I can see that the intersection of two planes must be a straight line, but is it possible to prove it from: "If a straight line passes through to points of a plane, then all points of this line lie in this plane"?

It is not possible to prove this. Indeed, in space of dimension 4, two planes can intersect (and usually do intersect) at only one point. Thus, what you want to prove in effect requires space to have dimension 3, but the property of lines that you use as the hypothesis does not impose this requirement.

What is the connection of your question with Exercise 8?

Alexander

I was just thinking that the explanation why the crease in the folded paper is a straight line (assuming it is folded "flat") is that it is the intersection of two planes, and I can see that it must be a straight line, but I'm not sure if this really would be a solution for Ex #8 if I can't prove it? Otherwise I'm not sure how to solve this problem (Ex #8).

P.S. I have many more questions about Introduction. I might be really a little slow, rusty, or just anal, but it may be also a need to really gain confidence and depth, even with some limited aptitude. I will try to post them during the weekend. Thanks.

Jay

Oh, I see! You are saying that in the *process* of folding, the crease is the intersection of two (or several) positions of the plane and must be a straight line. Well, it may be a valid explanation (as the remark to this problem says, there can be several correct explanations). In any case, this is a "physics" problem (by the way, just as the very introduction of basic geometry notions is akin to empirical science) and so it is not *that* important if the property at hands can be logically derived - the main question is if it is true.

It seems, when we really fold a piece of paper, we first bend it in space (so, no part of the paper remains flat in the process) and then flatten it on a desk creating the crease at the last moment. My explanation why the crease is straight would go like this. Suppose it is not. Pick any two points on the crease and, keeping the paper folded, cut it along the straight line passing through these two points. Then unfold the paper. You will see a hole with two straight edges connecting the same pair of points on the crease. This contradicts the uniqueness of a straight line between two points. Thus the crease must have been straight.

I'll be looking forward to your other questions.

Alexander

**Dear Chris, I am inserting into your post my comment in the bold face font. In fact I don't accept your answer as a correct one, and the main reason is that I don't understand anything in your explanations.
Please keep in mind that the exercises we are discussing are from a textbook intended for 7th-graders (unlike PCC-students, they don't study philosophy yet), and that these exercises are merely illustrations to those very few facts learned about straight lines in Introduction. Essentially all we know is that through arbitrary two points, there is a straight line, that such a line is unique, and that it lies in every plane containing these two points. Using this, one is invited to explain why the crease of a piece of folded paper is straight, and the explanation better be short and clear to a 7th-grader. Yours, Alexander.**

**P.S. Also, if it is possible, could you please avoid posting your logo under each post? It eats up a lot of space and contains information (like phone number, etc.) and math-looking graphics which have nothing to do with the questions we discuss, and may be therefore distracting to other readers. Thanks!**

Thank you Dr. Givental for you explanation. I actually came to a similar conclusion.

Before I was able to solve this problem, I was hitting my head against the wall for about six months until a recent translation of Hadamard's text (which you recommended in your list of further reading) was published which I purchased a copy of. Within it's introduction was the subtily simple definition: "Finally, the common part of two contiguous portions of a surface is called a *line*. This definition is equivalent to the following: *a* line *is the intersection of two surfaces.*"

It was at this exact point that the light came on. Well, after a few other conversations with people I believe I have a clear justification for why the paper produces "theoretically" a straight line.

When we fold a piece of paper in half it represents the theoretical half of two intersecting surfaces (not planes as paper isn't really flat), which in turn produce a curve. If we idealize these as we are prone to do in mathematics, we can see that this idealized form creates two continuing intersections which as we see on our paper, a line—the fold.

I further would argue that the intersection by definition of two surfaces produces a curve by definition. Since we are idealizing a "straight line" to mean "something that maintains the quality of 'straightness'", we can thus idealize this mathematical geometric model (perhaps not that different from Plato's forms) to our idealized plane. By definition, the intersection of two surfaces produces a curve/line. If we stipulate the surfaces are idealized and flat, then we can deductively derive our proof for the folded paper by arguing that by the sheer nature of mathematics deductive form, and that we define our initial premises to be true, that it is the case that our outcome must be a straight line. For if the intersection of the two surfaces (in this case the specialized case of planes) were not to produce a "straight line", then we would have to alter the nature of mathematics, thus proving our conclusion (the line produced by folding a piece of paper in half produced a straight line) true, via *reducio ad absurdum*.

I would also argue that by reversing the ruler, we can easily see if the line is curved, because the ruler is constant in space and by this very fact, it should curve in an equal proportion opposite to its initial curve, thus indicating if it produces a straight line or not. We may not necessarily, however, be able to indicate if such a curve exists because of the limitations of our eyes. However, in principle, I believe this answer to work. I wasn't sure if that was the intended answer you were getting at with your hint to the stared problem #7, but I think it is a suitable answer, nonetheless. As I was taught, a proof is merely a completely explained answer, whereby one explains all the logical steps one takes along the course of an argument to justify reaching the final conclusion by accepted means.

