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Questions tagged [geometry]

Branch of Mathematics about the properties of the shapes, their similarities and transformations in the space.

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An unpublished calculation of Gauss and the icosahedral group

According to p.68 of Paul Stackel's essay "Gauss as geometer" (which deals with "complex quantities with more than two units") , Gauss calculated the coordinates of the vertices of ...
user2554's user avatar
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Where does the term "reflection" come from?

Earlier today, I was asked why a motion of the plane that fixes a line of points is called a reflection and I was stumped for an answer. The best explanation I can think of is that the image of a ...
Numeral's user avatar
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12 votes
1 answer
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What is the origin of "two straight lines cannot enclose a space" axiom in Euclid's Elements?

This post is prompted by a recent question on MSE asking about "Axiom 10" of Euclid's Elements, as found in editions by Byrne and Conway: "Two right lines cannot enclose a space". ...
RobinSparrow's user avatar
4 votes
1 answer
207 views

What is the history of vector bundles and their characteristic classes?

The theory of vector bundles (and their characteristic classes) appears to have been standardized in the 20th century by all of the familiar names. Considering its substantial importance throughout ...
user19642's user avatar
3 votes
1 answer
362 views

Ancient drawing board in mathematics

According to Van Der Waerden's "Science Awakening", it was common for Ancient Greek mathematicians to use a board filled with sand to draw their figures, ie : But the ancients made their ...
Slereah's user avatar
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75 views

What is Laguerre's definition of the angle via the cross ratio?

I recently read an article which said Cayley showed that affine geometry could be developed from projective geometry after he learnt of Laguerre's definition of the angle using the cross ratio. This ...
Mozibur Ullah's user avatar
2 votes
2 answers
228 views

Ancient Egyptian geometry

When reading on the topic of Ancient Egyptian geometry by Ancient Greek philosophers, there is a certain sense that this is quite a thriving discipline that seems comparable to the type of geometry ...
Slereah's user avatar
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3 votes
1 answer
108 views

Which geometer first compared a length (one dimensional) to an area (two dimensional)?

What are sources placing a length (one dimensional) in proportion to an area (two dimensional)? The Greek geometers compared quantities of the same dimension: e.g. the area of a circle is in ...
SRobertJames's user avatar
8 votes
1 answer
1k views

DeMorgan's commentary on Euclid's Elements

Augustus DeMorgan wrote comments on Euclid's Elements, which capture many of the most important points. Heath quotes them extensively. I cannot find any source for the original: Where can I see ...
SRobertJames's user avatar
9 votes
2 answers
217 views

Who first considered signed area?

Who first suggested that the area enclosed by a closed path and that enclosed by that path traversed in reverse could be regarded as equal in magnitude but opposite in sign? Cauchy must have noticed ...
James Propp's user avatar
1 vote
0 answers
70 views

How did the Ancient Greeks conceive of the Platonic solids?

Now we generally think of the Platonic solids as being the regular convex polyhedra. And while the Ancient Greeks were aware of this solids as being particularly special, I don't believe that it is ...
Sriotchilism O'Zaic's user avatar
2 votes
1 answer
87 views

Negative coefficients in the barycentric calculus

The barycentric calculus of Möbius involves formal sums of expressions of the form $mP$ where $m$ is a real number and $P$ is a point, where $mP$ is to be thought of as $m$ units of mass located at $P$...
James Propp's user avatar
6 votes
1 answer
118 views

First occurrence of hyperboloid paraboloid

The ancient greeks considered surfaces such as cones, but did they study the hyperbolic paraboloid? What is the first occurrence of such surface in history?
coudy's user avatar
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2 votes
1 answer
349 views

The role of the Elements in the development of mathematics

The Elements are often regarded as the cornerstone of the axiomatic approach to mathematics. However, mathematical textbooks have served as the foundational pillars upon which writing style, language, ...
Leandro Caniglia's user avatar
6 votes
2 answers
1k views

First appearance of the "four triangles and a square" proof of the Pythagorean Theorem

A well-known proof of the Pythagorean Theorem is illustrated in the figure below: This figure shows a square with side lengths $a + b$, dissected into four right triangles (each with area $\frac 12 ...
mweiss's user avatar
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1 vote
1 answer
71 views

Kepler's Mysterium Cosmographicum with regular polygons: in which nesting order?

