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Compact Discs, washers and Aerobie frisbees are all disks with a hole in the middle. Is there a word (either mathematical or not) to describe this shape? I mean the specific case of a round hole in a flat disk such that the inner and outer rings are concentric circles, like below.

Some Disks with holes in the middle

--Edit: Accepted answer

I feel rather unqualified to select one answer as correct, so I'm going to choose the one that says "It depends who you're talking to". I hope that future readers will choose between the various helpful answers here depending on their exact object and their audience. After reading the etymology of annulus, I also hope that nobody ever tries to market an "Incredible Flying Annulus" to 13-year old boys.

Mari-Lou A
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Fillet
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    @Fillet Might I suggest a toilet seat, deflated whoopie torus, peak-a-boo pasties, hat brims, or deficit circles? +1 for the question. – user179700 Aug 25 '11 at 04:44
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    I would call it {z: r < |z - z0| < R} .. ooops, sorry, thought i was on math.stackexchange.com – wim Aug 25 '11 at 08:43
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    @user179700 Maybe this will become the new Rohrschach test. You see whatever your inner psyche wants you to see. I will be sending your list to your psychiatrist for evaluation. – Fillet Aug 25 '11 at 10:46
  • Or we can all it an Inception Disc. A disc within a disc. Imagine a disc within a disc within a disc... – EarnestoDev Aug 25 '11 at 12:58
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    86 votes for a 12 inch vinyl disk with a hole in the middle. Could this be a record? –  Sep 07 '12 at 02:56
  • @DogLover, Please look at the answers posted, I believe everyone has written disk, before deciding to edit a question that is nearly four years old. – Mari-Lou A Jun 18 '15 at 02:23
  • I'd call it a disc. – BladorthinTheGrey Dec 03 '16 at 23:21

12 Answers12

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There is an informal, nontechnical-English answer and a technical mathematical answer.

  • informally, it can be a ring (like a coffee ring, aerobie, or washer (the last one is questionable, could be 'washer-shaped'), or a disk or disk with hole in it for compact disk (because the hole is somewhat secondary).

  • technically, it is an annulus.

Ambo100
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Mitch
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In math it is called annulus. You also have the washer method, a method to calculate volumes using washers.

Edit:
On second thought, mathematically a "washer" is a 3D object.
It is worth mentioning Steven Pinker:

Few people think of a wire as a very, very thin skinny cylinder and of a CD as a very short one, though technically that's what they are. We conceive of them as having only one or two primary dimensions, respectively.

Alain Pannetier Φ
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Bogdan Lataianu
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  • Well, one of the dimensions is infinitesimal, so it's not clear that really counts as 3D. – Random832 Aug 25 '11 at 06:35
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    no, one of the dimensions is small compared to the others, but very much still a measurable finite size – jk. Aug 25 '11 at 07:45
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    Regarding the Pinker quote, I don't think that comparing a CD and a wire is quite correct. If he compared a wire and a beer mat, or a CD and an urban-legend drilled wire (http://www.snopes.com/business/genius/wire.asp), then I would agree. – Fillet Aug 25 '11 at 11:19
  • The "washer method" for integration implies washers with a height of dx (or dy), not with a height of an actual number. The linked page does not do a good job of explaining it, since it mixes it with numerical integration (in which the slices do have to have a real thickness). – Random832 Aug 25 '11 at 12:02
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    From the online Etymology Dictionary 'annulus 1560s, medical, from misspelling of L. anulus "little ring, finger ring," a dim. of anus (see anus).' – Ben Haley Aug 25 '11 at 12:53
  • And since nobody else has mentioned it, the plural of annulus is annuli. – Peter Shor Aug 25 '11 at 14:33
  • @Ben I was about to ask.. – Rei Miyasaka Aug 26 '11 at 09:21
  • @Random832 the way I see is the volume V=limit S_n, where S_n=sums of volumes of 3D washers. The heights of the washers become smaller and smaller when n becomes larger. But some authors might call it infinitesimal dimension. I know how it is defined as an integral. Thank you for your observation and I might edit the answer later. – Bogdan Lataianu Sep 12 '11 at 05:21
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    Any object existing in the real world must be three-dimensional in the literal sense. 2D objects are a mathematical abstraction. But we often refer to things that are very thin as "two-dimensional". Saying that this is mathematically incorrect is like criticizing someone for saying that his job is "hard work" on the grounds that, as he mostly sits in a chair barely moving, by the technical physics definition he hasn't done much work at all. – Jay Dec 07 '11 at 21:24
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Let's talk some mathematics, rather than just language. If our seeker is asking about 3D objects, I believe the shape name would still be considered as a torus according to basic definitions of topology (and in support of the answer given by @T.E.D., which was unfairly downgraded by some). In particular, it might be clearer to call it a "flat torus". Topology is a higher level abstraction than geometry and is somewhat affectionately called "rubber sheet geometry" in certain mathematically mind-warped social circles (to which I belong). In topology, you can perform "continuous deformations" to topological objects, so you can "squeeze down" the classical donut image of a "ring torus" into something that represents a flat disk, CD, or washer without doing anything that would make it "not a torus." Do note, however, that we have had to use words like "disk," "washer," and "ring" to explain and exemplify throughout this exploration and "annulus," along with "toroid" belong somewhere in the ontology.

