From Wikipedia, the free encyclopedia
Jump to: navigation, search
WikiProject Mathematics (Rated B-class, Top-importance)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
B Class
Top Importance
 Field: Geometry
One of the 500 most frequently viewed mathematics articles.

Bug in new wiki software[edit]

I'm thinking there is some bug with the new wiki software, I saw the text went away that Zundark is talking about, but couldn't figure out how to bring it back. I'm using internet explorer 5, and had no problems with the old software. Either that, or my dog ate it. <grin> -- BenBaker

There was such a bug on the first day. Should be long gone. For old version, you'll have to wait for the promised --Magnus Manske


Does anyone have any objections to the following text being added...?

A sphere can also be defined as a sphericon that is based on a polygon approaching a circle.

Proberts2003 19:31, 10 Apr 2004 (UTC)

I do Tosha 00:42, 11 Apr 2004 (UTC)

Which are...? Proberts2003 01:45, 12 Apr 2004 (UTC)

Ok, I think it is not directly relevent, one can mention sphere in sphericon, but not other way arround. (Otherwise you should include in this article ref to all geometric topics) Yet an other thing: yes it can be defined this way but it would be most wierd way to define sphere. Tosha 05:15, 12 Apr 2004 (UTC)

Water drops[edit]

The article currently says "water drops (in the absence of gravity) are spheres". I changed this to simply "small water drops are spheres" because photographs of rain show small spheres (see ). -- DavidCary 23:33, 27 Jun 2004 (UTC)

WRONG! — Preceding unsigned comment added by (talk) 21:40, 21 November 2013 (UTC)


I believe the claim that the topological convention is "the most common" is NPOV, as I believe it's a true statement. (Of course, people who work in the fields using the other definition will feel differently...I don't directly work in either field, so I'm going by my own experience. The topological definition is the only definition I've ever seen or heard; until today I wasn't aware there was another convention.) It would be good if the actual fields of research that use this convention were spelled out. (I honestly have no idea what these are.) I strongly object to the use of the term "geometrical definition" or "geometrical convention", because "geometry" is too broad a word; also, it conflicts with the convention used by differential geometers, who most certainly consider themselves to study "geometry". Revolver 00:08, 12 Jul 2004 (UTC)

Unless someone can provide at least one example of a peer-reviewed mathematical paper that uses the term "n-sphere" to mean (n-1)-sphere, then I'm going to remove all this stuff about different conventions. There is only one convention, as far as I'm aware. --Zundark 09:35, 23 Dec 2004 (UTC)


Can we split this up into two sections — Geometry and Topology — as I did for Ball (mathematics)? In the Geometry section have the metric-space definition (locus of points a radius from the center; boundary of a ball), the Eulcidean examples, the current Equations subsection, and all that jazz; and under Topology have

A sphere is any space homeomorphic to the Euclidean sphere described above under Geometry.

, definition of n-sphere, the fact that the boundary of a ball is a sphere one dimension down (for n>0), and definitions of homology sphere.msh210 21:17, 27 Oct 2004 (UTC)

Proof of compactness[edit]

the prove that a n-sphere is compact is not complete?

  • needs why complement has only innerpoints
  • why is it bounded?? (maybe due it's definition )

Split article?[edit]

Currently the material on n-spheres is split between this article and hypersphere. The situation is somewhat unsatisfactory. I propose separating the material along logical lines

  • the sphere article should focus the ordinary 2-sphere in Euclidean 3-space. This is probably what people expect when they type sphere into the search box.
  • the n-sphere article should discuss the general case, with sections on both geometry and topology. We can have a redirect from hypersphere to there (I prefer it this way since the term hypersphere is not in common usage in mathematics).

I'm happy to do this split if no one objects. Comments? -- Fropuff 02:29, 2005 Apr 16 (UTC)

Iff sphere has a link to n-sphere at its top (not just {{otheruses}}), I agree. That is, it should say something like For higher-dimensional spheres in mathematics, see n-sphere; for other spheres see Sphere (disambiguation).msh210 13:50, 17 Apr 2005 (UTC)

Perfect symmetry[edit]

Seems to me that the sphere is not perfectly symmetrical. For example, punctured 3-space is also self-similar under radial contractions, and so is more symmetrical. Ideas of perfection led to a lot of wrong-headed ideas about e.g. planetary motion. Maybe they are worth mentioning, but in my opinion putting "perfection" up front doesn't help. --JahJah 02:50, 22 August 2005 (UTC)


Definition number one from It's succinct, accurate, and readable. It accomplishes in one sentence what Wikipedia's entire article fails. To whom are you trying to explain a sphere? Who is your audience? If your son or daughter asked you what a sphere is, how would you describe it?

Also, ease up on the passive voice.

finding the surface area of part of a sphere[edit]

I think it would be very useful to have either a derivation for, or simply the integral one needs to use to find the surface area of part of a sphere. Also perhaps it would also be useful to do the same thing for the volume of part of a sphere. 18:22, 10 May 2006 (UTC)

I would like to echo and amplify this comment. After about an hour of google searching I didn't find an article I liked deriving the surface area of the sphere. It seems this and the volume derivation should be front and center. Particularly, an elementary derivation would be nice. Kaimiddleton (talk) 21:57, 25 December 2008 (UTC)
I empathise with you. As a pupil, it was never explained to me why the surface area of a sphere is what it is, although the volume formula was, as was the area of a circle. I offer a relatively simple explanation to those not so comfortable with double integrals and spherical coordinates. Unfortunately I cannot cite any references, as I don’t have any (I’ve never actually seen an explanation in any book). Perhaps someone else can verify and/or provide some diagrams. Gouranga Gupta (talk) 15:29, 27 April 2009 (UTC)


Yesterday I assumed that a sphere could obviously be coloured with 4 colours. Then I realized I didn't actually know this, and it wasn't obvious. Can anyone confirm it by adding a bit to the article?

Maybe you mean the Four color theorem? --Abdull 09:29, 30 May 2006 (UTC)

Removal of "Jade Sphere" Graphic[edit]

It seems to me that this graphic is irrelevent and adds nothing to the article. It seems more suited to an article concerning digital graphics/photoshop/layers/aqua. Therefore, I am removing it from the sphere article.

