Questions tagged [geometry]

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

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66 views

Origin of “Inverse Pythagorean Theorem”

There is a lot of information on the history of the Pythagorean theorem, but not much on its closely related cousin; the Inverse Pythagorean Theorem. Would appreciate any resources on the the history ...
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81 views

Does Hadamard claim that Pascal could have discovered non-Euclidean geometry?

In The psychology of invention in the mathematical field, p. 53, Hadamard makes the following claim: Is his point that there must, in any axiomatic theory, be undefined terms, and if you write the ...
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Fourth powers and quartic equations before Descartes

How did mathematicians interpret quartic equations and fourth powers before Descartes propose to perform elementary arithmetic on line segments? I ask this because it seems strange to me that ...
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Where can I find the complete papers of abstracts published by P. G. Tait in Proc. Roy. Soc. Edinburgh in 1880?

I am interested in looking up P. G. Tait's flawed proof of the four-colour theorem, published in 1880. The citation that I have seen is: P. G. Tait, On the colouring of maps, Proc. Roy. Soc. ...
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131 views

Who came up with the name “Manhattan distance”?

Who came up with the name "Manhattan distance" (for the distance between two points as measured by the sum of the horizontal and vertical distances, as opposed to the length of the straight ...
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105 views

What are the earliest known proofs that planimeters 'work'?

The dates of various physical implementations of planimeters are pretty well known. I'm interested in discovering when formal mathematical proofs were published that any given design does calculate ...
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122 views

Why is Freeth's nephroid called a nephroid?

A nephroid is an epycloid that can be generated by rolling a circle on the outside of a circle with doubled radius. It was called by Richard Proctor (1878) because its shape looks like a kidney (see ...
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163 views

How did polar coordinates come into existence?

So, I came here from Mathematics StackExchange where I posted this question. So, I want to know why polar coordinates came into existence. Why exactly did the mathematician who introduced them......
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56 views

Using Leonardo of Pisa's table of Chords

I saw a problem asking to use Leonardo's table of chords to find the arc length cut off by a chord with length 8 rods, 3 ft, and 16 2/7 unciae in a circle of diameter 10. I do not understand at all ...
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What geometry is involved in the architecture of Indian stupas?

Is there any interesting geometry involved in the construction of Indian stupas such as the Great Stupa at Sanchi? There are interesting geometric patterns in the architecture of Stonehendge, ...
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What is the basic idea behind calculation of area? [closed]

The system of calculating area in terms of square units is pretty philosophical and not very intuitive. It must have taken a great amount of time for humanity to arrive at such a convention and to ...
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76 views

When was the special relation between sides in a triangle with an angle of 60 or 120 degrees written for the first time?

The case of these triangles is special and close to the Pythagorean theorem, namely $c^2=a^2+b^2\pm ab$. This is a particular case of trigonometric relations and the law of cosines, and remains quite ...
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Why couldn't Robert Peary use his theodolite at the North pole

Robert Peary writes in his book The North Pole: The instruments used in taking observations for latitude may be either a sextant and an artificial horizon, or a small theodolite. Both these ...
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66 views

What is the earliest article in which Leibniz used 'matrices'?

The Chinese were using matrices ( fengcheng in the Nine Chapters of the Mathematical Art), long before they were used in Europe which suggests that possibly they were introduced by way of them. For ...
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What was the content of Emmy Noether's 'two-line' note on Homology?

I was surprised to learn that the note published by Emmy Noether in 1925 that suggested that Homology was better thought through as actual groups rather than numerically as Betti numbers consisted of ...
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122 views

Did physicists correct an error of mathematicians in counting twisted cubics in the quintic?

One problem in enumerative geometry consists in counting the number of rational curves of degree $d$ in the plane going through $n$ general points. If $n = 3d-1$, this number, denoted $N_d$, is finite ...
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136 views

What did Delaunay invent Delaunay triangulations for before computers were developed?