See also 2-dimensional torus.

John Tobler
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  • Welcome to EL&U @John. +1 to help you up the ladder. Thanks for your mathematical insights. How about "a z-axis projection of a torus" although I doubt this would be very useful to a mechanic in need of a washer. Also the consecrated term for a "seeker" is an OP, an abbreviation standing for "Original Poster". Good luck! – Alain Pannetier Φ Aug 24 '11 at 23:06
  • @John Wasn't sure which wiki link you wanted, but suggested (a wiki link to the 2-D torus link, which links back to torus page). We'll see if someone else approves it. +1 for nice answer. – user179700 Aug 24 '11 at 23:35
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    I now see that I (the OP) was asking 2 questions at once. With my examples I was referring to (very thin) 3-D objects, and with my picture I was referring to the 2-D shape. This is a helpful 3-D answer. – Fillet Aug 25 '11 at 06:13
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    Just to make a distinction: A *torus* is a special case of *toroid* where the shape that is rotated about a line is a circle or ellipse. When non-circular or non-elliptical shapes are rotated about a line, the resulting 3D object is simply a *toroid*. – oosterwal Aug 25 '11 at 12:18
  • @oosterwal: In Topology there is no toroid. All are called torus and they are essentially (for Topology, that doesn't care about distances) the same. – ypercubeᵀᴹ Aug 27 '11 at 20:45
  • I agree in purist principle, @ypercube, but "torroidal" has inserted itself into the vocabulary of (I hesitate to say) "applied topology," as in "toroidal space" and "toroidal topology". This interesting article refers to the OED's discussion that the word ‘torus’ defines the shape and the word ‘toroid’ denotes "an object having the shape of a torus." By now, though, we are rather far from the OP's question! – John Tobler Aug 29 '11 at 17:04
  • @ypercube -- Understood. However, in Topology a doughnut and a coffee cup are the same thing, and although both are also topologically identical to a DVD, neither will work in my DVD player. (It's probably an issue with the region code. ;-) ) – oosterwal Aug 30 '11 at 14:37
  • I agree with you, @ypercube (see "purist principle," note my difficulty in saying "applied" in front of "topology," and re-read my original post). However, this question did not confine itself to mathematical definitions. The OP included mathematical terms but did not require them. My answer to this interesting question was not selected; but, I'm no longer a total n00b in this group, thanks to the many kind responses from you and others! – John Tobler Aug 30 '11 at 15:30
  • "Flat torus" is not appropriate; in mathematics, it refers to a different object. (A CD is not really "flat" in the mathematical sense, because there are corners at the edge. A flat torus doesn't have any corners, and can't physically exist in a three-dimensional universe). "Annulus" is the correct mathematical term. Note: I am a mathematician. – Nate Eldredge Nov 18 '11 at 01:22
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Another name for this, I believe, is "annulus"

Tom Au
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The question is unclear about whether a 2-Dimensional shape is wanted, or a 3-Dimensional object which is flat, but has a finite thickness. The examples were all 3-Dimensional, but the drawings were 2-Dimensional.

Like rajah9 says, if you stack up enough washers or CDs, you will get a big tower. Trying to stack annuluses is like trying to build a tower of filled circles.

That means that there are two answers to the question:

  1. The 2-Dimensional shape in the drawing is an annulus. (Thanks to Bogdan Lătăianu, Mitch, and Tom Au).
  2. The 3-Dimensional object that you can throw across the room is an annular disk. A typical image-search for "annular disk" in google is this or this.
Fillet
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  • Good point-the question is unclear either it is 2D or 3D. Also, we have mathematical/general language difference. A 2D shape would be mathematical though. – Bogdan Lataianu Aug 26 '11 at 04:55
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Generally, a ring, or a donut.