Kepler tried to use regular polygons before using the 3D platonic solids in his Mysterium Cosmographicum. My question is: which regular polygons did Kepler try to use and in what nesting order?
Humberto José Bortolossi's user avatar
1 vote
0 answers
43 views

Origin of $V_a$ (median) notation

My question about median of a triangle. The English equivalent of the Turkish word "kenarortay" is "median". In English-language geometry sources (like books or web pages), the ...
scarface's user avatar
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1 answer
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A text or YouTube channel with a comparison between pre-Cartesian with post-Cartesian mathematics

Is there some good book or YouTube channel that make a good comparison/distinctions between the mathematics before René Descartes, with the mathematics after Descartes? In the short article "The ...
Ignacio Botaya Vera's user avatar
3 votes
0 answers
167 views

Question on Gauss's geometric interpretation of "spherical functions"

In the physics chapter of his biography of Gauss, W.K. Buhler writes the following: Expansions into series are frequent and important in potential theory. So it does not come as a surprise that ...
user2554's user avatar
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2 votes
0 answers
110 views

Was the small Desargues Theorem known to ancient Greeks?

My question concerns the classical Desargues Theorem and its simplest version The small Desargues Theorem: Let $A,B,C$ be three distinct parallel lines and $a,a'\in A$, $b,b'\in B$, $c,c'\in C$, be ...
Taras Banakh's user avatar
5 votes
1 answer
134 views

Leibniz conjecture that geometry is a form of algebra

Hermann Grassmann (1840s) verified the Leibniz conjecture that geometry 1s a form of algebra, showing that the geometric figures themselves are algebraic entities, because they are subject to definite ...
Babu's user avatar
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1 vote
0 answers
149 views

When and why Cantor-Hume principle was universally adopted in set theory instead of Euclid's principle?

In this answer and the comments Joel David Hamkins talks about a conflict between Cantor-Hume principle and Euclid's principle. He writes: This principle [Cantor-Hume] is often defended as a ...
Anixx's user avatar
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1 vote
1 answer
235 views

Did Riemann invent the Riemann curvature tensor?

I'm pretty sure they weren't using tensors in the modern sense at that point, but to what extent did Riemann lay out the structure or significance underlying his eponymous tensor? In his habilitation ...
Adam Herbst's user avatar
1 vote
4 answers
227 views

Why does Eratosthenes method for calculating the circumference of the Earth requires the city of Alexandria and Syene to be in the same meridian?

I'm reading a book where the author claims that in order for the method of angles and proportions used by Eratosthenes to work, the two cities would have to be located in the same meridian, or at ...
zlaaemi's user avatar
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0 answers
113 views

Why is bachelors' unknotting called as such and who discovered it?

Bachelors' unknotting is a way to show that all tame knots are isotopic to the unknot, by tightening a knot to a point. Why is it called 'bachelors' unknotting'?
Apoorv Potnis's user avatar
10 votes
3 answers
3k views

How did the notion of rigour in Euclid’s time differ from that in the 1920 revolution of Math?

I am reading about the 1900s revolution of math pioneered by figures such as Hilbert. I have seen many articles speak very fondly of these figures due to the fact they tried to study Mathematics ...
Babu's user avatar
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5 votes
0 answers
97 views

What was Gauss's algorithmic method to solve a certain "nearest neighbour search" problem in multi-dimensional euclidean space?

In his 1829 paper on a new formulation of mechanics, Gauss presented his principle of least constraint, which parallels previous formulations of analytical mechanics and provides a new point of view ...
user2554's user avatar
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7 votes
1 answer
644 views

Fibonacci and straightedge and compass constructions

In "Mathematical Thought from Ancient to Modern Times" Morris Kline claims (on page 209) that Leonardo da Pisa (Fibonacci) "showed that the roots of $x^3+2x^2+10x=20$ are not ...
Frunobulax's user avatar
2 votes
1 answer
133 views

Searching for book about non-Euclidean geometry that recapitulates the First Book of the Elements

I am looking for a specific book on non-Euclidean geometry that I read in my undergraduate years. The unique characteristic of this book is that the first part of the book started by re-proving in ...
ltcomdata's user avatar
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1 vote
1 answer
175 views

How to build a protractor without a protractor?

We all know how to use a protractor, it is taught in elementary school. However, I was wondering what type of knowledge is required to build one from scratch. For instance, was the understanding of $\...
YGranja's user avatar
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0 votes
0 answers
97 views

Is there any book dealing with the history of both astronomy and geometry?

I am preparing an introductory course on the history of astronomy and geometry for intermediate students. The intention is to teach Copernican heliocentrism, Kepler's laws of planetary motion & ...
workingclass_science's user avatar
3 votes
1 answer
186 views

What was the pre-Renaissance proto-perspective scaling technique for painting square tiled floors?

I read once (I don't remember where exactly) about an early technique (Early Renaissance) to draw a square tiling floor in perspective. The next row in the drawing is done multiplying the previous one ...
Humberto José Bortolossi's user avatar
2 votes
0 answers
374 views

When did we first know the $11$ ways of opening up a cube?

