# trapezoid

## Primary tabs

Defines:
trapezium, base, leg, height, median
Major Section:
Reference
Type of Math Object:
Definition
Parent:

## Mathematics Subject Classification

### We were missing this???

For those of you who were surprised that an elementary definition of radical was missing from PM, here is an even bigger surprise: Trapezoid was missing!!!

I think I noticed this earlier (back when I suddenly became active again at about June of 2006), but I did not know how to create graphics, it seemed pointless for me to make an entry on trapezoid without graphics, and it slipped my mind to file a request. After a while, the fact that a definition of trapezoid was missing slipped my mind completely.

Anyways, I have tried to list many properties about trapezoids that I know off the top of my head within the entry. I realize that I may have easily left out properties about trapezoids that someone would want in its entry, so I will leave "trapezoid" and its children open for editing for quite some time. On the other hand, I *implore* you not to change my statement about the PM convention. In order for the "trapezoidal rule" to actually use trapezoids, parallelograms (specifically, rectangles) *must* be considered to be trapezoids. Thanks a lot.

Warren

### Re: We were missing this???

Wow, that was a major thing to be missing! I'm gonna look up things like square, pentagon, cone, pyramid, cylinder, etc. and see if I can spot any other geometric shapes we might be missing.

### Re: We were missing this???

> Wow, that was a major thing to be missing! I'm gonna look up
> things like square, pentagon, cone, pyramid, cylinder, etc.
> and see if I can spot any other geometric shapes we might be
> missing.

Best of luck.

Just from my experience browsing entries, I can tell you right now that each of the things you listed is defined on PM, although they may not have their own separate entry. (For instance, I think "pyramid" is defined in the entry "cone". This actually makes sense topologically speaking.) The entry "polygon" would cover any polygon whose name is based solely off of the number of sides it has (although a lot of them, such as triangle, quadrilateral, pentagon, hexagon, and dodecagon have their own entries). Common three dimensional figures such as polyhedra, prisms, sphere, cylinder, and cone are defined somewhere on PM. I did have to add "antiprism" some time ago, but that was the only three dimension figure that I found that was missing.

I definitely do not mean to dissuade you from searching though. It is better that you look to see what is missing whenever you get a chance, just to be sure that we have all the basics. Better safe than sorry!

Warren

### Re: We were missing this: Rhomboid

> I definitely do not mean to dissuade you from searching though. It is better that you look to see what is missing whenever you get a chance, just to be sure that we have all the basics. Better safe than sorry!

I saw a post about "skew rhombus," so I looked up "rhombus" and sure enough, it has an entry. Then I thought of "rhomboid:" nothing here at PM. Mathworld has a single line about it while Wikipedia has a couple of paragraphs and an illustration.

### Re: We were missing this: Rhomboid

Are you going to add an entry on "rhomboid" then? If so, let me know if you need any help with graphics. Otherwise, I (or someone else) would be happy to add it.

### PM convention on trapezoid in relation to hyperbolic geometr...

> On the other hand, I *implore* you not to change my statement about the PM convention. In order for the "trapezoidal rule" to actually use trapezoids, parallelograms (specifically, rectangles) *must* be considered to be trapezoids.

Another reason for not changing my statement about the PM convention which is *much* dearer to my heart is that it allows Lambert quadrilaterals and Saccheri quadrilaterals (both of which are parallelograms) to be trapezoids. To the best of my knowledge, this convention is standard in hyperbolic geometry.

### Re: We were missing this: Rhomboid

With my big trip to Africa coming up next week, I don't have the brains for this. The airline screwed up my name on the e-ticket (it's not a hard name at all) and it's a major headache to fix it, plus packing, etc. Please feel free to add an entry on rhombus.

### Re: We were missing this: Rhomboid

Houston, we have a rhomboid! :-)

> Geometric shape. Definition, diagram, formula for area, instructions for drawing with ruler and protractor. Maybe some relevant quotations from Euclid (Elements I:22), maybe some bits about history of rhomboids, maybe some pseudocode for computer graphics, maybe some Star Trek references. MSC 51-xx.

Definition, diagram, and formula for area are done. (I find it odd that there is no area formula in the parallelogram entry. I filed a correction to that effect.)