A Torus is the proper mathematical name for that shape (if it is in actuality a three dimensional donut-like shape), but more folks know about donuts and rings than three-dimensional geometry.

T.E.D.
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    Mmmm... Donut. But shouldn't a Donut be three dimensional? If not, how do you get the filling in. Maybe 2-D Donut? Flat Donut? Squashed Donut? – Fillet Aug 24 '11 at 19:12
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    A torus is definitely not a flat disc, as described in the question. – Ben Voigt Aug 24 '11 at 19:18
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    -1: A torus is 3-dimensional. An annulus isn't. An example of a torus would be an inflated bicycle tube. That's not a flat disk. – Jimi Oke Aug 24 '11 at 19:54
  • Pardon me. Of course, your second paragraph isn't incorrect. Nevertheless, it doesn't correctly answer the OP's question. – Jimi Oke Aug 24 '11 at 19:57
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    +1 as the OP's CD, washer, and Aerobie are all three-dimensional. View them edgewise and you will see their height. My case of 100 CDs is about 6" tall, while my stack of 4000 annuli has no height at all. Yes, my CD is a torus, of course, of course. Mathematicians don't look for how "donut like" the object is, rather, the fact that it has a height will place it in the torus category. – rajah9 Aug 25 '11 at 00:58
  • @rajah9: is toroid a better description than torus? Merriam-Webster describes a torus as specifically being "generated by a circle rotated about an axis", and a toroid as being generated by "any closed plane curve". To generate a CD you'd have to rotate a thin rectangle about an axis. – Fillet Aug 25 '11 at 09:30
  • @Fillet - Perhaps. But unless you are speaking to mathematicians, nobody will have a clue what you are saying. That's why I'd stick with ring or donut. – T.E.D. Aug 25 '11 at 11:47
  • @Fillet, the M-W definition for torus is not the mathematical definition. (Actually, the mathematical domain is called "topology.") A coffee-mug shape is also a toroid, because you may transform it (without breaking it) from a donut to its desired state. (You can start with a donut/torus made of clay and shape it so that the hole became the handle and some part of the clay became the cup.) The coffee mug remains a toroid just as a planar three-angled, three-sided object remains a triangle no matter how wide or narrow the angles are. The "generated...circle" is a conceptual oversimplification. – rajah9 Aug 25 '11 at 12:28
  • For the record, topologists define torus as 2D, not 3D. It's the 2D surface of a "doughnut". – ypercubeᵀᴹ Aug 27 '11 at 20:51
  • @T.E.D. As Fillet pointed out, a toroid is an object whose annular shape is described by "revolving a plane geometrical figure about an axis external to that figure which is parallel to the plane of the figure and does not intersect the figure" whereas a torus is a specific annular shape described the same but specifically using a circle as the plane geometrical figure. The OP is describing a toroid based on a square or rectangle. – Doktor J Feb 25 '15 at 17:52
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Toroid Washer (see edit below)

mathworld.wolfram.com provides:

A surface of revolution obtained by rotating a closed plane curve about an axis parallel to the plane which does not intersect the curve. The simplest toroid is the torus. The word is also used to refer to a toroidal polyhedron (Gardner 1975).

Gardner, M. "Mathematical Games: On the Remarkable Császár Polyhedron and Its Applications in Problem Solving." Sci. Amer. 232, 102-107, May 1975.

A square rotated about a non-intersecting line that is parallel to the closest side of the square.

The mathworld.wolfram description of a toroid specifically states that the axis of rotation does not intersect the shape being rotated, but on the mathworld.wolfram page describing a specific case of a toroid, called a torus, three types of tori are described:

  • The axis of rotation for a ring torus does not intersect the rotated circle. ring torus
  • The axis of rotation for a horn torus lies tangent to the rotated circle. horn torus
  • The axis of rotation for a spindle torus intersects the rotated circle. spindle torus

(All images in this post come from the Wikimedia commons and have been released into the public domain.)

EDIT:

Based on the comment by @dannysauer: "Given that you're trying to describe a specific kind of toroid, adding an adjective to the base "toroid" seems quite reasonable." I assume that in this case square toroid or rectangular toroid would be the terms being meant.

