# Talk:Plane (geometry)/Draft

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 Definition:  In elementary geometry, a flat surface that entirely contains all straight lines passing through two of its points. [d] [e]
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## locus of points

In the last sentence of the lead, I would include that the locus is considered in a space whose properties are assumed to be known (from elementary geometry.) --Peter Schmitt 23:19, 12 May 2010 (UTC)

What do you prefer, "locus of points" or "set of points"? Boris Tsirelson 05:39, 13 May 2010 (UTC)
I did not think of it, but you are right: "locus" is a technical term, and "set" is easier to understand. --Peter Schmitt 10:56, 13 May 2010 (UTC)
Really, I am not completely understanding the logic of this lead. Why Hilbert's contribution (and even non-Euclidean geometry) are here? Why not in the section "Axiomatic approach"? And especially, the text "The first axiom regarding the plane is axiom I4: Three points A, B, C that are not on one and the same line determine always a plane α. He adds that this is expressed as "A, B, and C lie in α", or "A, B, and C are points of α". His axiom I5 is a subtle extension of I4: Any three points in plane α that are not on one line determine plane α", is it relevant enough? And, not completely unrelated question: what is your impression of my lead to "Line (geometry)"? Boris Tsirelson 11:55, 13 May 2010 (UTC)
By the way, all these "includeonly" inserted for the "Draft of the week", probably they should now be removed? Boris Tsirelson 11:58, 13 May 2010 (UTC)

## Definition via lines

I have moved the figure to this place. It is not ideal (and could be replaced later: Tom has departed, so we cannot ask him to change it) but sufficient, if the description is adapted to use A,B,C accordingly. --Peter Schmitt 11:18, 13 May 2010 (UTC)

I'll better make another picture, wait about two days. Boris Tsirelson 12:01, 13 May 2010 (UTC)
I did. Boris Tsirelson 08:08, 14 May 2010 (UTC)

## plane geometry

rectilinear or piecewise-linear or both? --Peter Schmitt 22:30, 13 May 2010 (UTC)

Surely not "rectilinear". Really, this text
"A plane figure is a combination of points and/or lines that fall on the same plane. In plane geometry every figure is plane, in contrast to solid geometry.
A rectilinear figure is a plane figure consisting of points, straight lines and straight line segments only. Rectilinear figures include triangles and polygons."
is not mine; I did not edit it at all, and in fact, I do not agree with it. Is it needed at all? Boris Tsirelson 10:30, 14 May 2010 (UTC)
Oops, sorry, I am cheating. I did edit it. Before me it was:
"A plane figure is a combination of points and/or lines that fall on the same plane.
A rectilinear figure is a plane figure consisting of straight lines only. Rectilinear figures include triangles and polygons. But anyway: Is it needed at all? Boris Tsirelson 10:44, 14 May 2010 (UTC)
To give the second paragraph some meaning one could state that plane geometry studies the properties of plane figures (and configurations).
And rewrite the second paragraph: Plane figures are sets of points, lines, line segments, ...
--Peter Schmitt 23:08, 29 July 2010 (UTC)

## modern approach

To exclude other uses of affine space, it should probably be real affine space. --Peter Schmitt 22:34, 13 May 2010 (UTC)

Fixed. Boris Tsirelson 10:34, 14 May 2010 (UTC)

## beyond mathematics

"made close to ... a finite part/subset of a plane", only --Peter Schmitt 23:06, 13 May 2010 (UTC)

Fixed. Boris Tsirelson 10:36, 14 May 2010 (UTC)

## Definition via right angles

What do you think of doing this as in "Line" (using triangles)? --Peter Schmitt 23:33, 5 June 2010 (UTC)