Differential equation[edit]

A sphere of any radius centered at the origin is described by the following differential equation:

What is a sphere of any radius? --Abdull 09:46, 30 May 2006 (UTC)

What is meant is all spheres, regardless of radius satisfy the equation if they are centered at the origin. So
satisfies the above equation, for all R. Sverdrup❞ 10:37, 30 May 2006 (UTC)


Shouldn't there be links to proofs of claims made on math pages? For instance, the claim that the sphere is the smallest shape enclosing a particular volume may seem obvious, but it would be good to see a proof.

Also, I added the word "locally" to the claim that "surface tension minimizes surface area", since there are actually examples of surface tension creating locally minimum, but not globally minimum, surface area. Flarity 06:33, 22 July 2006 (UTC) its all so easy

other usage[edit]

does S^n only mean a N sphere or does it have any other meaning? thanks --I got scammed 09:12, 6 September 2006 (UTC)

spherical triangle picture[edit]

i couldn't figure out how to take the picture out temporarily but, the spherical triangle picture is incorrect. see the discussion of the image.

I've moved it here. --agr 13:49, 29 September 2006 (UTC)
A spherical triangle (red).

2-sphere as 3-dim object[edit]

I think it's unnecessary to say that the 2-sphere is a 3-dimensional object. Given the logic (ie. that it can be embedded in R3), a loop, a point, etc. are 3-dimensional objects. But they are also 4, 5, and 6(etc)-dimensional objects as they can be embedded there as well.

Sorry, a 2-sphere is (somewhat non-intuitively) a 2-dimensional object. Tparameter (talk) 04:02, 14 December 2007 (UTC)

Removed vandalism[edit]

just removed some vandalism from the page (says "removed filth" in the edit log). I am not sure as to whether anything was deleted to put it in and I am not sure how to check. It would be helpful for someone to either do this for me or point me in the direction of instructions, so I know what to do in future. I just had the immediate response of removing the vandalism at the time. Aphswarrior 23:15, 10 January 2007 (UTC)

You got it right - you can check by looking at the page history. For help on reverting vandalism (or anything else that needs reverting), see Help:Reverting. --Zundark 09:27, 11 January 2007 (UTC)

Number of points to define a sphere?[edit]

What is the minimum number of point needed to define the surface of a unique sphere? Such as 2 points define a line, 3 points define a plane and a unique circle, my guess is that you need at least a fourth point not in the plane of the other three, but are more required? 14:28, 30 March 2007 (UTC)

Yep. One could think of a one parameter family of spheres which each contain the circle, a fourth point would define one memer of the family. Its a strangly high number consider as you only need 4 numbers center and radius to uniquely define a sphere. --Salix alba (talk) 22:03, 30 March 2007 (UTC)
You got to mind the difference between finite and infinite objects. Any point can be represented by a vector from the origin to the coordinate in question. As you said, a line requires two vectors to define, a plane three, etc. However, in the strictest sense of the terms, lines and planes are infinite objects. A line has no ends, while a plane has no borders. A sphere on the other hand is a finite volume, specifically defined to be 3-dimensional. As for the minimum number of vectors/points required to define it, the simple answer is two (this answer applies to lower and higher dimensional objects of the same family as well). The first vector defines the origin/center of the sphere (let's call it vector 'O'), while the second vector may rest on any arbitrary point on the surface of the sphere (Call it 'P'). By Subtracting O from P, and taking the magnitude (result is distance between surface and center), we get the radius of the sphere. But because this magnitude is constant for any P that lies on the surface (by the very definition of sphere), this vector is not actually required. Instead, a Sphere can be defined using only a vector/point to define the origin, as well as a single scalar to define the radius; hence the final answer is a vector and a scalar. Ghostwo 00:44, 13 December 2007 (UTC)


The article states the following formula under the paragraph equitations, Cartesian coordinate system :

Could it be that the formula for x and y are incorrectly reversed ? Both the wiki-page and mention the formula for x and y the other way around. As I am not a mathematician, I post it here so that somebody can verify it. Reaver121 19:56, 14 May 2007 (UTC)

According to Multivariable Calculus 5e p. 1136 (By James Stewart, Published by Thompson Learning), the parametric definition of a sphere in cartesian space is:
Thus, the article seems to be correct as of this writing.
Ghostwo 01:36, 13 December 2007 (UTC)

Straight line[edit]

A straight line may be defined as the locus of a point which moves radially from another point traversing as the circumference of the circle infinitely,when viewed perpendicularly. —The preceding unsigned comment was added by (talkcontribs) 11 June 07.

Change correct?[edit]

Is the change at 2007-07-20 07:33 by Section→Equations alright? = Philips Campus-ICT, Eindhoven, NL Maybe there's support here: Electron9 08:04, 20 July 2007 (UTC)

You can parametrise a sphere in this way, but its not the standard spherical coordinates. I've reverted the change. --Salix alba (talk) 08:29, 20 July 2007 (UTC)

For reference: in mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance from a fixed origin, the elevation angle of that point from a fixed plane, and the azimuth angle of its orthogonal projection on that plane, from a fixed direction on the same. The elevation angle is often replaced by the inclination angle measured from the zenith, the direction perpendicular to the reference plane.--Wikamacallit (talk) 13:19, 26 November 2009 (UTC)



  • 1. Kugel mit einem Radius von 6.000.000 Meter.
  • 2. Der Radius ist zugleich die Höhe zum Nordpol und zum Suedpol.
  • 3. Der Nordpol und der Suedpol sind Punkte und haben keine Fläche.
  • 4. Der Nordpol und der Suedpol haben die Gradzahlen 180 Grad und 0 Grad.
  • 5. Die gesamte Kugel dreht sich in 24 Stunden einmal um sich selbst.
  • 6. Die gesamte Kugel dreht sich bei 365/366 Sonnenscheindauern einmal um den Gasplaneten Sonne im Sonnesystem.
  • 7. Der sogenannte Äquator sind 40.075.000 Meter.
  • 8. Jeder Punkt zum Nordpol auf der Kugeloberfläche hat eine andere Höhe in Bezug zum Äquatordurchmesser. Der Mount Everest liegt zum Beispiel geschätzt bei der Höhe 3.000.000 * Meter. Der Punkt London liegt zum Beispiel geschätzt in der Höhe von 4.500.000 Meter.
  • 9. Der Punkt London ist gegeben.


