List irrationals that the Greeks could define via geometry and give name, if name is not already known.

Pi and "golden ratio" have geometric definitions and are irrational. Are there more? Preferably defined with one sentence geometric definitions.

I.e.
(1)Pi is the circumference of a circle divided by its diameter.
(2)Golden ratio is the ratio of the sides of a rectangle which when a square is cut off of one end leaves another golden ratio rectangle.

I thought there might be a "doubly golden ratio" rectangle if the square were cut out of the middle, leaving two "doubly golden ratio" rectangles on each side, but it is very rational rectangle (sides in 2 to 1 ratio).

"e" is certainly irrational, but I do not know any geometric definition for it.

I think a (3) may be possible in this list because I know that the centers of an infinite set of progressively smaller golden rectangles do form a named spiral but do not recall the name or if there is any irrational number associated with it.

Surely some one here can find and post (3).

I shall name sqrt(2) "bob" and submit it as number 3 ;) The ratio of the diagonal to the side of a square.

-Dale

(1)Pi is the circumference of a circle divided by its diameter.
(2)Golden ratio is the ratio of the sides of a rectangle which when a square is cut off of one end leaves another golden ratio rectangle.
(3)*Square root of 2 is the ratio of the diagonal of a square to a side.
(4)*Square root of 3 is the ratio of the diagonal of a cube to a side.
____________
NOTES:
(3)*Contributed by DaleSpam, but I think it better know by this longer name than his "bob." (4) is obviously extension of Dale's

This thread is a list of irrationals that the Greeks could define via geometry. Give name plus brief definition for Number 5.

Suggestion: Does anyone recall the spiral that is mathematically defined and found in nature (pattern of sunflower seeds, etc)? I am almost sure the center of an infininite set of golden rectangles also forms it and hope an irrational number is associated with it (in the mathematical definition).

See http://en.wikipedia.org/wiki/Fibonacci_number but it a sequence rather than a constant but it is related to the golden ratio and to many things in nature.

-Dale

See http://en.wikipedia.org/wiki/Fibonacci_number but it a sequence rather than a constant but it is related to the golden ratio and to many things in nature.-DaleThanks for the reference, Fibonacci name.
I went to your link - it was good, as are two sub links by the same author:

www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html

He designates the golden ratio "Phi" and notes that its square is exacly one larger than its self. This means that you can start with any number (but choose a one digit one to not take too long) then:
Step 1: add 1 then take square root.
Step 2: with results of step 1 as your 'start number" go to step 1

Exit this loop when the calculator has converged to the golden ratio, Phi. approximately 1.61803 39887 49894 ....

Much more there to keep the mathematically inclined occupied than here by far! I liked his history section in second link above.

But I did not find any new geometrically defined irrationals there in quick first look.

to oPhiolite:

In now locked pole I and you posted:
Originally Posted by Billy T:
Instead, assemble a bunch of women and determine the ratio of their total heights to their navel heights. The result will be the golden ratio - do not try with men. They are not golden. (This was one of my favorite college research projects.)

Originally Posted by ophilite, in response to mine above:
Has this been tested on homosexuals of either sex? Seriously.

I think that an interesting question, but my research in this area was focused on attractive women at parties with just enough attention to men to demonstrate that it is true that only women, in average, "are golden." The sexual preferences of my sample were not asked.

Reason I think your question may be very interesting is that I have read that the ring finger of homosexuals is usually as long as the middle finger. I do not know if this is true or not - I have no evidence one way or the other, but perhaps there are minor physical differences as the evidence that preference for the same sex is partially genetic and with some differences in brain structure/ patterns is increasing, but still very weak in my view.

The length of the ring finger is related to the amount of testosterone recieved in the womb, and some people thinnk that homosexuality can be linked to levels of hormones in the womb, but this is no definate conclusion.
The ring finger is usually compared to the index finger rather than the middle finger,
longer ring finger- "girly man"
same size - normal
longer ring finger - "macho man"
Keep in mind that the limp-wristed effeminate homosexual is only a stereotype.
Okay, something a little more on-topic,
You may be interested to know that in Da Vinci's man, which I forget the name, I believe that the ratio of his forearm to upper arm is phi, as is the ratio of his height to armspan.
If I remember correctly, phi is exactly (1 - sqrt(5))/2
e has many definitions, and one I have seen is the number e such that the area under 1/x from 1 to e is exactly 1.

