Flirting with Infinity

That would be my guess, yeah.

This is exactly what I wanted to talk about next. What happens when we begin to dabble with Infinity and Triangles?

We are going to start with a Right Triangle. The Right Triangle is composed of one internal 90 degree angle, then let us give the value of the other 2 internal angles 45 (thus, 90+45+45=180 total). The side opposite to the 90 degree angle is the hypotenuse, and the other two sides are called catheti.

So this is the Triangle we will start with. Now what I want you to do with this Triangle is simple. Select one of the catheti (a side which is not the hypotenuse) and begin extending it. Extend it 1 inch. You will notice that when this side is extended the hypotenuse must extend as well! The untouched catheti (side) remains the same distance always (thus the 90 degree, right angle always will remain constant, but the other 2 angles will increase).

Take this example of extending the selected side as above, and begin to lengthen it more and more. Now take this concept into Infinity!

As the side you selected increases in length toward infinity you will see a few things happen: The original Right Angle remains at a constant of 90 degrees. The untouched side, that touches the 90 degree angle and the hypotenuse, becomes 90 degrees (it was only 45). The hypotenuse and the extended side become parallel, yet simultaneously somehow actually still touch one another somewhere into infinity. A perfect paradox is created!.... I dare say: a mysterious 4th side is created. This triangle now has a total of 360 degrees, and is composed of 4 right angles.

[[ Simplified version: Simply take any triangle of your choice. Select any side of that triangle. Start increasing the length of that selected side. Adjust the other sides according to the new length of that side, so that you still have a triangle. Continue into infinity. ]]

Hope you had as much fun as I did with this

You can't just say "well take this shape and make it bigger until you get to infinity!" That's the equivalent of saying "count until infinity!"

You simply can't do that. There is no such thing as an infinintely big triangle. It's one thing to extend a line to infinity, but it's another thing to say that three discrete points are all mutually inifinitely far away from each other. Remember, a triangle is a polygon defined by three discrete points on a plane.

And why would the acute angles of the triangle get larger as the triangle gets larger? They don't. The sum of the internal angles of any triangle remains 180 degrees no matter how arbitrarily large the triangle is. Whether a side of an iscoceles right triangle is a millionth of a nanometer or ten trillion light years long, the sum of the internal angles is 180.

The only paradox is that you're not making any sense.

I have decided to post a visual so you can actually see what I am talking about. I have found visual aid helps dearly in such geometrical lectures. I certainly wouldn't just try to BS you, my friend. I am very serious about what I speak of, and wouldn't leave you or anyone else hangin'.

Let's see if this visual aid works:



(c) Thirteen Invitations

I think the line is being neglected because the difference between a line with infinite length and a circle with infinite curcumfrance should have the same differences as a line with a finite length and a circle with a finite curcumfrance. But maybe I'm just over-simplfying things. Math theory like this isn't really my strong suit, after all.

This is like the underpants gnome business strategy from South Park.

Phase 1: Collect Underpants
Phase 2: ???
Phase 3: Profit

I especially like how you copyrighted the magic of turning a triangle with one angle approaching 90 and the other approaching 0 to a rectangle with four distinct angles.

Here's a little calculus for you: As the limit of the larger of the acute angles approaches 90, the remaining angle approaches 0. Not 180.

By the way, just in case you're wondering, a short line segment with two parallel lines intersecting its endpoints at right angles doesn't count as a "triangle."

It doesn't count as any sort of polygon, because it only has two vertices. And there is no sum of its internal angles because it has no interior.

And you claim to have four internal angles in the bottom drawing, but you only actually have two. If you had four angles, you'd have four sides. And you would have a quadrilateral, not a triangle. But you don't even have that.

As you can see, we start with a basic triangle. As this one side is extended, the other line increases in length too. Their common vertex keeps getting further away. Once the length hits infinity that vertex is reaaaally out there. It looks as if these two sides have become parallel lines. But how in the hell can that be possible when they both meet... meet somewhere off into infinity? By definition parallel lines never meet, but that is only applicable under normal circumstances. Here is where the paradox comes into play... they are meeting, but simultaneously don't meet! And exactly, a quadrilateral! We now have a quadrilateral, yet, also a triangle. It's as if this vertex, that's now infinitely far away, is no longer "just a vertex", but now a line, as if it got stretched at the point of infinity! Thus, the 2 unseen 90 degree angles I spoke of in the diagram.

Oh, the angles are just "unseen" because they're infinitely far away.

Trust me, dude. You're not making any sense. There's no paradox. The parallel lines never meet. The angle between them is exactly zero, it's not a polygon, there are no "unseen vertices." There are no quadrilaterals that are also triangles.

If you look at the diagram I posted you can see the first 90 degree angle. Aside from that 90 degree angle there are 2 other angles. These angles values are being altered as the sides increase in length. One of these angles "disappears into infinity", and as this occurs, the angle still visible will become a 90 degree angle. So now there are two 90 degree angles that can be SEEN. The problem we are having is that "Unseen angle" that disappeared into infinity. You are claiming it has become 0 degrees. Is that possible? If this is so, then it is no longer an angle at all! An angle with 0 degrees is, AHA, a line. Just as a predicted: That vertex becomes a LINE! And perhaps this Angle of 0 Degrees, which now has become a Line as it enters Infinity, is the Line bringing about the birth of the quadrilateral (the unseen line, the 2 secret 90 degree angles).

In your deprecated "triangle" there are still three angles: two 90 degree angles and one non-existant 0 degree angle without an associated vertex.

Have you ever dealt with the concept of a limit? The limit is the fundamental basis of calculus.

You're literally just making up crap.

beat me to the punch..

Sooooooooo, whoever gave me a kudo saying "yeah rite ROFL" just now for making this statement, would you like to come forward?

Or do you prefer to insult other peoples' intelligence through kudos. Cause let me tell you, I may not be the sharpest tool in the shed at times, but if I'm gonna say something like that, I'll say it to your f***ing face.

Done.

You cant have a circle with a circumfrence of infinity. The end points would never meet if was ongoing forever. If a circle has a circumfrence then it must be measurable i.e not infinate.

When you speak of the 0 degree angle, some things come into play. The 0 degree angle is what's called a "degenerate angle". We would no longer have 2 rays, or sides, but 1. How does a 0 degree angle exist? Would it be anything other than a line? If an Angle can have 0 degrees, then SURELY a triangle can have two 90 degree angles! But as I stated earlier, this vertex, when hitting infinity, would become a line. Thus the unseen 90 degree angles.