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39

If the person is in a Möbius strip, then it seems we are assuming he is $2$-dimensional. Suppose he has with him two identical circles split into sectors of $120^{\circ}$, and each sector is colored a different color. Notice being $2$-dimensional, he can rotate this circle but not reflect it, so the two circles are identical up to a rotation. Now, let him ...


32

The concept of dimension is surprisingly subtle. The mathematics invented by Georg Cantor and his contemporaries famously showed that, contrary to intuition, it is possible to specify a point in $2$-dimensional space using only a single real number. To make this precise, we need the concept of a bijective function. What Cantor's mathematics shows is that, ...


29

Yours is a very interesting and subtle question, which often generates confusion. First let us give a name to the property you are interested in: a ring $A$ will be said to satisfy (DIM) if for all $\mathfrak p \in \operatorname{Spec}(A)$ we have $$\operatorname{height}(\mathfrak p) +\dim A/\mathfrak p=\dim(A) \quad \quad (\text{DIM})$$ The main ...


22

Your set of vertices satisfies all the terms of the definition, so it is technically a polygon by that definition. Some would call it a degenerate polygon. To disallow degenerate polygons, you will need to modify the definition, adding additional constraints. EDIT: in the original post, I claimed that adding the condition that there exist at least non-...


22

These things are not universally defined. In some contexts it would make sense to admit your example as a polygon, and in others it would not. An example of the first context would be a discussion of a computer algorithm for detecting whether a point was interior to the polygon, or for calculating the area or the convex hull of a polygon. One would ...


17

Quoting from Abbot's book, "Understanding Analysis": There is a sensible agreement that a point has dimension zero, a line segment has dimension one, a square has dimension two, and a cube has dimension three. Without attempting a formal definition of dimension (of which there are several), we can nevertheless get a sense of how one might be defined by ...


14

You have an intuitive notion of the symmetry of the numbers (let's say the integers for concreteness) around $0$, and we can formalize this in mathematics. A “symmetry” is usually understood as a mapping from a set to itself which preserves some sort of structure; typically the “distance” between the elements of the set. For example, a symmetry $f$ of the ...


12

Your ideal is generated by binomials, so one can be smart about it. There is an algebra map $\mathbb C[x_1,x_2,x_3,x_4]\to \mathbb C[s, t]$ such that $x_1\mapsto s^3$, $x_2\mapsto s^2t$, $x_3\mapsto st^2$ and $x_4\mapsto t^3$, and this map maps your ideal $\mathfrak p$ to zero, so it induces $\phi:\mathbb C[x_1,x_2,x_3,x_4]/\mathfrak p\to \mathbb C[s, t]$. ...


11

Negative dimension is actually much easier to talk about than complex dimension. Super vector spaces are a natural collection of objects that can have negative dimension; given a super vector space $(V_0, V_1)$ we can define its dimension to be $\dim V_0 - \dim V_1$, and this definition has many nice properties; see this blog post, for example. More ...


11

It depends how much structure you want to impose on numbers. If you consider the reals just as an affine space, or as a totally ordered set, then there is no good sense in which they can be considered to be "anchored" anywhere. You could pick anywhere and find just as many numbers on one side as the other. If you want to define a monoid structure (or a ...


10

I don't know if this is on Cover's list, but maybe it should be: For $n=2$ and $3$, any tiling of ${\mathbb R}^n$ by unit $n$-cubes has two with a complete facet in common. But it's not true for $n \ge 10$: see http://arxiv.org/pdf/math.MG/9210222.pdf


10

While goblin's answer is right, its a little specific. A more common notion is covering dimension, which is both applies in more situations, and is a little easier to understand. First we start with the notion of open set. An open set is one that does not contain any points in its boundary. For example, the set of points less than $1$ away from $(0,0)$ is ...


9

suppose $P_0, \ldots, P_n$ are $n+1$ polynomials, of degree less than $d$. Then by multiplying the $P_i$ among themselves up to $k$ times, you can build at least about $k^{n+1}/(n+1)!$ polynomials of the form $\prod P_i^{\alpha_i}$ of degree less than $dk$. But the dimension of the vector space of polynomials of degree less than $dk$ in $K[X_1,\ldots X_n]$ ...