  • 1. Die Höhe jeden Punktes auf der Geraden vom Mittelpunkt der Kugel zum Nordpol (Bsp. London).
  • 2. Die Entfernung vom Mittelpunkt der Kugel zum Punkt London auf der Geraden „dreht sich in 24 Stunden einmal um sich selbst“.
  • 3. Eine Art von Berechnung der Fläche von London auf der Kugeloberfläche. —Preceding unsigned comment added by (talk) 13:28, 12 December 2007 (UTC)

Google translation of above


    * 1 Ball with a radius of 6,000,000 meters.
    * 2 The radius is also the height to the North Pole and Suedpol.
    * 3 The North Pole and the Suedpol are points and have no surface.
    * 4 The North Pole and the Suedpol have the numbers 180 degrees degrees and 0 degrees.
    * 5 The whole ball rotates in 24 hours by himself
    * 6 The whole ball rotates at 365/366 Sonnenscheindauern once around the sun in the solar gas system.
    * 7 The so-called Equator are 40,075,000 meters.
    * 8 Each point to the North Pole on the ball surface has a different height in relation to the equator diameter. The Mount Everest, for example, is estimated at the height 3,000,000 * meters. The point is, for example, London estimated in the amount of 4,500,000 meters.
    * 9 The point is London.


    * 1 The amount of each point on the straight line from the center of the ball to the North Pole (eg London).
    * 2 The distance from the center of the ball to the point on the London straights "turns into 24 hours to herself".
    * 3 A kind of calculating the surface area of London on the ball's surface. 

--Salix alba (talk) 14:54, 12 December 2007 (UTC)

Hilbert Space[edit]

"[A] sphere is the set of all points in three-dimensional space (R3) which are at distance r from a fixed point of that space, where r is a positive real number called the radius of the sphere"? I'm sorry, but Merriam-Webster disagrees. A sphere is "a solid that is bounded by a surface consisting of all points at a given distance from a point constituting its center". Therefore, a sphere is not bounded by dimension, and, therefore, can only exist in infinite-dimensional [Hilbert space]. This article would be better moved to [3-sphere]. Junulo (talk) 18:13, 16 December 2007 (UTC)

This is about the mathematical entity, not the layperson's definition. That might be at Wiktionary. In either case, I'd say that comparing encylopedia articles with definitions is an apples and oranges endeavor. Tparameter (talk) 14:37, 17 December 2007 (UTC)

Surface Area is the differential of Volume[edit]

In a maths lesson I noticed that the equation for the Surface Area of a sphere is the differential of the Volume equation. This is not mentioned in the article but I assume this is not just a coincidence, could someone please try to explain this and mabe it needs adding to the article.

Mrpowers999 (talk) 13:56, 9 April 2008 (UTC)

Perhaps the inverse phenomena might make more senses: volume is the integral of surface area. Consider a solid sphere, of radius R, made out of series of concentric hollow spheres, Russian-dole style. If the hollow spheres fitted perfectly the volume of the solid sphere would be the sum of the volumns of the hollow spheres. Now the volume of each solid sphere is to the first approximation its surface area times it thickness δr, that is for a hollow sphere of radius r it volume will be and the sum will be . In the limit you get the integral . You get a similar result for circumference and area of a circle. --Salix alba (talk) 17:04, 9 April 2008 (UTC)

Moved formula of maximal inscribed cube[edit]

This edit feels like the solution to a homework exercise, rather than a fact about spheres which passes the basic WP:WEIGHT requirement for inclusion. I have reverted this edit. The text was:

For the largest cube possibe with all vertices tagent to the surface of the sphere.
2[√⅓(r2)]= L
r = The radius of the sphere.
L = One side of the insribed cube.

This formula is completely out of place without a wider context, and Wikipedia is not an indiscrimate collection of information. (Moreover, I should also mention the side issue that it is unintelligible to talk about a vertex being tangent to a sphere.) siℓℓy rabbit (talk) 00:43, 28 November 2008 (UTC)

Formula for volume and area[edit]


Request for some simple mathincluding[edit]

echoing and adding to above section, no derivations or wierd symbols or wierd fonts, just simple list of Volume =

Surface area =


Circumference at a distance r from the equator (eg, the circumference at a given latitude) I think the answer to this is 2(pi)R*cosine(L); L in degrees. It appears that excel 2003 flunks simple math cause the formula =COS(RADIANS(90)) (excel calculates in radians) returns ~~1e-17 where L is the latitude, a value that runs from 0 at the equator to 90o at the pole

how many spheres of diam D1 can pack on a sphere of D2, D1<D2 thanks, Cinnamon colbert (talk) 12:59, 15 May 2009 (UTC)

rationale for simple math section[edit]

I would say to the math inclined, it is almost impossible to overestimate how intimidated and confused people get by math; a simple section like this is needed, so people don't get lost in the longer sections. —Preceding unsigned comment added by Cinnamon colbert (talkcontribs) 13:21, 17 May 2009 (UTC)

Both formula are at the top of their relevant sections so should be easy enough to find. You addition also had some errors, pi is closer to 3.14 than 3.12 and the volume of a sphere is closer to 4 r^3 than 3 r^3.--Salix (talk): 14:13, 17 May 2009 (UTC)

Dear Salix:

I postulate that the article, as currently written is confusing, and that the simple math section will help a lot of people. The questions then are, (a) is it in fact confusing, and, if so, to how many people, and (b) how do we write the article correctly. I'm not sure how we resolve these questions; I invite your comments. Cinnamon colbert (talk) 12:47, 18 May 2009 (UTC) PS: if you have the formula for packing small spheres on a large sphere, I could reallly use that.

written by —Preceding unsigned comment added by (talk) 07:57, 22 August 2009 (UTC)

Complete proof of sphere value and surface area[edit]

Sphere area proof
Half sphera value dividing into 10 discs of radius r=1 and height h=r/10=1/10. From Pythagoras theorem And so each of ten discs radius: , , , , ..., .