In "Line" I use "orthogonality in disguise" because I am afraid the usual orthogonality would raise objections: "you cannot introduce angles before lines". But in "Plane" there is no such problem: lines may be introduced before planes. Do you really want to see "orthogonality in disguise" in "Plane"? In addition to the orthogonality used now, or instead of it? Boris Tsirelson 09:00, 6 June 2010 (UTC)
Not necessarily. I just wondered why you avoided right angles in Line, but not here. I can see it now. I am not sure if some comment on this should be added (and what) --Peter Schmitt 10:40, 6 June 2010 (UTC)
By the way, a number of my questions (in sections above) are waiting for your answer... Boris Tsirelson 21:21, 8 June 2010 (UTC)
I know ... it is just that I left the difficult parts (the lead) to be touched last ...
While reading the article it occurred to me that all but the first definition assume that "lines" are known (defined). But in "Line" all but the first definition assume that "planes" are known. Thus most combinations are circular. This has to be mentioned somewhere.
I still wonder whether the "algebraic equations" section fits well into this article, or if it should be moved elsewhere. If it stays the relation to two of the definitions should be pointed out. (Moreover, my impression -- I may be wrong -- is that, in an attempt to make everything absolutely clear, the description got needlessly complicated with too many different notations.)
As for (plane/rectilinear) figures: The need not be defined here, I think. One could mention that plane geometry is concerned with sets of points and sets of lines, their properties, and their relations to each other, and perhaps mention polygons, curves, ???, ...
--Peter Schmitt 15:30, 9 June 2010 (UTC)
"But in "Line" all but the first definition assume that "planes" are known." — I'd say it differently. In the school, as far as I know, pupil learn first the (easier) plane geometry and only later the (harder) solid geometry. After learning the solid geometry they know planes; they understand them as parts of the space, their intersections etc. But before learning the solid geometry they know "the plane", with only rudimentary idea that it is a part of the space. I would not say that "planes" are known to them. Boris Tsirelson 17:52, 9 June 2010 (UTC)
But anyway, what is the problem? Circularity is forbidden in the axiomatic approach, but quite usual in the non-axiomatic approach. Boris Tsirelson 18:42, 9 June 2010 (UTC)
"the description got needlessly complicated with too many different notations" — well, it could be better; but (almost) everything could be better. It takes about one third of the length of the article, near the end; it is natural that a more technical part follows a less technical part. Boris Tsirelson 18:03, 9 June 2010 (UTC)
"As for (plane/rectilinear) figures:" — please look now. Boris Tsirelson 18:13, 9 June 2010 (UTC)

• Definition via lines:
"Consider the lines DE for all points D on the line AB different from B and all points E on the line BC different from B."

I think that changing the order of phrases (or another change) makes it easier to read:

"Consider the lines DE for all points D different from B on the line AB and all points E (also) different from B' on the line BC."

Wouldn't

"through F that intersects the lines AB and BC, in distinct points."

be simpler than

"through F that intersects both the line AB and the line BC, and not at their intersection point B."?
Done. Boris Tsirelson 06:49, 30 July 2010 (UTC)
• Fig. 1. "Equation for plane. P is arbitary point in plane;" better "The equation for a plane. P is an arbitary point in the plane; "?
Done. Boris Tsirelson 06:54, 30 July 2010 (UTC)

Point-normal representation

I think \equiv is unusual here and should be replace with "=" or ":=".

Inserting an additional step would help some readers:

${\displaystyle \left(\mathbf {r} -\mathbf {d} \right)\cdot \mathbf {n} _{0}=0\Longleftrightarrow \mathbf {r} \cdot \mathbf {n} _{0}=\mathbf {d} \cdot \mathbf {n} _{0}\Longleftrightarrow xa_{0}+yb_{0}+zc_{0}=d}$

The same here:

${\displaystyle \mathbf {d} \cdot \mathbf {n} _{0}=\mathbf {d} \cdot {\frac {1}{d}}\mathbf {d} ={\frac {1}{d}}\mathbf {d} ^{2}={\frac {d^{2}}{d}}=d={\sqrt {a^{2}+b^{2}+c^{2}}}.}$
Done. Boris Tsirelson 07:03, 30 July 2010 (UTC)
• "From this follows the Hesse normal form": It is not clearly expressed that the Hesse normal form is characterized by the use of n0 among infinitely many equivalent equations.
Done. Boris Tsirelson 08:09, 30 July 2010 (UTC)

--Peter Schmitt 23:03, 29 July 2010 (UTC)

In Line (geometry) it was said in advance that only the first definition works in space. Similarly, it should probably be said that all but the first definition assumes "lines" to be defined. --Peter Schmitt 23:21, 29 July 2010 (UTC)

I am in trouble with Cartesian coordinates: do they assume lines, or not? And I believe, a title like "Definition via lines" says boldly enough that lines are assumed; for "Definition via right angles" the reader will automatically imagine angles, thus, lines (the latter is not quite necessary, but I do not want to dwell on this in the article). And again: this is the non-axiomatic approach; each definition uses other notions freely. And lines naturally precede planes in school math, don't they? Boris Tsirelson 07:13, 30 July 2010 (UTC)
You have some arguments, so I'll probably not insist. But for coordinates you need lines (unless you use modern approach, of course): How else would you introduce them? --Peter Schmitt 12:40, 30 July 2010 (UTC)
"unless you use modern approach" — this is just the problem. A non-mathematician can say that without lines I cannot introduce distances (since I do not know along which path to measure). And I have no reason to enter a long discussion of such points in this article. For the same reason I did the "orthogonality in disguise" in "Line": for not arguing whether "line" is a prerequisite for "angle" or not. I prefer to avoid any explicit statement of this kind. Otherwise, how should we choose between being correct, being understood, and being not off-topic? Boris Tsirelson 13:44, 30 July 2010 (UTC)
I am not sure whether some remark would help or not. Probably only non-mathematicians can really judge this ... perhaps (some day) a comment will show it. --Peter Schmitt 14:37, 30 July 2010 (UTC)

## Introduction

As for your questions regarding the lead:

Perhaps "flat surface — without projections or depressions — that"?