I will get to construction, though I might do it as a separate entry.

My copy of "The Elements" has the suckiest index ever! I am so glad that you supplied I:22. I might add a quotation.

I have no knowledge of the history of the rhomboid or pseudocode, and (possibly most lamentably) my knowledge of Star Trek is minimal.

### Re: We were missing this???

Actually, the fact the even elemenetary terms are missing does not
surprise me nor has surprised me for the last few years. For a long
time I have pointed out that there are all kinds of holes in the
coverage of PM which should be adressed. As I mentioned in a post on
Kaehler potential, I see little hope for adressing these issues
within the current modus operandi; rather what I think is needed is
a more concerted effort to identify what is lacking and more
collaboration between authors.

### Re: We were missing this: Rhomboid

She marked the Star Trek thing as optional, and generally she's not going to reject request fulfillment just for the sake of being mean-spirited.

The only Star Trek reference to rhomboids I can think of is the TNG episode "Samaritan Snare:" The Enterprise answers a distress call from the Rhomboid Dronegal sector. I think this was in dialogue only, no onscreen map of the sector was shown. Keep in mind though that I am not well-versed in the Classic series.

### Re: We were missing this???

I agree that figuring out basic concepts that PM is missing could take a very long time via the current procedure. Any ideas for setting up a system where people "hunt" for missing concepts? As for "more collaboration between authors", what did you have in mind?

Usually, when I find that something elementary is missing on PM, it is entirely by accident. Some exceptions are my Euclidean construction entries. Even though they are elementary, I kind of doubted that someone was going to go through the trouble of adding these, especially since use of graphics has been somewhat scarce until recently. (Why bother describing a construction when the reader cannot see the construction?)

### Re: We were missing this???

> I agree that figuring out basic concepts that PM is missing
> could take a very long time via the current procedure. Any
> ideas for setting up a system where people "hunt" for
> missing concepts? As for "more collaboration between
> authors", what did you have in mind?

Here is one idea. Choose a topic, find a *very good book* on that topic, widely available, make a wiki page saying you will follow such book and systematically read through the book looking for concepts missing in the encyclopedia. Write down the missing concepts/theorems in the wiki page, so others can find the list and contribute entries about them.

That should work, but it is a lot of *work*.
Alvaro

### Re: We were missing this???

I was in the process of constructing a detailed reply to your questions
when I had a computer bleep which wiped it out before I could send it
to you. Unfortunately, there is no way I can now reconstruct the
dozen or so suggestions I had from memory, so I will have to get back
to you later on this.

Sorry,
Ray

### Alvaro's idea

> Here is one idea. Choose a topic, find a *very good book* on that topic, widely available, make a wiki page saying you will follow such book and systematically read through the book looking for concepts missing in the encyclopedia. Write down the missing concepts/theorems in the wiki page, so others can find the list and contribute entries about them.

> That should work, but it is a lot of *work*.

I agree that this sounds like a good idea (although it will take ***tons*** of work). In my specific case, I would likely have to buy or borrow books. I only own a handful of math books which are sufficiently good and sufficiently widely available.

Another thought crossed my mind: Why a wiki? We can make collaborations right here on PM, and that seems to be an appropriate use. Why not utilize those instead of creating these pages externally?

Warren

### Re: We were missing this???

> I was in the process of constructing a detailed reply to your questions...

I must admit that I especially enjoy reading your posts. It is clear that you take a lot of time and effort when you type your posts. The "detailed reply" seems typical.

> ...when I had a computer bleep which wiped it out before I could send it to you.

I hate when that happens. :-(

> I will have to get back to you later on this.

I look forward to hearing from you.

Warren

### The Elements

> Here is one idea. Choose a topic, find a *very good book* on that topic, widely available, make a wiki page saying you will follow such book and systematically read through the book looking for concepts missing in the encyclopedia. Write down the missing concepts/theorems in the wiki page, so others can find the list and contribute entries about them.

Would The Elements be a good starting point? I imagine (and hope) that a *lot* of things in there are already on PM, but it would seem a shame to be missing something simple that The Elements covers.