All sphere value:


Close to integrated result:

Dividing into more peases gives more precise answer.

Surface area proof
First take a look here.
Each pyramid value is . And sphere can be divided into many such pyramids. And if h=r=1, then surface of sphere will be sphere value multiplied with 1/3. So surface area . —Preceding unsigned comment added by Matematikas1 (talkcontribs) 09:06, 30 December 2009 (UTC)

Neutron stars[edit]

"An image of one of the most accurate man-made spheres, as it refracts the image of Einstein in the background. This sphere was a fused quartz gyroscope for the Gravity Probe B experiment, and differs in shape from a perfect sphere by no more than 40 atoms of thickness. It is thought that only neutron stars are smoother." This is contradicted by the sourced information at Neutron star "Below the atmosphere one encounters a solid "crust". This crust is extremely hard and very smooth (with maximum surface irregularities of ~5 mm), because of the extreme gravitational field." Kernow (talk) 03:27, 14 May 2010 (UTC)

This is not a direct contradiction. The measure of smoothness of a sphere is relative to its diameter. 5mm in 20km is a smoothness of 4e6. The gyro is reportedly 10nm in 4cm, a smoothness of ... 4e6! GreenAsJade (talk) 11:18, 1 June 2010 (UTC)

This is my first Wikipedia comment, so sorry if I didn't get the format right. Anyway, in the description for the image, it reads "It is thought that only non-rotating neutron stars are smoother." I think this is a somewhat irrational statement because:

a) Celestial objects rotate. b) Neutron stars, as I understand them, rotate at an extremely high angular velocity due to the conservation of angular momentum, since they used to be larger stars.

This statement already has a citation needed. Does anybody have any arguments for keeping it? — Preceding unsigned comment added by (talk) 03:51, 18 November 2011 (UTC)

Physics "definition" of a sphere[edit]

The Article currently says

"As defined in physics, a sphere is an object (usually idealized for the sake of simplicity) capable of colliding or stacking with other objects which occupy space"

This seems to be failing to mean what it intendes to mean :)

A car, in physics, is capable of colliding with other objects that occupy space. So is the Earth. Neither is taken to be a sphere in physics...

GreenAsJade (talk) 11:21, 1 June 2010 (UTC)

Actually, the Earth is considered to be a sphere... but you're right that this statement needs some clarification.—Tetracube (talk) 16:55, 1 June 2010 (UTC)
It depends on what level of detail you are looking. The Earth is an Oblate Spheroid. I agree it may be a bad example of what is not a sphere though :)
I wanted to pick an entity that more typically under consideration than a car, in physics.

GreenAsJade (talk) 21:39, 1 June 2010 (UTC)

The sentence was introduced with this revision [1] as User:Phasechange's 3rd edit. His edit to Disk (mathematics)
In theoretical physics a disk is a rigid body which is capable of participating in collisions in a two-dimensional gas. Usually the disk is considered rigid so that collisions are deemed elastic.
is slightly better. It seems that he is trying to describe the properties of a sphere to be useful for physics rather than define a physical sphere. A small rewording should clarify the intended meaning. --Salix (talk): 18:10, 1 June 2010 (UTC)

area element[edit]

The article gives "the area element on the sphere" only "in spherical coordinates", can we also list this in x,y,z??? Jackzhp (talk) 23:32, 18 March 2011 (UTC)

In Cartesian x, y, z coordinates the volume element is simply dx dy dz but the bounds for the integration are quite awkward. An area element in cartesian coordinates could be found by slicing the sphere into thin bands vertically and then slicing in another direction, formula for the resulting segments would be rather awkward. For the sphere it is much easier to work using spherical coordinates which have the nicest formula.--Salix (talk): 00:25, 19 March 2011 (UTC)
The "volume" (i.e., area) form of the sphere can easily be written in Cartesian coordinates. It's just . However, correctly interpreting this expression is probably beyond the scope of the article. Sławomir Biały (talk) 15:19, 28 March 2011 (UTC)


It is well known that Archimedes proved that the area of a sphere equals that of a right cylinder of the same radius and height. However, I can't find any evidence that he proved it using an "area-preserving" transformation. So I've added a "citation needed" tag. Can anybody provide a reference? I'd be very interested to read it! best, Sam nead (talk) 02:53, 17 March 2013 (UTC)

Note that, by reference, I mean a reference to one of the works of Archimedes, or to a reliable historian. For example, Serge Tabachnikov says exactly this in a few places, but he doesn't give a citation... Sam nead (talk) 03:03, 17 March 2013 (UTC)

Well, this page suggests that I may have to eat my words. Still no references, however... Sam nead (talk) 03:15, 17 March 2013 (UTC)

Surface area of a sphere (simple proof)[edit]

The proof given in the article seems a bit complicated. A simple proof can be read in the discussion "Superficie della sfera" on the article "Sfera" in italian Wikipedia. — Preceding unsigned comment added by Ancora Luciano (talkcontribs) 09:59, 26 May 2013 (UTC)

Integral form for surface area[edit]

If I am not mistaken, the integral form for the surface area in the article is not precise. The integral presented only gives half of the surface area (since trying to integrate over 2pi in theta will result in 0), and so the correct form should be something like: (talk) 15:23, 1 June 2013 (UTC)

The article looks correct. We are using the first convention in spherical coordinate system where θ measures the angle from the positive z-axis. Hence θ runs from 0 to π, π/2 giving a point on the equator. Also note .--Salix (talk): 16:49, 1 June 2013 (UTC)

Poorly worded and circular introduction[edit]

... resembles the shape of a completely round ball? A sphere IS a ball, indeed the text has already said so! — Preceding unsigned comment added by (talkcontribs)

Not saying that it's ideal, but a ball is the solid object, whereas a sphere is the boundary. Chris857 (talk) 21:45, 21 November 2013 (UTC)

Spherical product?[edit]

Spherical product redirects to Sphere but it is not mention anywhere in the article, is it? --RokerHRO (talk) 12:20, 31 March 2014 (UTC)

Terminology Section[edit]