I also wanted to thank you for providing this site. I was having considerable difficulty with the text at start, due to my limited experience with formal proof and only a meager elementary education in mathematics. I am presently entering a two year study of elementary calculus, Vector Calculus, Linear Algebra and Differential Equations here in Portland at Portland Community College before I enter Reed College in 2012. I was hoping to get through formal geometry and then use that as the means to study Spivak, Courant and Apostole's Calculus Texts. I am presently studying formal logic on my own.

Thank you for your efforts in bringing *real* mathematics to students like myself. It has helped inspire me over the past 3 years to devote myself to teaching and providing others with a *real* 'education'.

-Chris

So, in a word, I'm doing my *typical* and "over-thinking" a problem designed for 7th graders & being too long winded in my description, therein? Perhaps, I'm assuming that I'm suppose to extrapolate more than what I'm given.

Does this explanation seem more reasonable?

1. A Plane is an infinitely flat surface. **"Flat" means "like a plane", hence the logic of this phrase is circular. Anyway, you don't use this in your argument, so there is no need to mention it.**

2. A Piece of paper resembles a (section **part** of a) plane.

3. When I fold a piece of paper in half it resembles the intersection of two planes (the folded sheet of paper is half of the 3-dimensional "x" that would be formed at the intersection of these two planes). The intersection of two planes forms a straight line along their intersection.

**The folded piece of paper lies in (one!) plane (that of the desk), and so it does not resemble the intersection of two planes.**

**I think your explanation (quite reasonable, although different from mine) is: Unfold the folded paper half-way; then the crease becomes the intersection of two (parts of) planes; thus it is a straight line, because two distinct planes intersect in a straight line. To justify the last statement, one can argue like this: If two planes have two points in common, then the line through these two points lies in each plane; moreover, no point outside of this line could lie in both planes, since in this case the planes would coincide. (The axiom of 3D geometry,
that any 3 non-colinear points lie in a unique plane, is used here.)**

Visualization: Take two planes. They will always be infinitely flat in all directions. Planes have no "thickness". If a line intersects more than one point of a plane that line resides in that plane. A plane can be thought to resemble a straight line that is dragged along another straight line across infinity. Therefore, if a plane intersects any other plane, it can only contain a row of collinear points (points residing in a straight line). Those points form a line, and that line forms the intersection of the two planes. For, if there were more than that single line, then the planes **must** lie in the same plane, and no crease would exist in the paper, i.e. no fold—else we would form a contradiction where a straight line could violate the principle "a line can intersect at most one point on a plane, else the line resides in the plane" assuming the line is straight.

Check:

4. Flipping the ruler over will reflect the image of the line, in the event it is curved. If the ruler is curved, by flipping it, we produce an inverted curve, similar to taking a letter and looking at it in a mirror. So, if it is curved, we will see the line curve in exactly the opposite direction upon the paper when we go to draw the line, after we have flipped the ruler over.

**OK, and … ? If you can see that the flipped curve is curved, then you can see that the initial one is curved, and so you don't need to flip it to find that it isn't straight. The question makes sense only because a straightedge can appear straight even when it isn't.**

Note: I believe the only piece of information I have added here is from Hadamard, "The intersection of two surfaces is a curve" [Me:"a plane is a special case of surface, which by the above reasoning, produces a straight line, which is a special case of curve"], but after re-reading this section, the information could be extrapolated directly from the text.

Also, you wrote "the exercises we are discussing are from a textbook intended for 7th-graders" and to me, being a seventh grader (and a 12th grader too) at one time, I never remember any problems even coming close to as abstract and rigorous as these. As a college student, in Calculus, I have had no problem even coming close to this kind of abstract, theoretical modeling.

So, perhaps the fault lies with I, but I can't see how you could answer this problem without first comprehending the relationship between the intersections of planes (or surfaces) and the resultant curves. I for one could not see that. Maybe I'm just that dense, but perhaps I'm just looking at the problem from a higher level's perspective or just misinterpreting what the question is asking. I really couldn't say.

**That's OK, but if you want to learn geometry, you need to move past Introduction (which, after all, merely contains a few definitions), focus on non-trivial geometric theorems, and solve more meaningful exercises (there are some 600 of them in the book). Good luck!**

[I have no idea what I was thinking when I wrote my last two posts. They are excessively vague. I do not think philosophy or Plato had anything to do with the matter, I merely was vague and poorly constructed the argument I saw in my mind. Why? I cannot say. I hope this answer appears more reasonable. If you find fault in it, please let me know. Thank you. -Chris]

Alexander,

Thank you for the corrections to my explanation. Your analysis seems more appropriate than mine pertaining to what I had initially conceptualized as I was leaving out the caveat that in my mind I was *assuming* to "fold a piece of paper and assess the line produced" was to look at the line created when you unfold it in order to answer the question (to rigorously prove), "Why is that line straight? etc." (i.e. the line produced by folding a piece of paper in half) and follow that specific line of reasoning. I never thought that the problem was or could be asked from a **non-unfolded** point of view. Thank you for clarifying that point.

