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A radially symmetric surface that extend outward from a single point and expands linearly at a rate of 2pi is called a circle (or a circular disk).
If it expands linearly at a rate of less that 2 pi, its a right circular cone.
What is it called if it expands linearly at a rate greater than 2 pi?
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posted
I don't think that's possible if you're only using 3 dimensions, and I don't know the names of many hyper shapes.
eta: I'm somewhat dubious that it's possible in higher dimensions, too, but I really don't have a good intuitive sense of what is possible in higher dimensions.
quote:Originally posted by mr_porteiro_head: I don't think that's possible if you're only using 3 dimensions, and I don't know the names of many hyper shapes.
eta: I'm somewhat dubious that it's possible in higher dimensions, too, but I really don't have a good intuitive sense of what is possible in higher dimensions.
It's definitely possible in 3 dimensions. Hyperbolic cones are possible in 3 dimensions but this is not one. A hyperbolic cone expands exponentially. The surface I'm talking about expands linearly, but at a rate > 2 pi. I'll see if I can find a picture. I've been coming the internet looking for a unsuccesfully looking for a name.
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posted
Maybe I don't understand what you mean by "expand linearly at a rate of X". You are saying that the area of the surface will increase linearly as the _____ increases. What goes in the blank? Linear distance from the center point? Curvilinear distance?
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quote:Originally posted by mr_porteiro_head: Maybe I don't understand what you mean by "expand linearly at a rate of X". You are saying that the area of the surface will increase linearly as the _____ increases. What goes in the blank? Linear distance from the center point, or curvilinear distance?
For a right circular cone, the circumference of the base of the cone = R*theta, where R is the slant length, so the circumference expands linearly as the slant length increases. If theta=2pi, you have a flat disk. The smaller theta is, the pointier the cone is. If theta>2pi, you a warped surface. I'm still looking for a picture.
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posted
Rabbit, I'm still unsure as to your meaning. Could you please answer my earlier question? By saying that there's a linear expansion, you are saying that there's a linear relationship between what two things?
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posted
I'm saying there is a linear relationship between, the circumference of the base and the slant length (distance from the tip of the cone to the base).
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posted
If that relationship is linear, and the shape is radially symmetrical, then I think you're stuck with an expanding disc or a right circular cone.
I don't see how it's possible for any shape to expand faster than the disc expands.
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posted
mph, I'm failing to find a picture on line. The simplest way to make a cone is to take a circular disk and cut out a pie shape segment. Then tape the cut sides together. Like this.
To make the the surface I'm talking about. Take the circular disk, cut a slit into the center. Then take a pie shaped section cut from another circle and tape it to the two cut ages. You get a kind of circle that can't lie flat, it warps or ruffles up.
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posted
Geometry is not my strong suit; is the shape you are talking about a circle drawn with its center at a saddle point? This seems to be the opposite of a circle with its center drawn at a maximum, which presumably forms a cone. Unfortunately I do not know what to call such a shape, but perhaps it could inspire someone's Google-fu.
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This is like trying to visualize the square root of -1. Maybe there's a solution on the imaginary plane, but I don't think there's any physical solution. Who knows, maybe the mathematics of it will enable scientific principles that haven't yet been discovered.
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I think I understand what you're looking for, unfortunately, I can't word it precisely enough for Google, or there are just no visualizations out there.
If I'm not mistaken, the shape you end up with is best described in spherical coordinates. Where for any given R, as Theta sweeps around the circle (2*pi) then Phi is moving in a sinusoidal fashion? And as your number (rate) increases, the amplitude will increase (and at a certain point, the frequency would have to double, assuming you want a continuous shape). I'm a visual person, and I'm having a little trouble with it, so I'm not 100% sure my description works completely.
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2pi is simply a maximum. This question is no different than asking what lies beyond any maximum. Answer: No such thing.
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quote:Originally posted by Lisa: 2pi is simply a maximum. This question is no different than asking what lies beyond any maximum. Answer: No such thing.
I disagree, it's almost exactly what aspectre linked, but that edge would follow the inside of a sphere. Seems like the larger your value of the rate of increase, the more options you have, which is a little unique.
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quote:Originally posted by Lisa: 2pi is simply a maximum. This question is no different than asking what lies beyond any maximum. Answer: No such thing.
Not true. What I'm talking about is related to the hyperbolic cone or pseudosphere, except that circumference of the hyperbolic cone expands exponentially with increasing radius.
What I'm talking about expands only linearly in radius. And it exists, unquestionably -- I just can't find the name.