From the article: "If a particular point on a sphere is (arbitrarily) designated as its north pole, then the corresponding antipodal point is called the south pole, and the equator is the great circle that is equidistant to them. Great circles through the two poles are called lines (or meridians) of longitude, and the line connecting the two poles is called the axis of rotation. "

Surely this is only true if a) the sphere is rotating (or is only capable of rotating on a fixed axis) and b) The arbitrarily designated poles happen to be at the ends of the actual axis of rotation. (talk) 13:11, 26 April 2015 (UTC)

No, the text is perfectly fine as is. Your reading of it is defective. You are confusing the real world Earth's north pole with a point chosen arbitrarily chosen as the north pole of an imaginary sphere. Similarly, a point in the country of Ecuador could be arbitrarily designated as the Earth's North Pole. — Preceding unsigned comment added by (talk) 03:14, 26 May 2015 (UTC)

But that's exactly my point, which I think you missed. To use your example, if a point in the country of Ecuador is arbitrarily designated as a pole, that doesn't change the fact that the earth's actual axis of rotation still runs from True North to True South, That being the case it would be erroneous to then refer to the line connecting our arbitrary "north" pole in Equador to it's corresponding "south" pole as being the "axis of rotation". (talk) 22:21, 3 June 2015 (UTC)


Gabby Merger insists on using this as the first paragraph in the lead.

A sphere (from Greek σφαῖραsphaira, "globe, ball"[1]) is a perfectly round geometrical or circular object in three-dimensional space that resembles the shape of a completely round ball. Like a circle, which, geometrically, is in two dimensions, a sphere is defined mathematically as the set of points that are all the same distance r from a given point in three-dimensional space. This distance r is the radius of a perfectly spherical ball, and the given point is the center of the sphere. The maximum straight distance through the sphere passes through the center and is thus twice the radius; it is the diameter.

I have several times tried to correct this passage, so let me mention some of the things that are wrong with it.

  1. "a perfectly round geometrical or circular object" - besides not being grammatical, what is the difference between being perfectly round and being circular? If the intention is to clarify then I suggest using parentheses ... perfectly round (viz., circular) ...
  2. "resembles the shape of a completely round ball" - it doesn't resemble this shape, it is the surface of a ball. This was objected to because not all balls are spherical,( ex. football, meatball) but it would be hard to call these red herrings completely round. One additional problem here is that the link is to the wrong page and should be to ball (mathematics) since the term is being used in a technical sense.
  3. "The [[Great-circle distance|maximum straight distance]] through the sphere" - this is just wrong! The great-circle distance is an arclength measured on the surface of the sphere (along a great circle in fact) and is not a straight line distance (in 3-space).
  4. In the rest of the article the term "sphere" is being used correctly (note, for instance, the discussion of the "volume enclosed by a sphere" and not the "volume of a sphere"). The lead should be a summary of what is contained in the article, but this one talks about something that is not there.

It has been claimed that I have made this an issue because "I don't like it". This is in part true, I don't like errors and I don't like sloppy descriptions. I have not included sources because these corrections are really not controversial. If a lead is a summary, as it should be, sourcing is generally not necessary (it should occur in the body of the article). But, if necessary -

Be sure to notice that we are only considering points on the surface of the sphere. In other contexts the term sphere is sometimes used to refer to a solid three-dimensional object; that is not was is meant here. The sphere is two-dimensional in the sense ...

- Gerald A Venema (2006), Foundations of Geometry, Prentice Hall, ISBN 978-0-13-143700-5, pg. 23

Other citations are readily available. Bill Cherowitzo (talk) 20:39, 14 July 2015 (UTC)

Ok, well I appreciate you taking the time to go over your concerns and issues, with some of the wordings and phrasings in the lede (which I did NOT originally create, by the way), and I do not discount the fact that you have a background in these technical matters. Yes, "ball (mathematics)", but the problem is that it simply says "ball" and most people perusing or reading Wikipedia articles won't necessarily know or understand that a "ball" is NOT (repeat NOT) always perfectly round or circular or spherical, but many times the word "ball" is a CLUMP OR PILE of something, that is UNEVEN, and only somewhat "round", but not necessarily a "sphere". Hence, why I felt that the original wording in the original lede (which was that way for many months and months and months) using the known and always-understood word "sphere" would be better. I'll consider what you say. And for now I'll try to follow a suggestion or point or two that you mentioned. Regards. Gabby Merger (talk) 21:45, 14 July 2015 (UTC)
Ok, now I'm back. I just changed the entire lede paragraph almost completely back to how it was, and did EVERYTHING you suggested above. It now reads like this:
A sphere (from Greek σφαῖραsphaira, "globe, ball"[2]) is a perfectly round (viz., circular) geometrical object in three-dimensional space that is the surface of a completely round ball. Like a circle, which geometrically is a two-dimensional object, a sphere is defined mathematically as the set of points that are all at the same distance r from a given point in three-dimensional space. This distance r is the radius of the mathematical ball, and the given point is the center of the ball. The maximum straight line distance through the ball, connecting two points of the sphere, passes through the center and is thus twice the radius; it is the diameter of the ball.
That should be alright now. Regards..... Gabby Merger (talk) 22:11, 14 July 2015 (UTC)
I would suggest getting rid of "(viz., circular)", because that makes it sound two-dimensional. Loraof (talk) 01:57, 15 July 2015 (UTC)
Also, I have a problem with the last paragraph in the lead:
In mathematics, a distinction is made between the sphere (a two-dimensional closed surface embedded in three-dimensional Euclidean space) and the ball (a three-dimensional shape that includes the interior of a sphere). However, in other contexts sphere and ball are used interchangeably and it is common to refer to the radius, diameter and center of the ball as the radius, diameter and center of the sphere.
The last sentence here suggests that the radius etc. of the ball is also referred to as the radius etc. of the sphere only outside mathematics. That's not true--in math the radius of the ball and the radius of the sphere are the same thing, just as in math the radius of the disk is the same thing as the radius of the circle.
I propose the following replacement paragraph:
While outside mathematics the terms "sphere" and "ball" are sometimes used interchangeably, in mathematics a distinction is made between the sphere (a two-dimensional closed surface embedded in three-dimensional Euclidean space) and the ball (a three-dimensional shape that includes the sphere as well as everything inside the sphere). The ball and the sphere share the same radius, diameter, and center.
Loraof (talk) 16:34, 15 July 2015 (UTC)