9

The most basic surprise, in my opinion, is that the ratio of the volume of the unit sphere to the volume of the cube circumscribing that sphere tends to 0 as the dimension of the space tends to $\infty$. In other words, a high-dimensional sphere takes up almost no space in the cube that circumscribes it. See pp.4--5 in http://www.cc.gatech.edu/~kingravi/ML%...


9

You are correct. There are several ways to show that it is impossible, and it basically boils down to the fact that no open subset of ${\bf R}^n$ is homeomorphic to an open subset of ${\bf R}^m$ if $n<m$ (as $U\cap V$ would be in your case). You can assume without loss of generality that the sets in question are connected. One way to show it is to use ...


8

There are formal ways to define dimension, indeed, lots of them, depending on the context. As already noted in a comment, your description of why a circle is one-dimensional is not really correct though. The one-dimensionality of a circle is not a function of the fact that a circle itself can be described by a single number (such as radius), but that to ...


8

In fact, there are many examples where this happens. And as you suggested, it comes from an algebraic point of view; namely in the area of what are called fusion categories. These are categories which come with quite a bit of data to begin with. In particular, they are monoidal, there is some notion of simple objects, have duals and evaluation $$\epsilon:a\...


8

Let $n = 2$, $C \subseteq \mathbf R$ a fat cantor set, $f \colon C \times C \to C$ a homeomorphism. As $C \times C \subseteq \mathbf R^2$ is closed, there is a continuous extension $F \colon \mathbf R^2 \to [0,1]$ by the Tietze extension theorem. Now $\lambda(C \times C) > 0$ and $F|_{C \times C} = f$ is one-to-one.


7

This should never be true for a reasonable definition of dimension (for example the dimension of a manifold). A lower-dimensional thing should have measure zero in a higher-dimensional thing, so removing it shouldn't change the dimension of the higher-dimensional thing. The correct version of the "naive equation" is that the Cartesian product of an $m$-...


7

If he has a friend then they both can paint their right hands blue and left hands red. His friend stays where he is, he goes once around the strip, now his left hand and right hand are switched when he compares them to his friends hands.


7

There are many generalizations of the usual notion of dimension, and they are there to capture different properties. Having said that, the intuition behind the dimension is that it describes the number of degrees of freedom you have, e.g. A point on a line has one degree of freedom. A point on a plane has two degrees of freedom. A point of two-dimensional ...


7

What is the rule or condition to be a 2D or 3D picture? This is a subtle question! In a certain sense, the picture is two-dimensional, since it's displayed on a computer screen. :) That's a trivially literal remark, yet not without mathematical content. A closely related question is, "Does there exist a physical object that looks like the picture from ...


7

Standard self-similarity Fractals are often constructed using a recursive procedure and self-similar sets in particular, are always constructed this way. I think that most folks working in fractal geometry would guess that your picture implies a recursive construction like so: Note that the first set is an initial seed. The second set is composed of ...


7

It is possible to force this perspective, but it would be incorrect to say that the mathematics "actually" is this way. To be more specific, yes, you can choose to consider the inclusions $j_n\colon \mathbb{R}^n\longrightarrow\mathbb{R}^\infty$ defined by $$j_n(x_1,\ldots,x_n)=(x_1,\ldots,x_n,0,0\ldots)$$ However, the objects $\mathbb{R}^n$ have their own ...


7

There is no topological space $X$ such that $X\times X\cong\mathbb{R}^n$ if $n$ is an odd integer. You can prove this using homology; see, for instance, this answer on MathOverflow. In particular, this seems like pretty good evidence that there is no reasonable notion of "$\mathbb{R}^{n/2}$" when $n$ is an odd integer. By similar homology arguments you ...


6

Chapter 8 of Eisenbud has a short history of dimension in algebraic geometry, even giving axioms for a theory of dimension. The historical order seems to be transcendence degree (think meromorphic functions on a Riemann surface), Krull dimension, then Hilbert functions. In particular, Eisenbud mentions that, though one might suspect differently, the most ...


6

Algebraic stacks are a far-reaching generalization of algebraic varieties. If an algebraic variety is considered as a stack, then its dimension as stack is the same as its dimension as variety. However there are many stacks that do not correspond to varieties, and some of these have negative dimension. Specifically, if $V$ is a variety and $G$ is an ...



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