### Re: The Elements

There is a wonderful electronic edition of Euclid available:

http://users.ntua.gr/dimour/euclid/index.html

If you click on the book on the main page, you get a table of
contents and if you click on an item in the index, you get a
list of the definitons or propositions in a particular book.
Clicking on a theorem gives you a proof of that theorem. For
instance, check out what they did with the proof of the
Pythagorean theorem:

http://users.ntua.gr/dimour/euclid/book1/postulate47.html

(This theorem being so famous, you can get to it from any place
on the website from the menu obtained by clicking on the bar
"Epiloges" on the top of any page.)
Moreover, in the version of the proof below the line, there are
all sorts of links provided. Clicking on a word in the statement
of the theorem (e.g. "orthogonia trigona") takes you to the
definition of the term. Within the proof, parenthesized
statements providing justifications for steps in the proofs
"(theor. a' 31)" takes you to the pace for proposition 31 of the
first book whilst "(k. en. b')" takes you all the way back to
the list of common notions to see the second common notion, which
is used to justify the step in the proof.)

I consider this a shining example of how to present a classic in
an electronic format. We could learn a lot about presenting math
electronically from them! For instance, I think that there is a
lot to be said for the way they present the mathematical text (in
this case, just words and figures, (because equations had not yet
been invented 2300 years ago!) using a full screen with big fonts
for the titles and statements of theorems and put the navigation
data in discreet little drop down menus on the top of the page as
opposed to here where the navigation data sits in big honking bars
on the side of the page which leaves not so much room for text, so
smaller fonts have to be used, especially since big margins further
reduce the width available further. If we want our work to one
day become a classic, maybe we should have a look at this
contemporary presentation of a classic!

### Re: The Elements

I agree that the site is nice, but alas, I am not fluent in Greek. I would be quite ecstatic if there were an English version of this site. (I would be pretty happy with a French and/or Spanish version.)

Any idea on that site's terms of usage?

### Re: The Elements

> Any idea on that site's terms of usage?

The content, of course is all in the public domian. (As a matter
of fact, they didn't even have anything like copyright in 300 B.C.)

The only think that is copyrighted is the layout and presentation
of the site. It clearly states on the site that this is owned by
Dimitrios E. Mourmouras. However, as the lawyers say, this sort
of copyright is thin. What that means is that while you can't just
copy his site wholesale to make your own website of Euclid's
elements, especially such features as the clickable book image
on the homepage, it doesn't mean you can't make a website on Euclid's
elements or reuse the content (which, after all, has always been in
the public domain).

### Re: The Elements

Knowledge of copyright is not my strong point, so I am glad to have this information. I was not even familiar with the term "public domain", but I can tell what it means from context.

I think that it would be cool to add the actual Greek in certain entries (so long as it is kept short). For instance, in "rhomboid", I have a translation of what Euclid had to say about them, but not the actual text. Since the text is in the public domain, I guess I could add it if I wanted to. Maybe the only reason I think adding the Greek would be cool is due to some nerdy tendency that I have. :-)

Despite the fact that the text is public domain, I think that, for any entry that uses the Mourmouras web site, a citation should be provided, and possibly a link as well. Mr. Mourmouras certainly put forth an excellent effort and deserves as much credit as he can get for that site.

Warren

### Re: The Elements

rspuzio writes:

> The content, of course is all in the public
> domian. (As a matter of fact, they didn't even
> have anything like copyright in 300 B.C.)

Unfortunately, though, the question of copyright of ancient texts is not that simple. The problem is that we don't usually have the original texts, only later copies (of copies of copies...), often indecipherable in places and often conflicting with one another. Attempted reconstructions of the original texts can be copyrighted, if they are of sufficient novelty.

The site you are talking about doesn't seem to say what text it's using. It may well be public domain (e.g., taken from a 19th century edition), but it may not be.

> Attempted reconstructions of the original texts can be copyrighted, if they are of sufficient novelty.

> The site you are talking about doesn't seem to say what text it's using. It may well be public domain (e.g., taken from a 19th century edition), but it may not be.