I suppose it bares mentioning that it is only through making these kinds of assessments to non-trivial problems and having those assessments further assessed by seasoned members of that particular discipline that a person actually grows and improves in his/her ability to be an active participant in the discipline in question. Thank you for giving me this opportunity to improve through the active participation on this wikidot site.

I also wish to thank you for noticing my mistakes in reasoning. Describing a **plane** in terms of "flatness" is like describing a **point** in terms of a **line**. Some terms must be left undefined and simply *described* because there is no way to provide an adequate definition without resorting to begging the question, which is just another way of saying "circular reasoning". Out of moderate curiosity, how (for the sake of the reader's understanding) would you *describe* the plane for the purposes of this proof/explanation? I included the poorly described, circular definition of **plane**, because I was trying to write to an audience whom may not otherwise know what a "plane" is and to completely clarify that distinction and indicate further a difference between the special case of a **plane** and **line**, and the more general case of a **surface** and **curve** (and their resulting intersections). I was told by my mathematics instructor that it is useful when writing these not to assume the person inherently understands the consequences of some of the terms and to indicate these differences. Would there be a better way I could have gone about indicating this without resorting to "defining" clearly "undefined" terms? Also, do you have any general suggestions when writing these kinds of proofs or *explanations* how a person can tell what information is necessary to indicate, and what is merely redundant information? Would it be useful to indicate information that is related to the proof, or will I be critiqued poorly for doing so? I'm trying to write this so I (as a student) could understand it even if I am only moderately familiar with the material (such as if I were taking a course in the subject for the first time).

I agree that there are well over 600 problems in the book and that focusing on the non-trivial ones is the most important (my current professor is always encouraging me to do the same with our Calculus materials, "focus on the non-trivial problems, because they will be the most rewarding to study, which is why I generally **only** assign them").

In the future, I'll make note to use more proof like descriptions to prevent ambiguity.

Again, thank you for the time you took to edit my work. I really appreciate it.

Christopher-Titus

It may seem like splitting hair, but I can’t see clearly enough what the notion of “superimposing” mean in geometry context. Let me try to explain.

Lets call the figure we are superimposing FS and the figure that the figure FS is superimposed on FR (in other words we are superimposing FS on FR). Then:

1. Superimposing means moving FS in such a way that all its elements will become identified with at least some elements of FR.

a. Example: a smaller square can be superimposed on a larger square. Is this true?

b. It follows that any figure can be superimposed on 3-dimensional space. Is this true?

2. Superimposing is not moving FS in such a way that:

a. Only some parts of FS will become identified with FR (like “laying flat” only a corned of one square on another square). Is this true?

b. None of the parts of FS will become identified with FR. Is this true?

3. Superimposing is impossible when FR cannot contain FS. Example: bigger square cannot be superimposed on a smaller square. Or a cube that will be “sticking out” of the sphere, no matter how we move the cube. Is this true?

If I’m taking a wrong path, and “superimposing” is simple an intuitive term, then I have a difficulty in understanding clearly the description of the property of the plane given on page 2.

Thanks

Jay

Hello Jay,

Here **superimposing** = **identifying with** = **laying over**.

So, *superimposed* is not a term at all, but is meant to be a regular word. Most readers would have no problems with *identified with*, but I was told that for younger students, a different meaning, such as in *identify yourself*, gets in the way. This is the only reason why *superimposed* is used (in addition to *identified with*).

Let's read the definition of congruent figures (page 1) with *superimposed* replaced, and apply it to your examples.

*A set of points, lines, surfaces, or solids positioned in a certain way in space is generally called a geometric figure. Geometric figures can move through space without change. Two geometric figures are called congruent, if by moving one of the figures it is possible to superimpose it onto lay it over the other so that the two figures become identified with each other in all their parts.*

I suspect that what causes your difficulty is the notion of "moving through space without change". It is indeed the most fundamental starting point of classical (since Euclid) elementary geometry. It is a *primary notion*, i.e. it cannot be accurately defined using previously defined terms - for the lack thereof - and thus it must be conveyed intuitively. The phrase * geometric figures can move through space without change* appeals to our experience with solid, rigid objects around us. We can move a desk from one place to another, and it will remain the same desk (except for its location). The more advanced term for "moving without change" is

**rigid motion**. Of course, thinking more thoroughly, we can easily confuse ourselves: rigid objects around us are not

*that*rigid - they stretch, vibrate, melt, decay, etc. Thus, "motion without change" is an

*idealization*where all these effects are considered negligibly small.

Let's look at some of your examples:

*1. Superimposing means moving FS in such a way that all its elements will become identified with at least some elements of FR.*

For congruence, "some elements" is not enough: "all parts" is required.

*a. Example: a smaller square can be superimposed on identified with a larger square. Is this true?*

This would require a "motion with change" (in scale). So, a smaller square is not congruent to a larger one.

*b. It follows that any figure can be superimposed on identified with 3-dimensional space. Is this true?*

Of course not: As a geometric figure, the whole space consists of only one part - the whole space. Moving it "without change" results in the same figure: the whole space. Thus the only figure congruent to the whole space is the whole space!