Like others, I'm not completely satisfied with the generic term toroid to describe the shape of a Compact Disk, since it covers so many other related shapes. Here are some other terms that may be more suitable:

A search on Google for the quoted text "axially bored cylinder" only returns eight results, mostly from patent descriptions. While descriptive and accurate, it's not common enough to be used in most applications.

The term cylindrical shell is much more common, especially among calculus aficionados, but like "axially bored cylinder" this term more accurately describes a tube than a disk with a hole through the middle. A cylindrical shell is a rectangular toroid where the height of the rotated rectangle is larger than its width.

A last term, that is also very common among the calculus folk, is one that appeared in the first few words of the original question. A washer is a rectangular toroid where the width of the rotated rectangle is larger than its height. This page on mathdemos.org has a number of great illustrations of "washers".

oosterwal
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    CDs and washers are Toroids, but I think that the term is too general, as any closed plane curve is allowed. For instance, a Gugelhupf (http://en.wikipedia.org/wiki/Gugelhupf) is a Toroid, but a Gugelhupf isn't a disk with a hole in the middle. – Fillet Aug 25 '11 at 13:08
  • @Fillet: +1 I totally agree. I'm not aware of a specific single word that describes only a "solid cylinder with a cylindrical hole bored through its axis." – oosterwal Aug 25 '11 at 14:27
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    Umm, reading the post we're commenting upon, it seems that "ring toroid" would describe only the specific situation where the axis of rotation does not intersect the rotated polygon. A Google search indicates not-uncommon usage of that term. Given that you're trying to describe a specific kind of toroid, adding an adjective to the base "toroid" seems quite reasonable. – dannysauer Aug 25 '11 at 14:44
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If you just remove a point from the middle, it's called a "punctured disk."

ncmathsadist
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I think part of this fascinatin' debate comes from the question of context. If the word you are seeking is for mathematically-inclined readers, torus describes the 3D ring shape, whether tending towards a donut or tending towards a CD. Annulus describes the planar figure, which the OP illustrated.

(As I have noted in other comments, my stack of 100 CDs is 6" high and clearly lives in 3-dimensional space. Annuli live in 2-dimensional space and have no height.)

Brewster Rockit: Space Guy 27Aug11 (Popular example of the planet of the Donut People being called Torus 8. Does anyone who hasn't read your SO question and responses get the joke?)

For those astronomically minded, annular disk would bring to mind Saturn's rings.

For the rest of literate, non-mathematical, non-astronomical humanity, I think washer-shaped works well, or donut-shaped if it has a bulge.

rajah9
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To make it simple: a flat donut.

Alexander
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    Not quite. A flat doughnut has rounded edges--basically a circle or ellipse that has been rotated around a point outside of that shape. In mathematics this shape is special case of a toroid called a torus. The item in question is a right cylinder with a bore-hole cut through its axis. ...or a rectangle rotated around a line that does not intersect that rectangle. This is simply called a toroid (which covers all shapes rotated about an axis. – oosterwal Aug 25 '11 at 12:09
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    I don't understand why the flat donut is wrong – Mark Sep 22 '11 at 07:55
  • oosterwal, I think you're wrong about the item in question, because none of the examples from the question have completely straight edges like a cylinder: "Compact Discs, washers and Aerobie frisbees" – Alexander Dec 02 '11 at 16:10
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I was thinking donut- or bagel-shaped disc.

Mark
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Isn't that just a...circle? Or am I missing something?

Michael Haren
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    A circle would describe something like a hoola hoop, not something like a CD. – dave Aug 25 '11 at 05:33
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    A circle only has one radius. What I drew has two: an inner radius and an outer radius. But whenever you draw a chalk circle on a board there will be two radii, and the difference is the width of the piece of chalk. So every representation you have ever seen of a circle was, to a pedantic mathematician with a microscope, an annulus. – Fillet Aug 25 '11 at 08:36
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    In fact, a circle is just the edge of a disk. – ncmathsadist Aug 25 '11 at 11:45
  • If we were on math.se I'd totally agree with @Fillet... – Michael Haren Aug 25 '11 at 13:11
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    @Fillet : that's why they're called representations, and not examples: they're imperfect images of an ideal – Kheldar Aug 25 '11 at 19:42
  • @Fillet - Such kind of explanation will not work in the real world. Limit it to the theoretical world and academia. – manojlds Aug 27 '11 at 20:08