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I understand that the word Euclidean comes from the name, Euclid, who is a greek mathematician.

However my question is, what should I generally understand or imagine or assume when someone says, Euclidean-something (like space, distance etc). How do you even use that word correctly in a sentence?

So far, my guess is, when people say Euclidean-something, that something has something to do with the set $\mathbb{R}^n$. This is only my guess from what I've seen so far.

I checked Wikipedia, but I didn't get a "satisfactory" answer, I understand this might be a vague question, since I've not mentioned what would be satisfactory to me, but to be honest I don't know that myself so any help is appreciated.

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    $\begingroup$ My personnal way of thinking is that Euclidean something when it comes to geometry represents the intuitive version of the considered concept. I.e. an Euclidean Space is a geometrical space behaving like what we are used to in "real" life, the Euclidean distance is probably the one that you will get if you ask someone to measure a distance between two points and so on. $\endgroup$
    – nicomezi
    Aug 13, 2020 at 11:28
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    $\begingroup$ Informally, a Euclidean space is one that satisfies the axioms of Euclid's Geometry. See, e.g., this. $\endgroup$
    – lulu
    Aug 13, 2020 at 11:29
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    $\begingroup$ It just depends on context. Usually it means something to do with Euclidean geometry but simetimes it refers to something else he did, e.g. the Euclidean algorithm. $\endgroup$ Aug 13, 2020 at 21:50
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    $\begingroup$ It's probably best understood in contrast to Non-Euclidean geometry: en.wikipedia.org/wiki/Non-Euclidean_geometry. Given a line and an extra point: (a) Euclidean space always has exactly one parallel line through the point, (b) hyperbolic space has infinitely many, (c) elliptic space has none. With, of course, a bunch of different implications based on those facts. $\endgroup$ Aug 14, 2020 at 0:31
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    $\begingroup$ For most terms you will see including the word "Euclidean", it is not an adjective but part of the name, just like "Riemann" in Riemann sphere. It just so happens that there are a lot of things named after Euclid. Whenever it is used as an adjective, you will have to use context to determine which concept including the word "Euclidean" that it refers to. E.g. if someone tags on the phrase "...in a Euclidean setting" when talking about geometry or topology they are just saying "we are working in $\mathbb R^n$". $\endgroup$
    – Samuel
    Aug 14, 2020 at 3:21

6 Answers 6

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Disclaimer: I'm basically just summarizing what I found on the English Wikipedia page for "Euclidean" through the lens of my experience, biases, and understanding of the math.

It seems there are three main categories of usage of the word "Euclidean" meaning "related to the ancient Greek mathematician Euclid of Alexandria". Essentially all usage stems back to things he wrote about in The Elements, a collection of mathematical facts and justifications. There are:

  1. Concepts related to his work on geometry.
  2. Concepts related to a method he described for finding the greatest common factor of two number.
  3. A couple assorted concepts tied to specific things from The Elements.

Geometry

In The Elements, Euclid described what we might think of as "standard" geometry for shapes in a flat plane or in three dimensions. This is called Euclidean geometry, especially when talking about the geometry of a flat plane, or the specific axioms and arguments Euclid presented in The Elements.

There are a bunch of related terms:

  • A "Euclidean triangle" (a term used on the page for triangle groups) follows the rules of Euclidean geometry in that its angles add up to exactly $\pi$ radians ($180^{\circ}$) (as opposed to a triangle on a sphere). There are other similar terms to distinguish a situation from non-Euclidean geometry, like "Euclidean angle" (used on Ask Dr. Math).
  • The Euclidean distance is the one we first learn about for points in $\mathbb R^n$, ultimately stemming from the Pythagorean theorem. It is closely related to the Euclidean norm for calculating the length of a vector.
  • A Euclidean ball is a ball (set of points up to a given distance away from a center point) defined via the Euclidean distance.
  • Euclidean space is usually something like $\mathbb R^n$ with the standard dot product for calculating angles and lengths (via the Euclidean distance). If it's not phrased as "Euclidean vector space", there may be certain restrictions like "distinguish between points and (Euclidean) vectors and don't choose which point is the origin".

and more.

Euclidean Algorithm

Something that doesn't need to be tied to geometry which appears in Euclid's Elements is the Euclidean algorithm for finding the greatest common factor/greatest common divisor ($\gcd$) of two numbers. This has a web of connections to other concepts in mathematics:

  • It is extended to the extended Euclidean algorithm to find coefficients $x$ and $y$ satisfying $ax+by=\gcd(a,b)$.
  • A Euclidean domain or "Euclidean ring" is an algebraic construct where the (extended) Euclidean algorithm still works, even if we're not dealing with usual numbers.
  • A Euclidean domain's Euclidean function is the measure of size you need for division and greatest common divisors.
  • And Euclidean division is the formal statement of how division works in a Euclidean domain (though Wikipedia doesn't have a source for that being the origin of the phrase). (Apparently, "the Euclidean division" could refer to a division of the Intermediate Math League of Eastern Massachusetts.)

