f(x) = sin(x) + sin(x*sqrt(2)) has no fundamental period.

Right?

Help me out here. I think I can define a kind of rational function as a locus of points, but my locus definition is still too long and confusing.

It all started with these doodles I like to make showing two points (A and B) and parallel lines going through those points. At the endpoints (A' and B') of any perpendicular to the parallel lines I'd connect the points A and B' and then B and A' so that they cross. Then I'd track the arc traced by the intersection of AB' and BA'. In this drawing to the left you can see the case where the lines from A and B are parallel, so no intersection... that is the the same point the function becomes undefined (I think) when you are working with it using algebra. In any case, this is great fun if you are in a long boring meeting. Try it some time.

At first I didn't know if it was a curve. Ha! I thought it'd be a line! But it's never *looked* like it was linear...

Then, I tried using a little algebra and found the parametric equation:

Xt=tm/(n+2t)
Yt=t(t+n)/(n+2t)

n/m is the slope from A to B (not reduced... though, I don't know if it matters.)

After solving for t in both parts of the parametric equation I found a general rational function. That part was quite a mess and I still need to check my answer. But, whatever it is, it's not a poly-function.

Bigger and better constructions held up this idea as some lovely rational curves emerged:




OK. So, now the question is, how can I define these curves as a locus. I want it to be shorter and sweeter than what I've written above. Are there and fun geometric applications for the rational function? (I mean a situation that mirrors the requirements I've described for these curves.)

Also, where can I learn more about this topic?

Thanks to anyone who can help!

You may often find your self wondering: Are there more ants or leaves in the world?



Here is the answer:

EO Wilson , a noted Harvard biologist and expert on ants, estimates that the number of ants in the world is between ten to the sixteenth and ten to the seventeen or: 100,000,000,000,000,000 ants Forests represent a third of the earth's land. On average they can support 200 trees per acre. 9.8 billion acres of forest in the world. (1995 estimate)

9,800,000,000 acres of forest X 200 trees/acre = 1,960,000,000,000 trees

The number of leaves per a tree varies widely. Most trees are pines with thousands of leaves. Some only have hundreds. let's use 1500 (low thousands) as a high estimate.

Thus we have:
294,000,000,000,000 leaves

vs.

100,000,000,000,000,000 ants

or around 340 ants per leaf.

In recent years the number of trees has fallen sharply due to clearcutting. The number of trees per acre and the number of forest ares I used were high. Even if we used a higher number of leaves like 7000 (the number found on the largest pine trees) the number of leaves would still fall far short of the many many ants in the world. Ants win hands down by two orders of magnitude.

Graph Theory Question:
I made this up after reading about trees

How many distinct (non-isomorphic) trees can be drawn for a graph of order n?

( Read more...Collapse )

I’m thinking about a variation of the traveling salesmen problem called the Euclidean traveling salesmen problem. In it the distances between the cities are a set of straight, coplanar lines. I’m also only concerned with a set of complete cities (so there is a road between every city)

Given a set of such distances my fist task might be to check that distances meet the geometric requirements for a Euclidean TSP.

If there are n cities there are n(n-1)/2 roads. But, how much work will it take to find out if a set of distances between cities is Euclidean?

If we have only one city no check is needed. With two cities no check is needed … but with three we need to check that triangle inequality is not violated.

The fourth city will require some kind of verification using the distance formula. There will be two possible locations for the 4th city.

The addition of a 5th city adds 4 new edges and each must be checked.

Here is a table of what will happen:

Vertices ---edges---checks
1-------------0----------0
2-------------1----------0
3-------------3----------1
4-------------6----------2
5-------------10----------6
6-------------15----------10
7-------------21----------15
8-------------28----------21
9-------------36----------28

At this point I want to stop and ask if my thinking so far makes sense?
The progression of these numbers seems to be the same as the number of edges, but for some reason it doesn’t start out the same way. Should I be concerned about that?

Okay, I’m still thinking about my constrained version of the traveling salesmen problem. Here is an easy little geometry problem:

Triangle ABC has sides:

AB = p
BC = q
CA = r

Triangle ABD has sides

AB = p (duh)
BD = s
AD = t

Express the two possible lengths of DC in terms of p, q r, s and t

I’ve done it one way, but it looks like a big mess.

I just found out about gauss primes! What a cool idea for complex numbers. If you have some number:

a + bi

Where a and b are integers and i = srt(-1) then you can call it a "prime" if it has no complex factors where a and b are integers except itself and 1.

So is 6 - 7i prime? Well, this questions seems to become: Are there any integer solutions to:

ac-bd = 6 bc + ad = -7

How can you tell??

By the way. 5 is not a prime number the factors are: (2 - i)(2 + i)