Now, about the property of the plane described on page 2:

**One can superimpose identify the plane on with itself or any other plane in a way that takes one given point into any other given point, and this also can be done after flipping the plane upside down.**

Of course, the plane (just as any other figure) is already identified with itself. The claim is that (unlike most other figures) it can be identified with itself in many different ways. For instance, mark any two points on the plane. Then the plane can be re-identified with itself in such a way that the first of the marked points becomes identified with the second one (and this re-identification can be made by moving the plane "without change" in space).

Euclid would express this idea by saying that **the plane lies evenly with all points on itself**.

At this point (:-), you should do Exercise 5, according to which the plane is not the only surface with this property. You should be able to find two more types of such surfaces. However, I bet, none of them will, like the plane, retain this property "after flipping it upside down".

I realize that all this might have been known to you, and your only question was about the meaning of "superimposing" per se. In this case, let me reiterate, that in itself, *superimposing* is not a geometric term. In the text, the default usage is in connection with the notion of congruent figures, i.e. in the sense of "identifying with … by a rigid motion." But any other usage is also possible, if it is clear from the context what exactly is meant.

Please don't hesitate to ask further.

Alexander

Thank you so much. I'd like to come back to this, most likely, and to Ex #5, but first I want to go to Ex #1. I will post a new question as soon as I could (sorry, I'm really working quite a lot now).

Jay

Alexander,

I find only the sphere's surface to maintain this plane's property, which is the other one?

By the way, Arithmetics for parents was a very pleasant read, I finished it in India, while visiting some schools over there. Now I am going over the planimetry, just started. When are you bringing us more books? I know they take a lot of time. Thanks for all you do.

You are welcome - and thank you for your comment!

Actually, "Planimetry" will go for a 2nd printing soon, but no new books are prepared - Sorry!

I am working on a linear algebra textbook and hope to finish it next year, but this is rather a college-level subject.

Alexander

Maybe the surface surrounding a cylinder that extends indefinitely?

I am working from the 1st edition of the book. Below are my answers and/or attempts/comments/questions for the exercises in the Introduction chapter. I seek to confirm the correctness of my conclusions, get help with the unsolved, and hopefully establish correct examples that others like myself can refer to when working from this book. I find it difficult to determine my correctness by referring to other sources since identical questions to the exercise aren't easily found and the pedagogy/chronology of the material differs.

## Ex. 1

*Give example of geometric solids bounded by one, two, three, four planes(or parts of planes):*

**One plane** - I cannot think of a solid object that occupies space yet only has one plane(that without thickness). If I consider geometric solids with one *surface* then a sphere comes to mind.

**Two planes** - Again as with one plane, I do not see a solid object that is constructed from two planes. If I consider geometric solids with two *surfaces* then a cone feels correct.

**Three planes** - Continuing the theme of the previous two examples, I do not see a solid object that is constructed from three planes. A three sided pyramid with no bottom may work, but any physical example of such an object has sides that are more than planes. If I consider geometric solids with three *surfaces* then a cylinder works.

**Four planes** - A pyramid.

## Ex. 2

*Transitive equality/congruence of geometric figures:*

If figure A is congruent to figure B, figure B is identical to figure A in all properties. If figure B is then congruent to figure C, figure C is identical in all properties to B. B already shown interchangeable with A, makes C congruent to A. A = B, B = C, C = A . This might also be explained by the underlying components of a geometric figure being founded on points and lines(segments). The congruence based on the congruence of the underlying segments in the figure and the identical position of their points with respect to each other.

*Edited: Truncated to Ex. 1 & 2, moving other exercises to individual posts.*

Dear Blakester,

Thanks a lot for your solutions. Some of them are correct, some others are not and deserve discussion - so, be prepared for a long one. It's better to deal with the exercises one-at-a-time. I am not very happy that all your questions are in the same post, so I'll try to make separate threads for different exercises. Let's begin with the first one here.

Your answer "pyramid" (more precisely, triangular pyramid also called *tetrahedron*) in the case of 4 planes is correct. Note however that the tetrahedron does not "consist of" 4 planes, but instead it is a solid *bounded* by 4 triangles, which are pieces of planes. (Actually the word "tetra-hedron" means "four-faced solid.") This is what the exercise asks for: to give examples of solids (not formed by, or consisting of, but) *bounded* by 1,2,3,4 planes.

Now let's go to the case of 1 plane. There is a geometric figure *formed* by 1 plane in space - it is the plane itself. Is that plane the boundary of any solid? Another hint: What is the solid figure bounded by 0 planes?

Alexander

First, a terminological remark. Figures that we call now *congruent* used to be called *equal*, and we still write A=B to mean that figures A and B are congruent.

Next, you write: "If figure A is congruent to figure B, figure B is identical to figure A in all properties." This is incorrect. Indeed, I have two "identical" chairs in my room. They are congruent, but not identical in **all** their properties, since I am sitting on one of them.