I agree with the removal of "circular", even though I suggested that modification. I had eliminated the term in an earlier edit. I see what concerns you in the second paragraph and think that the rewrite is a good start. I would like to see something a little stronger than "share" in the second sentence. These concepts are elementary and make sense for the ball, but they are really definitions for the sphere. When ball and sphere are confounded this of course makes no difference since they are taken to be the same, but when distinguished (which I am pretty sure is ahistorical) the terminology needs to be more clearly explained (but not belabored). Bill Cherowitzo (talk) 18:02, 15 July 2015 (UTC)

Okay, but I'm not sure what you have in mind for a replacement for "share". I'll go ahead and put it in as I wrote it above, with the expectation that you'll improve on it. Loraof (talk) 19:15, 15 July 2015 (UTC)
The word "circular" does NOT mean just "2 dimensional" in context. A sphere has been said to have a "circular base". And a "circular section". See here in where it clearly calls a sphere a circular object. Check this mathematical pdf paper out, on that matter of "three-dimensional circle" being a "sphere" etc... (Also, you can click here as a little example.) And a "circularity" about it, and a sphere has been called a "three-dimensional circle". (Just like a cube is a three-dimensional "square" etc). So "circular" is not narrowly limited as just something always only two-dimensional. That's a mis-notion. Not all 'circles' are necessarily "flat disks" in context. A sphere IS a "circle" in that it is "circular", "round", and the sun has been called the "golden circle in the sky" by people who fully know that it's a three-dimensional sphere. The point is that if you were to draw on a piece of paper a full moon, you'd be drawing a "circle" shape. Not a square or triangle or rectangle shape. This word correctly has been in this article for years, and I restored it, but with slightly different phrasing. Regards.... Gabby Merger (talk) 21:18, 15 July 2015 (UTC)

Sorry, but you are misreading these sources. A circle is a two-dimensional object. The intersection of a sphere with a plane (not tangent) is called a circular base or circular section because it is a circle and since it lies in a plane it is two-dimensional. You happened to pick a math paper by a good friend of mine, and I can tell you that he would be very embarrassed by what you claimed. The background for that paper is not even Euclidean geometry, and the terms are being used in analogy only. The other sources would not be considered reliable. People are free to use analogies when talking about full moons and golden suns, but this is artistic license, not a matter of definition and not a good idea to use if it gives the wrong impressions. Bill Cherowitzo (talk) 23:56, 15 July 2015 (UTC)

Even if you have a point about one of the references, I'm not "misreading" at all Jennifer's words here: "A sphere is a geometrical figure that is perfectly round, 3-dimensional and circular - like a ball." And I'm not sure why you consider it a "not reliable source". But even so, you said earlier that "round" and "circular" are redundant and mean the same, and that THAT was your objection. Not because it was incorrect. (Also, you misunderstood, because the "circular base" etc is to the point that there's a general CIRCULARITY with spheres, in given situations, that would not exist at all with cubes. I'm not misreading anything, because you miss my point completely in why I even cited that, and there's no need for your friend to be "embarrassed"...even if his paper was not directly dealing with the matter of spheres, or unless he and you miss the drift of what I was saying. But again, there's no misreading Jennifer's clear words, of "a sphere is perfectly round and circular". She's an instructor in the field.) Spheres are circular, and the word "circle" is not just limited (though it's been made to seem that way, wrongly) to just two-dimensional flat disks, depending on context. And also, it's not like the lead is saying "a sphere is a circle", though some have called it a THREE-DIMENSIONAL "CIRCLE". But simply saying "circular", as that's what "round" basically means, as even you earlier on conceded. Spheres are "circular and round", not just in analogy but in actuality. Jennifer Beddoe is not some cleaning lady who said this. She's a math educator. Spheres are circular objects, in that they're not triangular, they're not square, they're not rectangular. (Sighs...) They're round, they're circular. If you were to walk around a sphere, you'd be walking around in a...what? You'd be walking around in a (you guessed it) CIRCLE. Because spheres are (there's no debating this fact) CIRCULAR. Draw a sphere, on a piece of paper, you're drawing a circle shape. That's all that's meant, and what's obviously understood, for elaboration and clarity. Regards. Gabby Merger (talk) 01:45, 16 July 2015 (UTC)

Why would you think that Jennifer Beddoe is a reliable source? As far as I can determine she is a math (and science) tutor who wrote up some high school level study notes. Those notes appear to be derived, at least in part, from earlier versions of this Wikipedia page. I can't tell if she is a math educator, but she is certainly not an authority. Round and circular are related, but do not mean the same thing. Circular refers to circles (Webster definition: 1) In the form of, or bounded by, a circle. 2) Moving in or describing a circle, etc.) while round is a more general concept (Webster again: 1) Spherical, circular or globular. 2) Circular in cross section: esp., cylindrical, as a rifle barrel. 3) Having a curved outline or form, esp. one like the arc of a circle; rotund; not angular or pointed.) When I made the suggestion above, I was trying to come up with a more technical version of "perfectly round" and the correct word, "spherical", just wouldn't do in this context. I admit that when I said perfectly round and circular were the same I was thinking two dimensionally, but as Loraof correctly pointed out, this term refers only to two dimensional objects and not to the perfectly round sphere. To be precise, I should have said something like "perfectly round (viz., analogous to a circular object in two dimensions)", but this is far too verbose and I actually prefer just to omit the reference all together. One can say that "spheres are round, with circular cross-sections" and if you insist on including "circular" this would be an acceptable locution. One last point, if what I suspect about Ms. Beddoe is correct, this would be a perfect example of why it is important that Wikipedia use language correctly. Errors that are created here have a tendency to proliferate throughout the internet. Bill Cherowitzo (talk) 06:08, 16 July 2015 (UTC)