I was teaching my students about Delaunay Triangulation which is a method for dividing a surface into triangles. This triangulation method is the basis of most computer calculations that require a ...
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What does “given in species” mean in old geometry textbooks?

I recently came across the term "triangle given in species" in Hatton's Projective Geometry. Searching in archive.org turned up other examples (such as this) of 19th century texts, and it ...
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166 views

How did the “Wheel of Theodorus” become known as the “Wheel of Einstein”?

I've desperately searched the Internet, to no avail, to find a citation of how the "Wheel of Theodorus" became known as the "Wheel of Einstein" as claimed by Wikipedia and Wolfram.
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102 views

Did Galileo use an erroneous geometrical result in 'Two New Sciences'?

In Thm. 4, Prop. 4 of Galileo's 'Two New Sciences' (pg. 187, Crew Translation), Galileo says the following: "From a single point $B$ draw the planes $BA$ and $BC$, having the same length but ...
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History of points with coordinates notation

In this MathEducator StackExchange article, "Notation of points with coordinates", it's posed the question about what is the best notation for geometrical points and their coordinates: $P(3, ...
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Lessons from apparent paradoxes in geometric limits

1) Zeno's oxymoronic fleet, stationary arrow: One of the earliest infinity paradoxes, of course, is the flying arrow of Zeno which can't possibly be moving since it takes a finite amount of time to ...
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198 views

What mathematics did Isaac Newton learn at school?

Since Sir Issac Newton invented a lot of modern mathematics, what mathematics did he already know? Since he was standing on the shoulders of giants which giants was he speaking of? I presume he knew ...
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Literature on Mayan mathematics

I asked this question on math.se and they sent me here. It is well known that Mayan people used vigesimal (base 20) numeral system, and had had an advanced calendar system. Except for these facts, I'...
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Who invented the push-pull construction?

I learned about the push-pull construction from a video lecture by Freedman in which it is explained starting around 39:08. It is somewhat long and technical to describe in detail, but the main idea ...
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Were epicycloids from astronomy acceptable curves in Greek geometry?

My simplified historical understanding is as follows. Euclidean geometry accepted a limited number of geometrical objects (straight-edge and compass constructions, conics). Descartes' Géométrie ...
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How did Eratosthenes know the distance between Aswan and Alexandria?

In his well-known measurement of the Earth, and according to Cleomedes, Eratosthenes estimated in 5000 stades the distance between Aswan and Alexandria. Modern accounts state that he calculated the ...
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205 views

Why is one meter as long as it is?

The metre is defined as the length of the path travelled by light in a vacuum in 1/299 792 458 of a second Why is this so? Who decided that 1/299,792,458 of a ...
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Did Newton invent convex hulls?

The convex hull of a set of points appears recognizably in a 1676 letter from Newton to Henry Oldenburg describing Newton polygons. Is there an earlier precedent for convex hulls or is this their ...
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91 views

What are the context and motivation for Gauss's study of the “Pothenot problem”?

My question refers to p.307-335 of volume 8 of Gauss's werke, which are part of a section of volume 8 which deals with Gauss's applications of complex numbers for geometry. The pages mentioned are ...
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When did Dehn start to work on Hilbert's third problem?

According to this wiki article, Dehn solved Hilbert's third problem within a year. Did Dehn start to work on the third problem after Hilbert's talk? Since Dehn is Hilbert's student, they were likely ...
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535 views

What is the history of staircase or 𝜋=4 paradox?

The staircase 'paradox' has been discussed here and elsewhere a few times (search for staircase + paradox). My question is whether this puzzle has been discussed in the academic literature or ...
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101 views

When did the use of Sine and Cosine as functions become mainstream?

In the work of early physicists like Newton, everything is explained in terms of cumbersome (in today's standards) geometry. They don't talk about "cosines" of certain angle, but about proportions ...
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What was the chain of theories that led to relativity? [closed]

Can you briefly sketch the sequence of math theories that were necessary for Einstein to figure out a convincing background for relativity?
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200 views

When did mathematicians transition from peg and rope to straightedege and compass?