The moral is that in answering mathematical questions one should not rely merely on intuitive understanding of the terms (such as *congruence*) but should instead refer to their actual **definitions**. The notion of congruent figures is defined in $\S 1$.

What does it say?

### Ex. 1 - Another attempt, more fog

Alexander, taking your explanations in and more consideration to the writings in the book leads me to this(pardon that I write with much uncertainty in my statements):

**One plane** - It would seem that any geometric solid(cube, tetrahedron, cone, hemisphere, triangular prism, all the solids formed from N-sided(N>4) shapes) with at least one flat surface is bounded by a plane or part thereof.

Probably off on a tangent of bad reasoning… I am uncertain if the undefined geometric abstractions like the line or point can be considered to consume physical space and qualify as geometric solids(probably not, and it is stated that they aren't to be considered as existing on their own anyways). Their properties seem to be born out of the geometric solid's existence(clearly you did not start the book off with an explanation of the 'point'). I know the way I would represent my idea of them in my physical universe would consume space, but those physical representations I have of them are 3-dimensional and much more than the abstractions. If they do qualify, then in some way they are surrounded(bound? What is the boundary if you are moving out of the point or line itself into the plane?) by the plane in which they exist and they theoretically have a plane above/below and many intersecting the plane all around them. In section 1 of the book there isn't an explicit/italicized statement of the property(ies) of the geometric solid like exists for the plane, so these ideas are based on the sentence, "A geometric solid is separated from the surrounding space by a surface."

If I consider all statements in section one of the book up to the 'geometric figure' definition, notably: "The part of space occupied by a physical object is called a **geometric** solid." and "The geometric solid, surface, line, and point do not exist separately." I get the thought that geometric solid == must(it has to have surfaces, lines, and points?) be a 3-dimensional object, but the first statement clearly states that we are talking about *space* and not the physical object. This brings another question, does the geometric figure consisting of part of a single plane consume space? It is a specific area/space of this plane bounded by lines and theoretically has a plane above/below, but in my physical world I think all objects have thickness which the plane has none, though I may be mistaken.(This makes me think of gases, but I think it can be reasoned that they have thickness/volume even if I can't touch it with my hand in their gaseous form.)

**Two planes** - Following the rationale in my one plane examples above, can we consider every geometric solid with at least two flat surfaces(cube, tetrahedron, triangular prism, the N-sided solids) as being bounded by two planes? If limited to geometric solids bounded by only two planes and nothing else, I don't have any in mind. If a combination of two parts of planes(disks) and a curved surface are allowed, then a cylinder feels right.

**Three planes** - Continuing the theme of my 1,2 plane examples, I do not see a geometric solid that is solely bounded by 3 planes or parts thereof. There is this pyramid like shape, if you consider it to not have a bottom, but I don't think it's possible to make an exclusion like that. Using a combination of parts of three planes and a curve I arrive at a U(imagine there's a line between the top tines)-shaped tube. If this were seen upside down it would be like some train/auto tunnels, not the half-circle/arch variant, think more right angles.

**Four planes** - With respect to my misuse of the term "formed". The example you give of the use of "formed" applies to geometric figures. Does that imply that it can be said that any geometric figure is *formed* by its underlying geometric solids, surfaces, lines, and/or points?

Switching to the case of the tetrahedron geometric solid(which I believe could be called a figure in a context that refers to a specific instance in space). This geometric solid consists wholly of the space between the intersection of the four planes, which makes me think that it exists because the four planes exist. Yet this should not be considered formation, just being bounded, correct? I believe my thought here might connect to the questions related to the existence of the geometric object in the 'One plane' section above. Is it that the plane doesn't exist at all prior to the geometric object, and the plane is really an idealization that extends/projects a plane surface on the solid indefinitely into space? Is the notion to be understood: the geometric object exists without being formed or caused to exist by any of these other elements thus far known, it is just there without further explanation?

### Ex. 2 - answer and more questions/clarification

The congruence of geometric figures is defined as: "Geometric figures can move through space without change. Two geometric figures are called **congruent**, if by moving one of the figures it is possible to superimpose it onto the other so that the two figures become identified with each other in all their parts."

As set forth the term "geometric properties" is not used at all in the definition of congruence and where I'm headed with this question I can see how involving it could make the definition much more complex(assuming this thought and what I'm about to ask is at all valid). Clearly I don't have a handle on all that constitutes the *geometric properties*, assuming it is possible to enumerate them or know them all. Can this **congruence** be expressed in terms of *geometric properties*, possibly by excluding the properties that set a geometric figure's position in space? I.e. geometric figures with identical geometric properties, excepting their space positional properties being a definition of congruence.

I do understand what you are saying with respect to the properties(my all encompassing "all"). You have two chairs in different locations in space, one has a property of being actively used, the others use property set to false. The 'use' property being in a set of properties other than "geometric properties", but in the "all possible properties" set.

Dear Blakester,

If you want me to help you, you should pay closer attention to the questions I am asking and try to answer them very neatly and concisely (remembering that mathematics strives for precision and economy of thought).