Even your own cited Webster definitions of "round" have the word "circular" in them, plainly. And that's really the point I was making. There's a circularity with spheres, obviously. As far as Jennifer, again, she's a certified teacher in the subject, and not just some woman who decided to slap a webpage together as a hobby. Why wouldn't she be an "authority"? It seems that you discount her as such because of (pardon the pun, lol) of a CIRCULAR ARGUMENT that simply because she doesn't agree with you apparently on this wording or whatever, or because she worded something in a way that (all of a sudden) for some reason you don't like, she's therefore "not a reliable source". (Or maybe you're the type in general who's too uptight with "reliable sources", frankly speaking, if they don't fit a narrow criteria that goes even beyond WP policy. Even WP policy says to NOT be overscrupolous with that. She's a math educator, and "" was never put on a list of Wikpedia as an "unreliable source".) But as I pointed out earlier, your initial objection to it was NOT that it was inaccurate but that it was REDUNDANT. And sorry, but Loraof is just wrong on this, period. Spheres are circular, and "circle" is NOT just "two-dimensional" as has been the wrong notion of many for far too long. Especially when talking about the word circulAR even. (Circle is BROAD, though many have tried to make it where that's not the case, and make it overly narrow, and THAT's an example of wrong information in a pipeline...) It's gotten ridiculous. To the point where even your own cited definition of "round" has the word "circular" in it, and you still find some way to discount the point!!!!!!!!!! I find that, frankly speaking, disingenous on your part. Your own webster citation for "round" proved my case, and you can't even see it for some reason, because of rationalizing the obvious point away. Says: round (Webster again: 1) Spherical, circular or globular. 2) Circular in cross section: esp., I'm not gonna around in "circles" (pun intended) with you on this. Regards. Gabby Merger (talk) 07:25, 16 July 2015 (UTC)
Gabby, please obey WP:BRD -- you boldly inserted your edit about it being circular, it has been reverted, and the next step is to discuss it, with the status quo ante prevailing unless a talk page consensus is reached to the contrary.
As you point out, Webster's says that "round" means "spherical, circular or globular". This distinguishes between "spherical" (the 3-dimensional case), and "circular" (the 2-dimensional case). It does not say that the two are the same; it distinguishes them by the use of "or".
You said above 'circle is NOT just "two-dimensional", as has been the wrong notion of many for far too long'. Wikipedia is not a forum for you to impose your personal preferences of what is wrong.
If one were to ask 1000 mathematics professors whether a sphere is circular in technical wording, all 1000 of them would say "No". I'm reverting what you have wrongly put in. Don't put it back, because that would be another violation of BRD. Loraof (talk) 14:08, 16 July 2015 (UTC)
hello. The matter was already discussed and referenced. No "violation" of bold revert anything. Just bold elaboration that is sourced and obvious and clear. And what's done against it by you is frankly speaking, uptight impositions. Please stop accusing me of violations, etc, and assuming bad faith, because I won't tolerate it, as a lack of civility. And there are math teachers etc that clearly say that a sphere is "circular" because "round" MEANS "circular". There's a reference and I already went into heavy detail and points and the point again was that "round" is "circular" too in general, and there's definitely a "circularity" with spheres, in specific situations... ...even webster says "round" means "circular". Good day. ( Gabby Merger (talk) 17:04, 16 July 2015 (UTC)
Addendum: You say a sphere is "ball-shaped", correct? That's stated in the lead itself, clearly. Well look at what says about "ball-shaped": . ball-shaped - having the shape of a sphere or ball, ball-shaped - having the shape of a sphere or ball; "a spherical object"; spheric, spherical, orbicular, global 'circular', 'round' - having a circular shape. Want another one? You know what WEBSTER'S DICTIONARY CALLS A "SPHERE"?? 3: [Noun] An orbicular body, or a circular figure representing the earth or apparent heavens.. You're gonna call Webster "unreliable"? Stop edit-warring and disrespecting sourced and grounded modifications and edits. I understand your points and your position, but I doubt "1000 mathematicians" would dogmatically say that a sphere is NOT "circular" or that it has no circularity to it in context, at all. Some would, some wouldn't. But the sources are here. Sphere is a circular object. You draw a sphere on paper, you're drawing a circular shape. Why would that be the case, if spheres are not circular at all? In any point or matter with it? Will you deny that fact? Your concern for accuracy is appreciated. But please don't impugn me or my motives and discount the sources (which are NOT "unreliable", but math teacher and webster itself) that say "circular" or "circle" in the matter of sphere. Spheres do have a circularity with them (obviously), and that’s stated clearly in the works and literature, either indirectly or sometimes even directly, even if the term “circle” is used generally for 2-dimensional geometrical contexts. Thank you. Gabby Merger (talk) 17:24, 16 July 2015 (UTC)

Gabby, you seem so intent on this that you are losing your perspective. Just a couple of comments. Anything that is circular is round, but not everything that is round is circular (rifle barrels come to mind). As I said before, round is a more general concept, they are not the same thing, as you insist. General purpose dictionaries are notoriously bad when it comes to technical definitions in mathematics, and on-line versions are the worst. is useless for precise definitions, and the later versions of Webster have become unreliable as well (in my 5th Ed. of Webster's Collegiate, circular is not used in the definition of sphere). Also, you are cherry picking these definitions to make your point and not trying to understand their intention. Of course you are going to find circular as a descriptor in the definition of round. "Round" is not a technical term and circular is certainly one kind of being round, but that doesn't make them the same! When you draw a sphere on paper, you are not drawing a sphere; you are drawing a representation of a sphere, and yes, you are drawing a circle - but an object and its representations are not the same things, and words that describe a representation need not apply to the actual object. Finally, you seem to have more information about Ms. Beddoe than I was able to obtain and a certain faith that all math instructors express themselves in an absolutely correct manner. This faith is misplaced, I have taught enough of them to know how false this belief is. Math instructors, like everyone else, can and do make mistakes. I have made mistakes ... and when I have, I have admitted them and tried to fix them, but this is not always possible when your words are published by someone else. I have always tried to instill in my students a healthy skepticism of what they read on the internet and I think that this is a lesson that you could profit from. Bill Cherowitzo (talk) 19:46, 16 July 2015 (UTC)