"e" is certainly irrational, but I do not know any geometric definition for it.A geometric argument for "e":

We choose a base "a" as the number such that the tangent line at (0,1) of the function y=a^x has exact slope 1. "e" is equal to this unique "a".

We choose a base "a" as the number such that the tangent line at (0,1) of the function y=a^x has exact slope 1. "e" is equal to this specific "a".

Wouldn't it be a good idea to prove that your "a" is unique?

...The ring finger is usually compared to the index finger rather than the middle finger,
longer ring finger- "girly man"
same size - normal
longer ring finger - "macho man"...Thanks for the correction. I looked at my fingers when trying to recall what I had read for prior post. My ring finger is almost as long as the middle finger and much longer than the index one, so I assumed it was the middle finger that was to be compared with the ring finger. (Second fact about my fingers probably explains why as college student I pulled up the blouse of every attractive female that would let me and often had a tape measure in my pocket.) :cool:

Wouldn't it be a good idea to prove that your "a" is unique?Soz, wasn't meant to be a proof only the definition :) I've edited the above post.

{e can be}Defined in the following way:

We choose a base "a" as the number such that the tangent line at (0,1) of the function y=a^x has exact slope 1. "e" is equal to this unique "a".Just thinking "out loud", to understand (and perhaps help others):
First I note that y = 1 if x = 0 at least if a not 0 (and perhaps even then for continuity), function y=a^x passes thru Cartesian coordinate (0,1) OK.

Now the slope of this function is (dy/dx) = x a^(x-1) is it not? (I am old and rusty, but think that correct.) The "tangent line" = slope, does it not? So at x = 0 the slope would seem to be zero for all a. If this is the case, how can I chose an "a" to make the slope unity?

I am not saying you are wrong. I am hope you do have a geometric definition for "e" but I do not see it yet. What have I missunderstood / done wrong etc?

Just thinking "out loud", to understand (and perhaps help others):
First I note that y = 1 if x = 0 at least if a not 0 (and perhaps even then for continuity), function y=a^x passes thru Cartesian coordinate (0,1) OK.

Now the slope of this function is (dy/dx) = x a^(x-1) is it not? (I am old and rusty, but think that correct.) The "tangent line" = slope, does it not? So at x = 0 the slope would seem to be zero for all a. If this is the case, how can I chose an "a" to make the slope unity?

I am not saying you are wrong. I am hope you do have a geometric definition for "e" but I do not see it yet. What have I missunderstood / done wrong etc?Your differentiation is wrong. It's an exponential function and not a power function (I was a bit unclear). The derivative of a an exponential function f(x)=a^x where a is some real number is:

f'(x)=a^x*ln(a)

As it depends on ln(a) you can't really use this to define "e" but it is a geometric argument. :)

So: f(x)=e^x ==> f'(x)=e^x*ln(e) ==> f'(0)=1 :D

...(I was a bit unclear)...No you were clear. I was just stupid and remembering the form of differentiation for y = x^a and failing to note that that was not your function. I warned you I was rusty. :D

I think you agree, your suggestion is not quite what we want - You said:

"As it depends on ln(a) you can't really use this to define "e" but it is a geometric argument"

In simple terms, would say that one could plot y = a^x for lots of different "a" and measure the slopes to determine an approximate value of "e" but this is not really a geometric definition of it. Do you agree?

Close, but no cigar!

Still waiting for No. (5)

As it depends on ln(a) you can't really use this to define "e" but it is a geometric argument. :)

You don't need "e" to define the natural logarithm, just define log(x) as the integral of 1/t from 1 to x. "e" is then defined as the unique point where log(x)=1 (NeonBlack mentioned this above).