Let's start with Ex. 2. It's good you understand my comment about your mention of "geometric properties". However there is no mention of "geometric properties" in this exercise - so, let's stick to what is there. Using the definition of congruent figures given in the text, can you show that if figure A is congruent to figure B and figure B to C then figure A is congruent to C? If "yes", then please write **one** sentence that shows this. If "not", then think how to express your difficulty in one sentence.

Now, about Ex. 1. Again, you are making a very elementary and straightforward question unnecessarily complicated.

Of course, the first paragraph in the book - about physical objects, their surfaces, etc. - iis not a definition of solids, surfaces, etc.

Definitions are conventions describing precise meaning of technical words in terms of previously defined words. At the foundation of such tower of definitions we run out of "previously defined" terms. So, such **primary notions** as *solids, surfaces, curves, planes, lines, points* cannot be formally "defined." Their meaning is conveyed in this and several further paragraphs of the book in an intuitive fashion.

That paragraph merely describes in what way the common-sense notions of "solids", their "surfaces", etc. become **idealized** when used in a mathematical context. E.g. mathematical lines are infinitesimally thin and infinitely long (i.e. can be extended in either direction as far as one wishes, which is an idealization of our images of stretched threads or rays of light), although no physical object can be infinitesimally thin or infinitely long.

The exercise asks you to do a very basic thing: given a plane, imagine and describe a geometric solid (a mathematical, idealized one) whose boundary is this very plane (and nothing else!) - Two words, please.

To help you, I asked you also another question that you haven't answered yet: Given a "geometric figure" consisting of 0 planes, describe the geometric solid whose boundary is this (empty!) figure. - Three words, please.

Alexander

Warning: I will violate your one sentence and 5 word constraint below. Not done in spite, I only seek to understand.

**Ex. 1**

*Re: geometric solid with empty geometric figure as its boundary:* All of space? That doesn't necessarily make it clear to me. The first two sentences of the Intro stating properties/notions of geometric solids don't make me confident that space itself can be considered a solid and that a solid can exist that has no surface(or is it that space is the surface of itself?).

*Re: geometric solid whose boundary is a single plane:* In your question you set forth "(and nothing else!)". In the question for the exercise, I feel like this constraint, "exactly" is not established and that was the rationale behind my second attempt at the exercise. Is the wording intended to imply this and I don't see that?

I don't have a two word answer, maybe three words? half of space. Changing my mindset on this exercise and looking beyond the solids that are common to my human experience, I have formulated these answers:

One Plane - Let there be a figure consisting of a single plane in space. This single plane bounds two solids that exist on either side of the plane, and each of these solids has exactly one surface which is this plane.

Two Planes - Let there be a figure consisting of two planes that intersect much like an "X" or "+". This divides all of space into four quadrants and each of these quadrants is occupied by a solid that is bounded by a part of each of the two planes. Further these quadrant solids are only bounded by these two surfaces.

Three Planes - Let there be a figure consisting of three planes in space, and these planes are oriented in such a way that they all intersect one another and the three lines? that run through the points of plane intersection are straight and never intersect each other. This describes a triangular tube through space, which is a solid bounded by one part of each of these three planes. Further this solid is bounded only by these three surfaces.

Four Planes - The tetrahedron was covered previously.

**Ex. 2**

*Re: answer the question in one sentence or explain myself in one sentence:* I will make an attempt at answering using the definition of geometric figure congruence, but it won't be one sentence:

Pursuant to the definition of congruent geometric figures, figure A can be superimposed onto figure B(the other figure), such that the two figures will be identified with each other in all parts. If figure B is also congruent to figure C, figure B already shown able to coincide with figure A, then it can be reasoned that figure A can also be superimposed onto figure C and they will be identified in all parts. Therefore figure A and figure C are congruent.

I feel that this leans on a common notion of a transitive property. It does not feel rigorous, but I'm not seeing any other tool yet available to make this connection that A clearly would be congruent to C because of this chain of congruency.

Dear Blakester,

Your answers to Ex. 1 are correct: The whole space, half-space, "quarter-space" (the official term is *dihedral angle*, see Book II) are solids bounded by 0,1,2 planes (or parts of planes) respectively. Your example for 3 planes is also correct, but is based on a very special position of the planes. If you start with 3 randomly chosen planes (all 3 will meet at one point), what solids will you get?

(*Hint:* The name is *trihedral angle*.)

In Ex.2, your use of more than one sentence probably indicates that you are trying to give an answer before you actually found one.

The problem indeed is about "transitive" property of congruence: if A is congruent to B and B to C, show that A is congruent to C.

Your answer uses the phrase "it can be argued". "Can be" is not good enough; **what is** that argument that shows that A is congruent to C, i.e. can be superimposed onto C by moving the figure through space without change? (I am sure you see the answer, you just need to put it in words in one sentence.)

Alexander

Re:Ex. 1 - 3 randomly chosen planes: I see something that would seem similar to a tetrahedron, but without the fourth plane bounding/limiting the space on the interior of the solid. Cone-like without the curved surface.