Other Ideas

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  • $\begingroup$ Granted that the OP asks about, "Euclidean-something (like space, distance etc)", I wish there were some way to highlight the first section on Geometry (or maybe cut the rest). $\endgroup$ Aug 14, 2020 at 0:27
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    $\begingroup$ @DanielR.Collins I did put Geometry first, and I think it's important to clarify when/where "when people say Euclidean-something, that something has something to do with the set $\mathbb R^n$" breaks down. But if you want to play around with the heading sizes or reorder things, by all means. $\endgroup$
    – Mark S.
    Aug 14, 2020 at 0:41
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    $\begingroup$ I once competed in the Euclidean division of the IMLEM. Crazy to see it pop up here. $\endgroup$
    – isaacg
    Aug 14, 2020 at 13:19
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Basically, Euclidean means "globally flat".

In the 1800s, non-Euclidean geometries were discovered. In these geometries, there is implicit curvature in the space (so they're non-flat).

Of course, by this time $R^n$ had been invented, and it was realized that these curvature calculations ended up with zero curvature on $R^n$ (so they're flat). It makes sense to generalize the term to $R^n$, even though most of what Euclid wrote applies to $R^2$ and $R^3$.

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    $\begingroup$ Though there are other kinds of geometry that are flat but not Euclidean. Projective geometry, for example. $\endgroup$ Aug 14, 2020 at 7:37
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There are specific concepts that bear the name "euclidean" in a precise way.

For instance, a euclidean vector space is a real finite dimensional inner product vector space; a euclidean ring is a ring that satisfies some form of euclidean division.

Then there are "looser" concepts of euclideanity: when you say something is euclidean, depending on the context, it can refer to various things.

The euclidean plane, for instance, refers to $\mathbb R^2$, with more or less structure depending on the context (e.g. the euclidean metric, or norm, etc.)

Euclidean space refers to $\mathbb R^3$ or more generally $\mathbb R^n$, again with more or less structure

There's also euclidean geometry as was pointed out in the comments and in another answer, which corresponds to Euclid's axioms for geometry (as opposed to non-euclidean geometries); the euclidean algorithm (for finding a gcd); euclidean division (in the usual context of numbers or in the context of polynomials for instance).

I'm not sure there are that many more places where the term is used; but sometimes there will be a specific, precise, mathematical definition, and sometimes it'll just be in reference to something Euclid did, in a loose way.

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  • $\begingroup$ This is also a great answer, in simple English. +1 $\endgroup$
    – William
    Aug 19, 2020 at 10:45
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This is a guess.

Euclid is primarily known for his work "the elements", which is probably the most famous math book ever written. In this book, he basically builds an axiomatic theory of geometry like no one did before. This theory of geometry is what we call Euclidean-geometry and all definitions within this theory or natural extensions such as to higher dimensions are called Euclidean-distance, Euclidean-Space etc.

Conversely, non-Euclidean-geometries are geometries which are built on different axioms than the one Euclid built his on, and thus non-Euclidean-things are definitions or natural extensions within non-Euclidean-geometries.

And people think Euler is the king ha.

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I agree that Eucledian ... usually can be though of as what perceive as the intuitive notions of 'space'.

However, more precisely, I think that when one refers to Eucledian space one refers to a vector space of n-tuples $\mathbb{R}^n$ of real numbers (the vectors) with the normal dot product defined on this space (it is therefore more specifically an inner product space). Using Eucledian as qualifier for other things (e.g. distance) then refers to properties of this vector space (e.g. norm of difference between two vectors as the distance between two points in the space).

This more modern definition is (as far as I know) in agreement with Euclid's original theory.

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Euclidean geometry is geometry which obeys all five of the postulates in Euclid's Elements. There are other geometries which obey the first four but not the fifth, so it is that fifth which is the crucial postulate for distinguishing Euclidean geometry from the others. This is his parallel postulate, viz that if a transversal $BE$ cuts two lines $ABC$ and $DEF$ so as to make the sum of the interior angles on one side (angles $ABE$ and $BED$) sum to less than two right angles, then those two lines will meet (if extended, if necessary) on that side. That is, $BA$ (extended) and $ED$ (extended) meet.

Some other postulates which are equivalent to it (in the context of geometries which satisfy Euclid's first four):

  • Given a straight line $l$ and a point $P$ not on $l$, there is a unique straight line through $P$, parallel and coplanar to $l$. This is often called Playfair's postulate but was also suggested by Proclus
  • Rectangles exist
  • Not all similar triangles are congruent
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