BTW: These objects are considered radially symmetric because the distance from the center point to the outer edge is the same everywhere. The object I can't name is topologically equivalent to a circular disk or a cone.
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posted
Are you sure that it has been formally named? Perhaps you should just coin your own term, say "supercircle" or "inverse cone".
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a circle doesnt expand at a common rate or a common difference so the only way it expands is exponentially because if the r=1 it pi and if r=2 it is 4pi and r=3 it is 9pi r=4 it is 16pi if r=5 it is 25pi if r=6 it is 36pi if r=7 it is 49pi and so on so that might be your problem
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quote:Originally posted by dlaxd: a circle doesnt expand at a common rate or a common difference so the only way it expands is exponentially because if the r=1 it pi and if r=2 it is 4pi and r=3 it is 9pi r=4 it is 16pi if r=5 it is 25pi if r=6 it is 36pi if r=7 it is 49pi and so on so that might be your problem
Nope. First off, you are calculating area not circumference. Second it isn't increasing exponentially, its increasing quadratically (i.e by the square of the radius). Area increases with distance squared. That's a simple fact you can derive from dimensional analysis.
What I am talking about expanding is the circumference. There is a linear relationship between the circumference of a circle (cone or "rabbitoid" and its radius, hence the first derivative of the circumference with respect to radius" is a constant. The first derivative of a hyperbolic cone with respect to the distance from its center to the edge is an exponential.
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posted
Exponential means increase like a^x where x is the independent variable. Quadratic is a special case of geometric or polynomial increase, which has the general form x^n. Exponential is much faster.
For all a>1 and n>1, there exists some x such that a^x > x^n.
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quote:Second it isn't increasing exponentially, its increasing quadratically (i.e by the square of the radius).
Um, how is that not exponential increase?
Are you serious about that question?
The area of a circle increases by r^2. It would be exponential increase if it increased by exp(Br).
Let me give you an example. If when you move one unit outward in a hyperbolic cone, the circumference increases from c to 2c, when you move two units outward, the circumference will increase to 4c. When you move 3 units outward, the circumference would increase to 8c.
As you go each additional unit outward the series goes 2,4,8,16,32,64,128,256,512,1024 ...., That's exponential growth
If you are looking at the area of a circle. A circle of radius one has a circumference of pi. A circle of radius 2 has a circumference of 4 pi. A circle of radius 3 has a circumference of 9 pi.
As you add each additional unit to the circle the area goes 2,4,9,16,25,36,47,64,81,100. That's quadratic growth.
They look similar for the first four steps, but by the time you've gone 10 steps, the exponential it 10 times larger than the quadratic. The more steps you take, the greater the divergence.
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quote:Originally posted by King of Men: Exponential means increase like a^x where x is the independent variable. Quadratic is a special case of geometric or polynomial increase, which has the general form x^n. Exponential is much faster.
For all a>1 and n>1, there exists some x such that a^x > x^n.
Correction, Geometric growth is the same as exponential growth. It is not a synonym for polynomial increase.
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posted
Are you sure? Geometric seems to me to describe such contexts as the area of a circle or a square, which indeed grow polynomially and not exponentially.
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Ok, I sit corrected. At any rate, exponential and polynomial, of which quadratic is a special case, are very different.
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Ok, I've uploaded some pictures of the "rabbitoid". view one, view two, view three. The circumference of this object increases by 18 units for each unit increase in the diameter (or roughly 3 times faster than a circular disk. The preferred conformation of this thing is with three nodes (views one and two) but it can be forced into other configurations like the one in view three where you can see that the circumference is very nearly 3 times that of a circular disk. This basic shape stays the same if you add more rows. You don't get more nodes.
I crocheted these. I got sucked in by [A Field Guide to Hyperbolic Space: An Exploration of the Intersection of Higher Geometry and Feminine Handicraft. I mean who could resist a title like that. I've ordered my copy but it hasn't arrived yet. I'm impatient so I found a few research publication on hyperbolic space and crochet and started making things. This is a hyperbolic cone or pseudosphere. In contrast to the "rabbitoid", the number of nodes in the hyperbolic cone increases as you increase the number of rows. In the pictured hypercone, each row is twice the length of the previous row. This gives it the unusual property that each new row contains the same number of stitches as all the previous rows combined. The object took the same amount of blue and green yarn.
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quote:A Field Guide to Hyperbolic Space: An Exploration of the Intersection of Higher Geometry and Feminine Handicraft.
That sounds really cool. I've always been somewhat fascinated by knitting and crocheting, but never really got into it because I don't enjoy wearing those kinds of things.