Take a box in the shape of a cube and cut $7$ of its edges so that all the faces stay connected and flatten it out onto a table. It turns out, there are $11$ distinct meshes that emerge: https://math....
Rohit Pandey's user avatar
3 votes
1 answer
164 views

Origin of the term "affixe"/"affix" in the geometric treatment of complex numbers

In current French mathematical tradition, when introducing complex numbers, it is common to hear about "complex plane of Argand-Cauchy". What is particular in French treatment, it is the ...
Alexey's user avatar
  • 261
4 votes
1 answer
2k views

Why is the letter $b$ used to represent the y-intercept in the equation of straight line?

The slope-intercept form of a non-vertical line is $y=mx+b$. I have been told that the slope is called $m$ because it is the first letter of the French word for mountain. But why is there the letter $...
user107952's user avatar
2 votes
0 answers
103 views

How to give a meaningful interpretation to Gauss's notion of "oriented area" of self-intersecting geodesic polygons?

Gauss's notion of "oriented area" of figures characterizes the notion of two-dimensional content in a way that enables self-intersecting geodesic polygons (the term "geodesic" ...
user2554's user avatar
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1 vote
2 answers
261 views

Discussions of why the Greeks "squared the..." geometrically

The Wikipedia article on squaring the circle has a relatively good history of their efforts on this particular problem, but it fails to mention why the Greeks were interested in this methodology on a ...
Maury Markowitz's user avatar
1 vote
0 answers
53 views

Research on Pre-Columbian Polynesian geometry?

Has there been any historical/cultural anthropological research into how Polynesian cultures understood geometry before contact with Europeans? In part what I am interested in is how did the dealing ...
Q the Platypus's user avatar
3 votes
0 answers
183 views

Why is this notation used to define points in (elementary) analytic geometry?

I have always found strange that in elementary analytic geometry points are defined by their names followed by their coordinates, for example: "Find the distance between $A(5, -3)$ and $B(2, 1)$....
user477398's user avatar
4 votes
0 answers
123 views

Explanation request for the terminology and notation employed by Gauss in his major 1843/6 treatise on Geodesy

Background: In his 1827 treatise on differential geometry, Gauss in his "theorema egregium" proved that the curvature of a surface is an intrinsic invariant; it doesn't change under ...
user2554's user avatar
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4 votes
2 answers
477 views

How was longitude determined in the 1700s?

I'm going through the journals of Alexander Mackenzie (ca 1790) and I came across this passage: I gather that he's determining his latitude and longitude but I'm not clear on what units he's using, ...
Lukas Bystricky's user avatar
1 vote
1 answer
99 views

Did the ancients have the concept of dimension?

Plainly, they knew what a circle and sphere were and also a square and cube; but did they discuss the idea that a sphere was analogous to a circle but in 3 dimensions or similarly the analogy between ...
releseabe's user avatar
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8 votes
2 answers
3k views

Why did Columbus think the Earth was much smaller than it is?

Columbus set off on his westward journey to Asia, believing the Earth was much smaller than it is. There was some apparent disagreement about the size, and Columbus was staking his life on it. Why ...
MWB's user avatar
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1 vote
1 answer
238 views

Why 360° is assigned to circle full turn ? Not any other number? [duplicate]

Please look at this question https://math.stackexchange.com/posts/comments/9011243?noredirect=1 A user comment this so I thought of asking here You mean why did we decide on using 360 degrees? I don'...
Suresh Chandra Pal's user avatar
5 votes
1 answer
154 views

What is the origin of the "problem of Brahmagupta" of constructing inscribed quadrangle with given sides?

I am looking for a source of the following construction problem: Construct an inscribed quadrangle with given sides. I know it under the name problem of Brahmagupta, but I do not know any evidence ...
Anton Petrunin's user avatar
1 vote
0 answers
104 views

Were units of area/volume always in terms of squares/cubes?

Throughout our known history of geometry were the units representing areas and volumes always in terms of squares and cubes respectively? Take ancient Egyptian formulas as an example, the fact that ...
Michael Munta's user avatar
0 votes
0 answers
43 views

Historicity of Euclid. Looking for the references of Euclid in ancient texts that has survived [duplicate]

The general consensus is that Euclid was a real historical figure. Wikipedia https://en.wikipedia.org/wiki/Euclid concludes on the hypothesis that Euclid was not a real person, "This hypothesis ...
Biswarup's user avatar
0 votes
1 answer
450 views

What is the History of Coordinates?

Can someone give a history of coordinates? When did coordinates first appear, which was the first ever coordinate system? In which field were they used?
user avatar
1 vote
0 answers
263 views

Analysis of Moscow Mathematical Papyrus problem no. 10

This problem is giving me headaches for quite some time now I guess mainly because I am not a mathematician, but I would very much like to know if there is anyone here that has any deeper insight into ...
Michael Munta's user avatar
1 vote
0 answers
104 views

Help understanding Egyptian circle

I was reading this Wikipedia page searching for the Egyptian area of circle and there is a following picture there: Trying to understand what is meant by this since it is under the "Area" ...
Michael Munta's user avatar