Done. Boris Tsirelson 07:21, 30 July 2010 (UTC)

I agree with you that talking about Hilbert in the introduction distracts from the purpose of this article to intuitively and heuristically describe the idea of the plane. In particular, citing isolated axioms (with numbers) does not fit.

I move that paragraph to Talk:Euclidean geometry; some day it can be used there. Boris Tsirelson 07:27, 30 July 2010 (UTC)

"doubts arose about the exactness and the limitations of the Euclidean definition of a plane." Rather "the limitations were recognized", maybe "doubts that they model physical reality"?

Non-Euclidean geometry is only tangentially related. Euclidean axiomatics should be accurate irrespective of possible non-Euclid geometries (and their possible physical relevance). I move all that to Talk:Euclidean geometry, too. Boris Tsirelson 07:36, 30 July 2010 (UTC)

Perhaps the 3-plate method could be used in the lead, too?

I am afraid, it would be a dangerous precedent, to mix math and non-math. For now, our articles are mathematical, but we tolerate a bit of non-math matter in a kind of appendix (at the end), boldly marked as non-mathematical. True, non-math appears in the intro as "the common notion of a flat surface"; but it is reduced to the necessary minimum. To say more in the intro would invite non-math matter to feel a first-class citizen across the article. Boris Tsirelson 07:42, 30 July 2010 (UTC)

--Peter Schmitt 23:46, 29 July 2010 (UTC)

(non-mathematics) I see what you mean but I do not completely agree. It is not important, however.
(For me, this article tries to connect the intuitive idea of a plane with its mathematical concept, and the 3-plane method formalizes the idea of a surface that can be moved "in itself" (but is not a sphere). Moreover, if I remember it correctly, Paul Lorenzen (Protogeometry) uses it to motivate the axioms.) --Peter Schmitt 12:34, 30 July 2010 (UTC)
I feel, you'd like to considerably extend the discussion of the 3-plate method. The problem is that I am not prepared to do so. In particular, I have no idea of Paul Lorenzen (Protogeometry). This should be quite a project, involving invariance under subgroups of the motion group. Maybe, in a separate article? You often write that an article should not contain too much. Boris Tsirelson 13:36, 30 July 2010 (UTC)
Really, why not? Write the article "Three-plate method" and let me approve it! Boris Tsirelson 13:55, 30 July 2010 (UTC)
I already said that I do not consider this as "important". I only wanted to tell you that I have a slightly different view.
And yes: The article should not become too long. I never meant a detailed discussion of it (1-2 sentences at most). But it is alright as it is. --Peter Schmitt 14:42, 30 July 2010 (UTC)

## Leftovers

I assume that you did not leave the remaining "\equiv" on purpose?

Sorry. Not. Hope now all are exterminated. Boris Tsirelson 13:20, 30 July 2010 (UTC)

Did you see the comment in #plane geometry?

I am not sure I understand your point there, but look now. Boris Tsirelson 13:26, 30 July 2010 (UTC)

--Peter Schmitt 12:44, 30 July 2010 (UTC)

Any more leftovers? Boris Tsirelson 13:47, 30 July 2010 (UTC)

I hope not. I just nominated the article for approval. We shall have enough time (until 3 Aug) to react if something is discovered. --Peter Schmitt 14:32, 30 July 2010 (UTC)

## Without edges

Boris, isn't without "a boundary" a better way to express this? --Peter Schmitt 23:00, 31 July 2010 (UTC)

Maybe. I was in doubt: edges, boundary, border or rim. I know that the appropriate mathematical term is "boundary"; but this phrase is, by intention, non-mathematical. Boris Tsirelson 05:28, 1 August 2010 (UTC)
I did not doubt that you know "boundary". But I think that its "plain language" meaning fits quite well here. On the other hand "edge" might suggest another image -- like the edge of a cube in its surface instead of the limit (end) of it. "Unbounded" could also be used. --Peter Schmitt 08:10, 1 August 2010 (UTC)
Yes, I see: "edge" was worse. No, "unbounded" is also worse, since in math it is rather "not within a ball". Boris Tsirelson 09:01, 1 August 2010 (UTC)

## Constables

Two constables are quite active these days (each one did many edits on each of the days 3, 4, and 5 Aug.) Nevertheless this article is still not frozen. Also, no objections are made. Because of the weather? or another problem? Boris Tsirelson 06:27, 5 August 2010 (UTC)

The Constables were frozen doing other stuff and overlooked it! Hayford Peirce 17:55, 5 August 2010 (UTC)

## APPROVED Version 1.0

Congratulations! Hayford Peirce 17:59, 5 August 2010 (UTC)