But lately my daughter has been wanting to learn those things. We could learn it together, with her making articles of clothing and me making useless awesome crap.
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quote:Originally posted by The Rabbit: The circumference of this object increases by 18 units for each unit increase in the diameter (or roughly 3 times faster than a circular disk. The preferred conformation of this thing is with three nodes...
So, is it coincidence that you got three nodes when you increased the circumference three times faster than a disk, or are they actually related one to one like that? Intuitively, it makes sense if it is, but I don't have the math skills to prove it.
posted
I'd guess that an individual rabbitoid could be reshaped to have many different numbers of nodes.
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That's true, but there would have to be a minimum (and I'm also assuming that each node is congruent to the others). Looking at rabbit's version, I don't think I could force that to have only two nodes, for instance. Maybe I'll play a bit with crocheting my own versions and see what I come up with.
posted
I can in fact arrange this thing in a wide variety of shapes, but the three node shape is the one it naturally falls into. I can't force it to have only two nodes, but its reasonably stable with four or six nodes or a variety of other shapes. These shapes are wahnsinnig fun to fiddle with. I'm reasonably confident that if I made one with a circumference that was 4 times that of a circular disk that four nodes would be the most stable configuration, but I need to try it to see.
Maybe my next attempt should be a "rabbitoid" with two nodes, like rabbit ears.
My current project is a crocheted mobia strip.
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From what I've been able to decipher, the "rabbitoid" is a large angle euclidean cone. I wish I could find a mathematician to verify that but I'm 98% confident that's what it is.
I tried crocheting one with a circumference of 4piR and it does in fact have two nodes. Like rabbit ears. Here's a photo. (I despoiled it's perfect radial symmetry by adding a fluffy tail and face. It was Easter. How could I resist ). I'm not sure whether to call it an "bunnoid" or a "Coney".
I also made this hyperbolic mobia snake which among other interesting properties, has only one side so it's inside is its outside.
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posted
No, it's definitely not hyperbolic, its a euclidean space. I've actually way ahead on the hyperbolic coral reef project. I crochet a dozen hyperbolic corals before I tried the rabbitoid. The differences between the large angle euclidean cone and a large angle hyperbolic cone become really evident when you are crocheting them.
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posted
I have a BS in Pure Math from MIT, but I'm not a professional mathematician, nor expert in this area, so I don't know the answer to your question.
I do know, however, that you need to manage to /state/ your question more clearly if you expect an answer. Having read through the whole thread, I think that you may now be able to.
I tried Googleing, but found only hyperbolics.
OTOH, neither what I found nor you were clear enough for me to tell the difference between the two - even in crocheting terms.
posted
Here's the best I've been able to do so far: (My math professor friend didn't know, & I haven't found anyone else who does.)
The surface swept out by a circle of radius r, r>0, is a plane iff the circumference, c, = 2pi*r.
If c < 2pi*r, it's a cone.
If c > 2pi*r, it doesn't exist, strictly speaking, since no such circles can exist in Euclidean space (space of zero curvature).
For /something like/ a circle to exist in that latter case, its circumference must be warped in some way, to increase its length as required.
We can certainly call these things surfaces, but they're fundamentally different from the kinds of sufaces we're used to.
They're inherently warped, if not floppy.
There's no preferred warping. (which I take to be the notion behind the introduction of the concept of node).
For such things to exist as the more familiar kinds of surface, they must do so in a space that has negative curvature.
These are called hyperbolic spaces.
I still haven't found any treatments of surfaces in this latter class that differ in the function, c(r), that relates c to r.
One can imagine many such functions (keeping c > 2pi*r): o c = 2pi*r + k (k a constant >0) o c = 2pi*r * k (k a constant >1) o c = 2pi*r ^ k (k a constant >1) etc.
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posted
Correction: We've all been missing the fact that the distance from the center of a crocheted piece to the (~circular~) edge is not necessarily the radius of the ~circle~.
It's definitely not for the cone. It is for the plane - and it at least kinda is for the Rabbitoid case.
So the analysis for the cone case should be like that for the Rabbitoid: A circle with c .LT.(*) 2pi*r cannot be drawn in Euclidean space.
Looking at this from another direction, Rabbit's question was really: What's the surface generated as a function of r = f(3-D distance from the surface's origin (cone point) to the circle(?) at r)?
(*: If I use "<" here, I get an error, "Sorry, we do not permit the following HTML tag or attribute: Parenthesis in HTML tag".)
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). I'm not sure whether to call it an "bunnoid" or a "Coney".