So it might be possible that that site itself infringes on someone else's copyright? (I can't learn about these nuances of copyright if I don't ask these types of questions.) Even if the web site is a case of copyright infringement, it seems unclear whose copyright has been infringed upon, unless that person/group can prove that Mourmouras used his/her/their source to make the web site.

Given that the web site is O.K., I think that Ray's argument that just putting your name on something is flimsy for copyright holds water. Of course, I don't know enough about copyright to say for sure.

I once had a professor who had an expressed copyright on his lecture notes! I thought that was pretty weird.

### Re: copyright (was The Elements)

Wkbj79 writes:

> So it might be possible that that site itself
> infringes on someone else's copyright?

It would be possible for such a site to infringe on someone else's copyright (although it's unlikely in this particular case).

> Even if the web site is a case of copyright
> infringement, it seems unclear whose copyright
> has been infringed upon, unless that person/group
> can prove that Mourmouras used his/her/their
> source to make the web site.

If the text on the site were copied verbatim from a single source, that would be obvious on comparing it to the source.

> Given that the web site is O.K., I think that
> Ray's argument that just putting your name on
> something is flimsy for copyright holds water.

I would guess that Mourmouras is only claiming copyright on the website, not on the text. But it's possible that this is his own text of Elements, not just a copy of an earlier one, in which case he could claim copyright on the text too.

### Re: The Elements

> Maybe the only reason I think adding the Greek would be cool is
> due to some nerdy tendency that I have. :-)

I think of it not as a matter of nerdy coolness, but as a matter of
being useful for scholarship. This way, anyone interested in
own words available and can form an opionion as to the interpretation
and conclusions presented as based upon that text.

The reason that I had such an easy time reading Euclid in Greek was
that I had something of an old-style humanist education which involved
years of studying Latin ang Greek starting in the eighth grade and
reading Virgil, Cicero, and other ancient authors. By the time I
went to college, I already had quite a background, so I quite
naturally took a lot of classes in the classics deprtment -- in fact,
had I not wound up in mathematical physics, it is quite likely that I
would have turned out as a classics major. Also, there was some
overlap between these interests. For instance, when I was taking a
class in Mediaeval Latin, for a final project I did a study of
Boethius' treatment of perfect numbers and remember trudging over
to the NYPL branch library to track down a copy of Nichomacus'
introduction to number theory (this is the source Boethius used and
is just as much about numerology as number theory).

The reason for this digression is that one of the things I noted
then is that there are quite a number of historians of math and
classical scholars interested in ancient math. If we do as good
job presenting Euclid, including quotations in Greek and links
to Mourmouras' site and similar places, then PM could also be a
useful resource for these people. Also, from a historical point
of view, remember that Euclid's book is very imporetant --- until
well into the nineteenth century, it still served as the primary
textbook for geometry in many places (for instance, see Lewis
Carrol's defense of Euclid as a textbook) and served as a model
for such works as Newton's Optics and Principia.

By the way, I am going to have to look up a package for writing
Greek text. Not only is it aggravating to write more than a word
or two using math mode, the $spacing~is~all~wrong$ for natural
language as opposed to formulae. Such things must exist because
I've read plenty enough math books in Greek including a wonderful
introduction to ring theory. (We will eventually have to introduce
them to the term "symperiphora kyklikou daktyliou") Of course,
there, it is Latin letters that play the role of exotic symbols
for use in equations!

### Re: The Elements

> I think of it (adding the Greek text) not as a matter of nerdy coolness, but as a matter of being useful for scholarship. This way, anyone interested in knowing exactly what Euclid had to say about rhomboids has Euclid's own words available and can form an opinion as to the interpretation and conclusions presented as based upon that text.

> ...there are quite a number of historians of math and classical scholars interested in ancient math. If we do as good job presenting Euclid, including quotations in Greek and links to Mourmouras' site and similar places, then PM could also be a useful resource for these people. Also, from a historical point of view, remember that Euclid's book is very important...

Excellent points.

With regards to your point on informing readers what Euclid had to say, I can totally vouch for this. While doing research for my master's thesis, I stumbled across a relevant source "Despre Inele Ciclece" (Romanian for "On Cyclic Rings"). To the best of my knowledge, this is only available in Romanian, Russian, and French. Although my advisor was content with the French translation, I opted to translate the original Romanian. It turned out that my translation yielded statements that were more mathematically precise. That added to the fact that my translation involved one less translation led us to believe that my translation was more accurate.