You don't need "e" to define the natural logarithm, just define log(x) as the integral of 1/t from 1 to x. "e" is then defined as the unique point where log(x)=1 (NeonBlack mentioned this above).Area under a curve could be fine / OK as a "geometric defintion" (one the Greeks could have done) but you need to define curve, your log(x) geometrically to play this "Greek irrationals" game.

Can you define "e" basically with only straight edge and compass?

PS - glad you are reading - hope you know any (not just "e") "greek irrationals" you can add to our currently short list of four.

Area under a curve could be fine / OK as a "geometric defintion" (one the Greeks could have done) but you need to define curve, your log(x) geometrically to play this "Greek irrationals" game.

Can you define "e" basically with only straight edge and compass?

PS - glad you are reading - hope you know any (not just "e") "greek irrationals" you can add to our currently short list of four.

About as close as I can think of to defining e without explicit caculus is graphing functions such that
f(x)=x^a

Where a is any number. e is defined when the slope at any point on f(x)=x. In this case x=e.

You could at least approximate e in this way. Though I understand that even defining e this way does not make it an absolute greek irrational, I'm just trying to give a different, more geometric answer.

Area under a curve could be fine / OK as a "geometric defintion" (one the Greeks could have done) but you need to define curve, your log(x) geometrically to play this "Greek irrationals" game.

Can you define "e" basically with only straight edge and compass?

The curve is just 1/t. I don't think you'd be able to construct it in a finite number of steps with compass+straightedge, but you can certainly find any point on it that you like, i.e. given a length t you can find 1/t.

PS - glad you are reading - hope you know any (not just "e") "greek irrationals" you can add to our currently short list of four.

If you accept the 1/t graph and defining log as it's area, you might also accept Euler's constant gamma http://mathworld.wolfram.com/Euler-MascheroniConstant.html certainly this, and e, never occured to the greeks though.

There aren't many "named" irrationals out there. The interesting ones often come from limiting processes that the greeks didn't really do anything with (exception Archimedes and pi).

The squareroot of most integers is easy to construct - certainly you can express all odd and half the even numbers (those divible by 4) as the difference of two squares, so by rearranging Pythagoras': b^2 = a^2 - c^2 you have that side b is the squareroot.

This is getting a little too confusing for me. My thoughts were not clear when starting the thread so I now think we should discuss the "rule of the game." I also think we should turn the "acceptance into the list" task over to some one better versed in math than me. I do not mean to slight anyone, but know Shmoe would be better qualified to do this function than me.

On the rules:
I am sure I made a mistake by asking for "named" irrationals, and possibly when I later mentioned "compass and straight edge." It seems to me that a critical question now facing us relates to use of areas in the "definitions."

NeonBlack first gave the same definition of "e" that Shmoe latter gave and I only realized this when Shmoe pointed it out to me. (I was too distracted by the "fingers part" of post and failed to understand, perhaps even to read, the last line of Neon's post.)

If area can be used in the definition, then Zephyr's suggestion, which involves the difference between areas of squares, either makes our list becomes infinitely long, or reduces it to three, if we throw all the irrational roots of integers into one group called "square root irrationals."

My view is that we can use areas in the definition provided they do not require a exact measurement of area to define the area. Thus, I am inclined to accept Zephyr's difference in areas as only the lengths of the sides is "measured" but reject the idea that we define "e" when the area itself is exactly 1. I am hoping others will give their views.

jdheiden's approach to defining "e" requires that a slope be measured as exactly 1. I am reluctant to accept this also, but again want others views.

In my view, it is OK to define an area by the ruler measuring the side, especially of squares, equilateral triangles etc. where the other sides are just to be the same as the original measured one. Or perhaps no measurement is used and we only use a side as the "unit of length" to be reproduced, as needed, with the compass. The definition of "golden ration" does not seem to me to require any measurement, but the evaluation of it numerically, even approximately, would.

I think it OK if the irrational can be defined geometrically, even if not evaluated even approximately. For example: make a square and the root 2 is the diagonal in the unit length of the side. What are your views?