the theory of circles of partition by introducing the notion of complex circles of partition. This work generalizes the classical framework, extending from subsets of the natural numbers as base sets to…
An interesting problem in classical differential geometry is to find methods to prove that two surfaces defined by different charts actually coincide up to position in space. In a previous paper we proposed a method in this direction for…
Consider an arrangement of $k$ lines intersecting the unit square. There is some minimum scaling factor so that any placement of a rectangle with aspect ratio $1 \times p$ with $p\geq 1$ must non-transversely intersect some portion of the…
Assembling parts into an object is a combinatorial problem that arises in a variety of contexts in the real world and involves numerous applications in science and engineering. Previous related work tackles limited cases with identical unit…
In this paper, we present a mathematical model for the angular projection of a rectangular arrangement of points in a grid. This simple, yet interesting problem, has both a scholarly value and applications for data extraction techniques to…
We introduce an elementary transformation called flips on tilings by squares and triangles and conjecture that it connects any two tilings of the same region of the Euclidean plane.
We develop a recursive formula for counting the number of rectangulations of a square, i.e the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations,…
We propose an explanatory and computational theory of transformative discoveries in science. The theory is derived from a recurring theme found in a diverse range of scientific change, scientific discovery, and knowledge diffusion theories…