BTW, what is the policy on adding translations of papers to PM? I imagine that, since I translated "Despre Inele Ciclece", I would have the copyright my translation, but I could be wrong.

Back the The Elements! I think that I will e-mail Mourmouras and ask about terms of usage in the site. Hopefully, by the time I hear back from him, you will have found a package for writing Greek text.

Also, I have noticed that, not only is the spacing all wrong, but also that either I do not know the code for producing certain Greek diacritical symbols or the code does not exist in TeX. The latter would highly surprise me, but after Google searching and looking in many places, methods for producing more accurate Greek diacritical symbols elude me.

Finally, I guess that I must amend the reason for my opinion why adding the Greek would be cool: I have always been interested in foreign languages and etymology. While this is nice on a word-by-word basis (pahio has been adding entries and briefly indicating the etymology of the word), a longer text would be nicer.

Warren

### implementation of Alvaro's idea

> In my specific case, I would likely have to buy or borrow books. I only own a handful of math books which are sufficiently good and sufficiently widely available.

Steve (user sjm1979) has kindly lent me his copy of Euclidean and Non-Euclidean Geometries: Development and History (third edition) by Marvin Jay Greenberg. This was the textbook for my undergraduate courses in Euclidean and hyperbolic geometry, so I imagine that it is sufficiently widely available, and I know that it is sufficiently good.

I decided that I should start with this book instead of The Elements. To be honest, I feel that Ray (user rspuzio) would be more suitable to take on The Elements because he can read the Greek, whereas I cannot. On the other hand, I do not intend to "assign" this book to Ray. If he does not want to look through it and/or does not have the time, then hopefully someone else (including me) may take a look at it.

From the title of the book, it seems that I should be able to add or give suggestions for additions to PM with regards to the history and development of these geometries.

Sadly, the index does not have "metric" or "distance". I had hoped to be able to fill Chi's request for metrics in models of hyperbolic geometry by reading this book, but that might not be possible. If no one else fills the request, I will make sure to fill it at some point in time.

I will not be able to start posting (in a collaboration) on my findings in this book until Friday night at the earliest, as I have other fish to fry. (I plan on "attending" the conference call though.)

Warren

### Re: implementation of Alvaro's idea

Greenberg's book is great. I am not sure how many schools use it as the textbook for the axiomatic treatment of geometry, but I highly recommend it. Another one that I have been told is a good source is Hartshorne's "Geometry: Euclid and Beyond".

### revenge of The Elements

In posts (which were written some time ago) to which this post is attached, we were discussing the copyright of a Greek rendition of The Elements that Ray had found. I did not know this, but vmoraru has posted a link to a site called "Euclid's Elements" on PM and describes it as being public domain. The link to vmoraru's description is:

http://planetmath.org/?op=getobj&from=books&id=22

And the link that he supplies is:

http://aleph0.clarku.edu/~djoyce/java/elements/toc.html

I am a little confused by this, as there is an expressed copyright on the page at the second link. I would guess that the copyright is on the diagrams and on the layout, not on the actual content. Please let me know what you think, as I would really like to have an English translation of The Elements that I would feel comfortable using as a resource for PM. Thanks.

Warren

### Re: revenge of The Elements

Wkbj79 writes:

> I would really like to have an English
> translation of The Elements that I would
> feel comfortable using as a resource for PM.

Although it's only the first six books (and part of book XI), the following from Project Gutenberg may be of interest:

http://www.gutenberg.org/etext/21076

This was published in 1885, and the author appears to have died in 1891, so copyright should not be a problem (and Project Gutenberg explicitly says that it's not copyrighted in the United States).

### The Elements, terms of usage

With respect to the web page at:

http://users.ntua.gr/dimour/euclid/index.html

a week ago, I wrote:

> I think that I will e-mail Mourmouras and ask about terms of usage in the site. Hopefully, by the time I hear back from him, you will have found a package for writing Greek text.