Ok, Bill, I appreciate your comment, and points, and seeming (to some degree) concessions, though I do think there's still a wrong notion to be honest, and also somewhat of a misunderstanding, because even if we go with your premise (and that of many others) that "circle" can only always means two-dimensions, and in geometric contexts, ok, maybe fine, that still does not negate the stated fact that there's a circulARITY in certain situations and points with a sphere. Like the point of "circular sections" and "base", etc...and also the point about walking around a sphere. Let's say not the earth or moon, but just a giant sphere, of maybe 1000 feet diameter etc, and if you were to theoretically walk around it, you'd be walking around in a circle. Why? Because (again) all perfect spheres are circular in that sense. Not saying (if we're talking strictly about the used geometric meaning) that spheres are geometric "circles", per se (even though a case can be made for a broad application of that word for spheres), but that there's circularity with spheres, to some real extent. You'll discount online dictionaries regarding this matter, this and that, conveniently, but meanwhile those same refs are considered "reliable" for other articles on Wikipedia and other terms. That seems too selective there. And the point about drawing a sphere on a piece of paper, of course you're not drawing or MAKING an actual "sphere" per se on a flat piece of paper, but the mere fact (and this was my point) that to represent a sphere, even with a three-dimensional look, you'd have to, all the time, be drawing a circle shape on that paper. A circular object. A circular figure. Wikipedia is for all people, not just for post-graduate and expert types, or even for college students. There are some arguably solid sources (it's not like this was on "" or something, but, and also Noah Webster, toto definitions, etc.) that said clearly "spheres are circular". At least in some real obvious sense. I'm sure that your friend in the PDF is probably more of an expert than Jennifer, but Jennifer Beddoe is a math teacher and instructor, and does heavy work for the reputable and widely-used learning website "". (Another example of her words and work regarding "symmetry in math" click.) Not some joe schmo on Yahoo questions or something. By the way, the pdf has two authors. Culbert and Ebert. Not just one. And you said that he was your good friend. But that pdf page has two experts writing it. So which one were you referring to? Culbert? Or Ebert? Because you said in an earlier comment that "he" would be embarrassed. Just wondering who specifically you meant.... Regards. Gabby Merger (talk) 20:17, 16 July 2015 (UTC)

Gabby this has gone far enough. Since you inserted this on 12 April 2013, it has been reverted at least five times. You are wrong and you do not have any support for your position. Jennifer's entry is no better than a blog and is considered unreliable by WP standards. Not one of the definitions in TOTO supported your claim. I have corrected the statement in the article. For your information, although it is irrelevant, my friend is the senior author of the paper and an Emeritus Professor of Mathematics at the University of Delaware. He trumps Jennifer. Bill Cherowitzo (talk) 21:07, 16 July 2015 (UTC)

Bill, not sure why the hostility and why you would say that the TOTO definitions "do not support my claim". When it says clearly for sphere "a circular figure". How is that "not supporting my claim"? Also, as far as who "trumps" whom, that's a matter of taste or opinion, and Jennifer is in the field (as is clearly seen and proven) and that's NOT a "blog". And Webster is definitely not either. That comes off as a bit elitist, to be honest. And that's against WP policy or recommendation. As far as your friend, you brought him up, and I was just wondering, though, both Ebert and Culbert are "Emeritus Professors". They both are. So I was asking who it was specifically you were referring to. It looks like it was Ebert. Anyway, it's not fair or WP valid to discount Webster and Jennifer Beddoe simply because you disagree with them. And it's a "circular argument" (get over it, it's a pun), to say "they're unreliable". Various sources can and should be used for WP articles. Not just super technical and uptight and elitist ones. That's been a complaint against Wikipedia for a long time now, by the way. But even so, Jennifer Beddoe and is not some Facebook blog or something or "" or "yahoo questions". You're putting her solid work that's on in the same exact category as that, and that's just plain demonstrably wrong, illogical, unfair, and inaccurate. It's a reliable source, even if as not as high or technical as maybe some others, and even if you don't consider it in the same category as one of Ebert's pdfs. Good day........ Gabby Merger (talk) 21:36, 16 July 2015 (UTC)


  1. ^ σφαῖρα, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
  2. ^ σφαῖρα, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus

Dubious definition of axis of rotation[edit]

In the Terminology section, it states that you can take any arbitrary point of the sphere as a pole, take its antipodal point as the opposite pole and the line between the two points is the axis of rotation. This is only true in the case where a) The sphere is rotating and b) the two poles are chosen such that their positions on the sphere are unchanged by the rotation - ie. not arbitrary.

The article correctly states (although in a very awkward way) that any diameter of a sphere may be an axis of rotation. This is a mathematical property of spheres and has nothing to do with spinning physical objects. Please sign your comments and put new ones at the end of the page. Thanks. Bill Cherowitzo (talk) 20:32, 27 July 2015 (UTC)

numbers of small sphere in a large sphere[edit]

I have calculated the the numbers of small sphere in alarge sphere by my formulaدکترغلامعلی نوری Dr Goal.A. Newray 09:05, 12 September 2015 (UTC)

prediction of numbers of small sphere enclosed in the larger sphere, by Dr Goal.A.Newray`s formula.

— Preceding unsigned comment added by غلامعلي نوري (talkcontribs) 09:05, 12 September 2015 (UTC)

This looks like Sphere packing in a sphere, your results seem to differ. We cannot publish results here which have not been peer reviewed, please see WP:OR.--Salix alba (talk): 18:01, 12 September 2015 (UTC)

Assessment comment[edit]

The comment(s) below were originally left at Talk:Sphere/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Seems much too brief, there is plenty more which could be said about the sphere. User:Salix Alba 16:52, 14 February 2007

The language is couched in academic high brow, most people wanting answer to this question will want a far simpler and easier to understand lesson, eg thier are no explanations to formula what it means and so on, formula for pi would be a start.

Otherwise good work, somebody might like to add a simpler section but please don't call it PI for idiots. —Preceding unsigned comment added by (talk) 11:00, 16 September 2009 (UTC)

Last edited at 11:02, 16 September 2009 (UTC). Substituted at 02:36, 5 May 2016 (UTC)