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The celestial phenomenon have always been a source of wonder and interest to people, even as long ago as the ancient Egyptians. While the ancient Egyptians did not know all the things about astronomy that we do now, they had a good…
Multiplication and exponentiation can be defined by equations in which one of the operands is written as the sum of powers of two. When these powers are non-negative integers, the operand is integer; without this restriction it is a…
We prove the High Girth Existence Conjecture - the common generalization of the Existence Conjecture for Combinatorial Designs originating from the 1800s and Erd\H{o}s' Conjecture from 1973 on the Existence of High Girth Steiner Triple…
The arrival of Modern Humans (MHs) in Europe between 50 ka and 36 ka coincides with significant changes in human behaviour, regarding the production of tools, the exploitation of resources and the systematic use of ornaments and colouring…
The early Renaissance artist Albrecht Durer published a book on geometry a few years before he died. This was intended to be a guide for young craftsmen and artists giving them both practical and mathematical tools for their trade. In the…
Recently it has been discovered that on a stone tablet over 3800 years old, the Plimpton-322 table, are carved the geometric relations that exist between the sides of 15 right triangles chosen in a very special way. Due to its property as a…
Lecture given before the Royal Academy Vienna that summarizes the state of knowledge about the mathematics of the ancient Egyptians, up to 1884. Contains all relevant references to classical Greek texts, and the 'latest' archeology results.…
How were surfaces evaluated before the invention of the sexagesimal place value notation in Mesopotamia? This chapter examines a group of five tablets containing tables for surfaces of squares and rectangles dated to the Early Dynastic…
We illustrate Archimedes' method using models produced with 3D printers. This approach allowed us to create physical proofs of results known to Archimedes and illustrate ideas of a mathematician who is known both for his for his mechanical…
Recently the author has studied rings for which products of flat modules have finite flat dimension. In this paper we extend the theory to characterize when products of modules in $\mathcal T$ have finite $\mathcal T$-projective dimension,…
The architectural complexes composed by the two main pyramids of Giza together with their temples are investigated from an inter-disciplinary point of view, taking into account their astronomical alignments as well as their relationships…
For h=3 or 4, Egyptian decompositions into h unit fractions, like 2/D = 1/D1 + ... +1/Dh, were given by using (h-1) divisors (di) of D1. This ancient modus operandi, well recognized today, provides Di=DD1/di for i greater than 1.…
The Great Pyramids of Egypt hide mathematic information unknown up to date. The measurements of the three Great Pyramids of Egypt at Giza show that Egyptians knew how to calculate the circumference, the volume and the area of the sphere…
Let K be an imaginary quadratic field, and x the Dirichlet character corresponding to the extension K/Q. Let m=2n or 2n+1 with n a positive integer. Let f be a primitive form of weight 2k+1 and and nebentype x, or a primitive form of weight…
Egyptologists and historians of mathematics around 1930 did an admirable job in showing that problem 14 of the newly discovered Moscow Papyrus from around 1850 BCE amounts to a general and exact calculation of the volume of a truncated…
Omar Khayyam's treatment of cubic equations by intersections of conic sections has often been read as an anticipation of analytic or coordinate geometry. This paper argues that such a reading obscures the conceptual structure of Khayyam's…
We establish a conjecture of Defant, Hopkins, Poznanovi\'{c}, and Propp concerning the dimensions of toggleability spaces for products of chains, shifted staircases, type-A root posets, and type-B posets. Generalizing this result, we show…
An ancient engineering firm worked successfully in the construction of one of the Seven Wonders of the Ancient World. The engineers used inclined planes, bags of sand and shafts of columns and architraves as wheels and axels.
We give an exposition of Plucker vectors for a system of joint axes in projective 3-space. We use Plucker vectors to analyse dependencies among joint axes, and in particular show that two rotational joints rigidly joined by a bar and each…
We show that on an arbitrary collection of objects there is a wide variety of higher order architectures governed by hyperstructures. Higher order gluing, local to global processes, fusion of collections, bridges and higher order types are…