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We review Euler's work on spherical geometry. After an introduction concerning the general place that trigonometric formulae occupy in geometry, we start by the two memoirs of Euler on spherical trigonometry, in which he establishes the…
Some of the most challenging problems in Timaeus' cosmology arise from the geometry of a universe without any void. On the one hand, the universe is spherical in shape; on the other hand, it must be entirely filled with the four basic…
Throughout human history, people have used sight to learn about the world, but only in relatively recent times the science of light has been developed. Egyptians and Mesopotamians made the first known lenses out of quartz, giving birth to…
A reformulation of the three circles theorem of Johnson with distance coordinates to the vertices of a triangle is explicitly represented in a polynomial system and solved by symbolic computation. A similar polynomial system in distance…
In a previous article, we discussed a paradox in Timaeus' cosmology: that there is no void inside the universe, even though it is entirely filled with polyhedra-a mathematical impossibility (Brisson-Ofman 2025). In the present article, we…
Since the dawn of civilization, humanity has grappled with the big questions of existence and creation. Modern cosmology seeks to answer some of these questions using a combination of mathematics and measurement. The questions people hope…
Muography is an imaging technique that can be used to examine the interior structure of large-size objects. The technique is based on measurements of the absorption of cosmic-ray muons passing through the object under study. Muographic…
An almost forgotten gem of Gauss tells us how to compute the area of a pentagon by just going around it and measuring areas of each vertex triangles (i.e. triangles whose vertices are three consecutive vertices of the pentagon). We give…
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…
This study reconstructs the origin of a constant, here called $\Xi$ (Xi), as a primary scaling factor in Old Babylonian mathematics and astronomy. $\Xi$ arises from the practical necessity of precise measurements on the sky or a circle,…
The large scale geometry of the late Universe can be decomposed as R$\times {\Sigma}_3$, where R stands for cosmic time and ${\Sigma}_3$ is the three dimensional spatial manifold. We conjecture that the spatial geometry of the Universe's…
The pyramids of the Giza plateau have fascinated visitors since ancient times and are the last of the Seven Wonders of the ancient world still standing. It has been half a century since Luiz Alvarez and his team used cosmic-ray muon imaging…
We consider the problem of enumerating integer tetrahedra of fixed perimeter (sum of side-lengths) and/or diameter (maximum side-length), up to congruence. As we will see, this problem is considerably more difficult than the corresponding…
In this paper we discuss the orientation of the Egyptian pyramids at Giza with respect to sunrises and sunsets, using SunCalc.net software. We can see that Khufu and Khafre pyramids had been positioned in a manner that, from each pyramid,…
Gromov introduced the notion of a pyramid as a generalization of a metric measure space, based on the idea of the concentration of measure phenomenon. In this paper, we introduce the concept of a direct sum of pyramids, which naturally…
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…
This is an overview of Nasir al-Din al-Tusi's Treatise of the quadrilateral, an invaluable 13th century document on spherical geometry which was translated into French in 1891. The title we are using here is the one given by the translator…
The pyramids of ancient Egypt are said to have shone brilliantly in the sun. Surfaces of polished limestone would not only have reflected diffusely in all directions, but would also likely have produced specular reflections in particular…
The Chinese Pyramids are huge ancient burial mounds. In the satellite images we can see some complexes where the main buildings are the pyramidal mounds of an emperor and his empress. Here we discuss a possible sunrise/sunset orientation of…
The manuscript provides formulas for the volume of a body defined by the intersection of a solid cone and a solid sphere as a function of the sphere radius, of the distance between cone apex and sphere center, and of the cone aperture…