This is a copy of the e-mail that I wrote to Mourmouras:

Dimitrios,

My name is Warren Buck. I am an active member of PlanetMath, which is an online encyclopedia that deals specifically with mathematics (available at http://planetmath.org ). A fellow member of PlanetMath found your web site on The Elements (the one at http://users.ntua.gr/dimour/euclid/index.html ). We were ecstatic to find this site. We were wondering about its terms of usage. We would like to incorporate portions of the Greek text of The Elements (which might be considered public domain anyways) into some of our encyclopedia entries that deal with Euclidean geometry. If this is permissible, we would cite your web site as our source and include a clickable URL for it in every encyclopedia entry for which we do this.

We noticed that your have an expressed copyright on the site. Are you claiming copyright of the Greek text of The Elements? Please let me know. Thanks in advance.

Warren

Dear Warren,
Thank you for your message and the kind words about my site.
I am not claiming copyright of the Greek text, but only of the web pages (figures, scripts, and html structure).
You may however feel free to incorporate to your encyclopedia portions of the Greek text under the terms you mention.
I might also mention that the Elements contain much more materials than just Euclidean geometry (number theory, irrationals, etc).
http://www.physics.ntua.gr/Faculty/mourmouras/euclid/

-- Dimitrios Mourmouras
http://www.physics.ntua.gr/en/faculty/Mourmouras_Dimitrios.htm

This makes my day, and it's only 9am here! :-)

Warren

### Re: The Elements, terms of usage

It certainly does counterbalance the negative reply from the
biography site a few months back.

Given this reply, I will put some effort into Greek
internationalization. I have already located some TeX packages
for writing Greek text, will try them out, and figure out what
is needed to make them work on PM. When I sort this business
out, I will place an account of how to type Greek in the
internationalization site document.

> I might also mention that the Elements contain much more materials
> than just Euclidean geometry

Such as the good old Euclidean algorithm!

In mediaeval terms (which ultimately go back to Pythagoras through
Plato), the Elements is a textbook for two subjects of the
quadrivium, namely geometry and arithmetic. As for the other two
subjects, music and astronomy, Euclid also wrote books on them,
namely "Elements of Music" and "Phaenomena". I suppose that a
good part of the reason why Euclid is most known for his geometry
is that his books on the other subjects were superseded by other
texts such as Ptolemy's Tetrabiblos (technical) and Aratus'
Phaenomena (popular) for astronomy, Diophantus' Arithmetica (technical) and Nichomacus' Introduction to Arithmetic (popular)
for arithmetic, and the works of Aristoxenos and Boethius
for music. (It should be remembered that, in this context,
music referred to the mathematical theory of acoustics than to
how to sing, play an instrument, or compose songs.)

Speaking of Diophantus here is an earlier scholium by Planudes
on the same problem which led Fermat to come out with his last
theorem --- "Thy soul, Diophantus, be with Satan because of the

### Re: The Elements, terms of usage

Given this reply, I will put some effort into Greek
internationalization. I have already located some TeX packages
for writing Greek text, will try them out, and figure out what
is needed to make them work on PM. When I sort this business
out, I will place an account of how to type Greek in the
internationalization site document.

In case you missed it: http://www.math.ucla.edu/~ynm/greektex/

### Re: The Elements, terms of usage

No, I didn't miss it. That is one of the sites I looked
at for information on one of the two popular packages for
TeX in Greek. Now I need to look at the documentation more
carefully, try out the packages, etc.

### Re: The Elements, terms of usage

Have you used this package?

### biography site? (was The Elements)

> It certainly does counterbalance the negative reply from the
biography site a few months back.

This statement naturally caught my eye. Apparently we cannot use it, but I would still like to know *what* biography site. Thanks.

Warren

### Re: biography site? (was The Elements)

I am pretty sure (unless I am wrong) that Ray was referring to the MacTutor history archive:

http://www-history.mcs.st-andrews.ac.uk/index.html

A

### Re: biography site? (was The Elements)

Thanks Alvaro. Actually, I've come across this site before. It is an excellent resource. Too bad we cannot use it. (Sounds like a Lord of the Rings quote: "You cannot wield it! None of us can!")