Re:Ex. 2 - Single sentence solution: The exercise states, "Show that if a …". I have approached this as if 'show' means prove the general case, the rigorous kind. I think your suggestions imply a solution that is limited to the application of the definition of congruent figures which would: impose A and C, some sort of implication that they would coincide in all parts, and therefore they are congruent. If this is the case, I am not realizing how this alone builds proof without connecting the transitivity dots some how. I think this transitivity is a common notion, but that hasn't been brought up in the text prior to this exercise.

**Ex. 1:** You are right, this is the figure called trihedral angle. The same was as we say that a triangle has 3 angles, we can say that a tetrahedron has 4 trihedral angles (one at each of its 4 vertices), as well as 6 dihedral angles (one at each of its 6 edges).

(Sorry for being carried away: This is a subject of Book II :-)

**Ex. 2:** The property of *transitivity* holds true for some relations, but can be false for some others. For instance, it is false for *non-congruence*: if A is **non**-congruent to B, and B is **non**-congruent to C, it is still possible that A is congruent to C. (By the way, could you give an example where it is?) As for *congruence*, the property is almost obviously true, and one should be able to prove this using the definition.

By the way, you seem to be misreading the definition of congruence: It is not that "one can be superimposed onto the other" that counts (many non-congruent figures can also be superimposed), but "superimposed by doing what?" - that's what is most important!

Alexander

Re: **Ex. 2**:

- When you say, "By the way, could you give an example where it is?", I think you are asking for an example of a case where A = C given A != B and B != C. Assuming that's correct, then if A can be superimposed onto C such that they identify with each other in all parts, then this is a case where A is congruent to C.

*misreading/understanding congruence/"superimposed by doing what?"*- I am uncertain of what the correct response is. Move through space without change? Identify in all parts? I thought I understood the concepts of rigid motion/isometries/superimposing a figure onto another figure by moving it through space without change such that the figures become(or not) identified with each other in all parts.

As far as the the proof of the figure congruence transitivity. I don't think that the definition of congruence of geometric figures directly(in words) states this _obvious_ truth. I can draw from my own common notion or that it follows from the definition that when two figures are identified in all parts, they are one. Within this geometry they are then indistinguishable/interchangeable from/with one another because they have the same geometric properties. That makes it easy to see reflexivity and symmetry, and step through to transitivity. Is what I'm saying here at all valid? If it is, in my head it rests on a notion that doesn't seem to be written.

Dear Blakester,

I am sorry, but most of what you say does not make sense to me. I don't know what you mean by "common notion". We are not dealing with philosophy or theology here - our subject is mathematics. Exercises are there to test you understanding. So, imagine that you are on an exam:

you find an answer to the question you'll get an A, you don't find it you get an F.

The question was to prove that if A is congruent to B and B to C than A is congruent to C. You say it is obvious. "Obvious" means "something easy to prove." So, on math exams, if something is easy to prove, never say "it is obvious" - or you'll get an F - just prove it!

In your answer, I pointed out a very specific flaw: you said "one can prove (or argue) that … ", instead of proving it. So, to get an A, you were supposed to fix the flaw, not to go away from it.

I asked you to give an example of 3 figures, A,B,C such that A is non-congruent to B, B is non-congruent to C, but A is congruent to C.

You rephrased my question, but didn't give any example of any figures; so far it's an F.

If you know you haven't solved a problem yet, then there is no much point to submit an answer - you'll get an F.

Continue thinking and go on to other questions. When you solve the problem, then you submit your solution for evaluation.

So, I propose that we move on in this more exam-like environment - I think it will be more efficient.

Regards,

Alexander

*Sumizdat*> In your answer, I pointed out a very specific flaw: you said "one can prove (or argue) that … ", instead of proving it. So, to get an A, you were supposed to fix the flaw, not to go away from it.

Certainly not intentional, just trying to find my way.

*Sumizdat*> I asked you to give an example of 3 figures, A,B,C such that A is non-congruent to B, B is non-congruent to C, but A is congruent to C. You rephrased my question, but didn't give any example of any figures; so far it's an F.

It was not clear to me that "example" meant a non-abstract instance of such. I thought that you wanted me to state a logical?/general case(It was obvious, but hey I thought it was intentional.). The specific example you wanted:

Figure A = a plane, figure B = a dihedral angle, figure C = a plane. By the definition of figure congruence, A is not congruent to B and B is not congruent to C, and by the same definition A is congruent to C.

*Sumizdat*> If you know you haven't solved a problem yet, then there is no much point to submit an answer - you'll get an F. Continue thinking and go on to other questions. When you solve the problem, then you submit your solution for evaluation.

I was submitting questions/thoughts for discussion in my last two posts, they weren't intended to be solutions. I didn't know that I was only to submit text with a high probability of being a solution. This exercise being so fundamental I can't imagine progressing without an 'A'. I may have identified(the other kind :) ) the blocking point in my thought that prevented accepting the proof below. My previous understanding of the congruence put the image(I know the book doesn't use this term on page 1) of one figure onto the other as if there were now 2-layers, not that you would differentiate the layers; It did not fully realize the meaning of 'identified', which literally defines them to be considered the same in the relation(I was going to write, "considered one in the same", but I believe this would be another misspeak since they are still distinct figures and they're only being compared.).