In the 19th and 20th centuries, the student of classical Greek geometry used "straight edge and compass". A. Seidenberg uses the terminology "peg and cord" in proposing that altar construction ...
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What is Poincare's “Fourth Geometry”?

In Science and Hypothesis, Poincare cryptically describes a "Fourth Geometry." Can anyone clarify what he is talking about? Is there a standard name for this geometry? The Fourth Geometry.—Among ...
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I want to know the source of why Archimedes' broken chord theorem is Archimedes good

I found "The Arabian scholar Abu'l Raihan al'biruni" has attributed to Archimedes the theorem of the broken chord in web. But I couldn't find a book or papers that this Arabian scholar wrote.
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Were people aware of the “mistakes” in Euclid's Elements before the start of the formalization of Mathematics?

For example, in proposition 1, Euclid assumes that the instersection of the two circles exist, when he shouldn't have. This, among many other things, was corrected quite recently (by Hilbert and ...
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How did the use of the word “origin” become commonplace in geometry?

My understanding is that in Cartesian geometry, all coordinate axes of an n-dimensional space may intersect at one point. I would like to know how that point--whether (0, 0), (0,0,0), ... -- came to ...
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120 views

Which is the earliest written record of hexagonal tesselation of the plane?

I am wondering which is the earliest record of the fact that the plane can be tiled by regular hexagons (in addition to triangles and squares, which may be slightly more obvious). Had a look in the ...
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193 views

Where in Gauss's nachlass did he pose the problem of connectedness of a surface?

On p.98 of the book "Mathematics of the 19th Century: Geometry, Analytic Function Theory", the authors mention a note written by Gauss in 1840: In 1840 Gauss wrote a note in which he introduced the ...
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When was 4D space “conceived of”?

In Measurement by Paul Lockhart (Harvard Press), he says (p.351): the classical geometers (as far as I know) never even conceived of four-dimensional space, whereas adding another variable is ...
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When Indian mathematicians learn of Euclid's Elements?

Transfer of mathematical knowledge from India to Europe (such as a positional number system with zero) allowed Europeans to develop arithmetic. But was there also a reverse direction (probably via ...
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What were the typical ways students were taught the elements when it remained the prime textbook of mathematics?

In modern textbooks, students are greeted with plenty of exercises. Usually they are also organized in such a way that you have examples and pointers to what concepts are most important. The elements ...
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Old geometry terminology

I was reading Ramsey's 1927 paper "A Contribution to the Theory of Taxation" and came across the following paragraph: "We have $\lambda_1 = \mu_1,\ldots,\lambda_m = \mu_m$, $m$ hyperplanes ($n-1$ ...
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Meaning of a cryptic sentence by Gauss on “the mobility of figures in the hyperbolic plane”

G. Waldo Dunnington writes in pages 189-190 of his biography of Gauss: Among the axioms of geometry which do not depend on the parallel postulate are those which secure the free mobility of a ...
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What made Euclid/Heron define line as a length without breadth and point as that which has no part?

A point is that of which there is no part. And a line is a length without breadth.$^1$ If above definition on point, expresses on point as to be indivisible length, as seems to be expressed in ...
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What theorem of Sophus Lie on the number of geometries is H. Poincaré referring to?

In this quotation from Henri Poincaré's essay "Non-Euclidean Geometry" published in Nature in 1892 (No. 1165, Vol 45, p. 406), he refers to a theorem of Sophus Lie. Does anyone know a source for this ...
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Who did the drawings in Hilbert's and Cohn-Vossen's “Anschauliche Geometrie”?

Hilbert's and Cohn-Vossen's wonderful book "Anschauliche Geometrie" ("Geometry and the Imagination") from 1932 contains a lot of great illustrations which, given the time of publication, must have ...
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Historically, how did René Descartes's works affect the invention of calculus?

When the "Cartesian coordinate system" was discovered By René Descartes, Algebra and Geometry were connected. How exactly did that affect Newton and Leibniz in the invention of what we know ...