The solution:

Given figure A is congruent to figure B and figure B is congruent to figure C, it follows that A can be superimposed onto C such that they will identify with each other in all parts and thus A and C are congruent by definition.

This is a stretch of a single sentence. I think the portion after the comma could be limited to, 'thus A and C are congruent by definition', but I'm not confident if being that terse is permissible.

*Sumizdat*> So, I propose that we move on in this more exam-like environment - I think it will be more efficient.

I value and welcome any help that you are willing to offer. I certainly desire more than 'A' or 'F', but I will have to accept what you will give.

Dear Blakester,

By proposing an A-F routine, I am merely trying to make our discussion more constructive. Of course, in a classroom, one can speak more freely, but if a student gets reguarly carried away from the subject, the teacher needs to adjust the mode of conversation.

For your solution to the Ex. 2 (about congruence), you still get an F, since you merely repeated the statement. You say "it follows" but you didn't explain *why*.

This exercise is not really so important; to "unblock" your thought, let me just tell you the solution. You are given three (different!) figures A,B,C.

If A can be moved through space and get identified with B at the end, and B can be moved through space and get identified with C at the end, then by **composing** these two movements, i.e. first performing the first one and then the second, the figure A will be moved through space and get identified with C at the end of the process.

About non-congruence. You were supposed to **dis**prove the claim: "*For all* figures A,B,C, if A is non-congruent to B, and B is non-congruent to C, then A is non-congruent to C." By thinking that this can be done by an abstract, "general" argument, you were making a logical mistake. To **dis**prove a claim that **for all** ABC something is true, one needs to show that **there exist** ABC for which that something is false, and for this **one example** of such ABC would suffice, but no abstract argument would help.

Your specific example is correct - so, you'll get an A.

Congratulations! - and let's move on.

Alexander

### Blakester wrote:

From the axiom stating that for any two points there exists a straight line between them and this line is unique, it follows that if two lines were to share two or more points they would be the same line. Thus two straight lines can intersect at only one point.

### Alexander wrote:

Your answer sounds perfect to me. Namely, you showed how the required property of lines follows logically from the property stated in the text. Exercises 4 and 6 are supposed to be answered in a similar fashion **referring to properties of lines and planes described in** $\$ 4.$

### Blakester wrote:

*Surface other than plane that can be superimposed on itself and identify ((at)) all points:*

A curved surface like that on a sphere could be superimposed on itself.

### Alexander wote:

Your answer seems correct, but your formulation of the question is incorrect.

**Every** surface can be "superimposed on itself and identify at all points" — because it is already identified with itself!

The question was: To find a surface such that if *I* mark on it two points of my choice then *you* should be able to superimpose the surface onto itself in such a way that the first of my points is identified with the second one.

Besides the plane, spheres indeed have this property. Can you find one more kind of surfaces that have this property?

The only other surface that comes to mind is the curved surface on a cylinder. I don't know the proper term for it.

Re: The formulation of the question. Not excusing my error, I had been paraphrasing/truncating the actual questions since I didn't think it was right to post them verbatim without your permission. Would it be ok to post your actual questions? My intent is to have the data in one place without having to reference the book to see the question. Also should the book change, it's clear what question was answered.

Dear Blakster,

Your answer is correct: beside the spheres, the cylinders (or cylindrical surfaces - either term is OK) have the required property.

Of course, to discuss a question from the book, it's better to repeat it verbatum. Rephrasing is also fine, as long as it is equivalent to the original.

Regards,

Alexander

Exercise 4: *Referring to section 4, show that a plane not containing a given straight line can intersect it at most at one point.*

It is postulated that if two points of a line lie in a plane then all points of this line lie in this plane. It follows that any line not lying in a given plane can intersect the given plane at most at one point, otherwise the line would lie in the plane.

Exercise 6: *Referring to section 4, show that for any two points of a plane there is a straight line lying in this plane and passing through them, and that such a line is unique.*

From the definition of a straight line it is known that the two given points in this plane have a straight line through them and this line is unique. Further, straight lines have the property that if two of their points are in a given plane, the entire line lies in this plane. Thus it has been shown that for the two given points in this plane, there is a line in this plane through them and this line is unique.

Exercise 7: *Use a straightedge to draw a line passing through two points given on a sheet of paper. Figure out how to check that the line is really straight.*

The straightedge can only be used to draw a straight line because it itself provides a straight line on its edge. After constructing the line through the two points on the paper, the ruler should be flipped over such that the edge you are using for the straight line stays in place, but the body of the straightedge now lies on the opposite side of the line. With the straightedge in this position and still aligned with the two points, the line can be constructed again. Should the second construction not coincide in all parts with the first, it is not a straight line.

This works because the two uses of the straightedge position the same points on the straightedge on opposite sides of the same points of the line being constructed. If the straightedge is not straight its shape/defect will be evident because the two constructions won't coincide and the defects from each construction will be opposites.