Geometry and the measurement of infinity

lacar1601

lacar1601

Well-Known Member
I propose two conceptually related brain-busting questions in geometry. I myself don't know the answers to these questions, but it would be interesting to see if someone here can provide some hard proof for the answers. If nothing else, I encourage everyone to post a silly but creative (and relevant) response even if the answer is wrong.

WARNING: Thinking ahead.




1) BACKGROUND- In analytical geometry, we know that a line can be composed of a set of all points that satisfy some linear relationship. Since a line extends into infinity, we can say that there are an infinite number of points that can make up that line.
Now take the concept of a line segment, which is a line bounded between two distinct, finite points. Even a line segment consists of an infinite number of points because points, as dimensionless entities, must fill every nook and cranny of that line into the infinitesimal.

SETUP: Suppose we have 2 line segments, AB and CD, with CD having longer length than AB.
Twolines.png

QUESTION: Which line has more points?


2) BACKGROUND- Suppose we have a rectangle of some known area (say, 64) that is bounded by the x-axis and 2 vertical segments that are equidistant from the y-axis as shown.
DiracGraph.png

Imagine that we can change the size of the rectangle by moving the two vertical edges along the x-axis, except that we also impose the constraint that the area of the rectangle must remain the same as it was originally. Then as we move the two vertical edges closer together, the height (y-direction dimension) of the rectangle increases to maintain the area.

SETUP: Suppose we have two such cases, with one rectangle having an area of 4 and another having an area of 5. Imagine that we keep squeezing both rectangles as described above so that their heights keep getting larger and larger until the two vertical lines join.
QUESTION: Which rectangle would have the greater height?
 
Though I can't provide proofs, here's my take:

1. Neither. Each has infinitely many points over an infinitely small area, so both have infinity points, and infinity doesn't have a defined way to compare to itself. To get a concrete answer you'd have to impose a minimum distance points could be located from each other, in which case it'd of course be CD.

2. I'm not sure I understand this one. If the area has to be maintained, then the lines can never really touch. In that sense neither could ever be "taller", since they'd both approach infinity, which again can't really be compared as far as I know.

So I guess my answer to both is "neither" :p
 
JHawk said:
Though I can't provide proofs, here's my take:

1. Neither. Each has infinitely many points over an infinitely small area, so both have infinity points, and infinity doesn't have a defined way to compare to itself. To get a concrete answer you'd have to impose a minimum distance points could be located from each other, in which case it'd of course be CD.

2. I'm not sure I understand this one. If the area has to be maintained, then the lines can never really touch. In that sense neither could ever be "taller", since they'd both approach infinity, which again can't really be compared as far as I know.

So I guess my answer to both is "neither" :p
Click to expand...

/thread
 
Here's my take: Quantum Physics has stated that you can either measure the speed of a particle in a vacuum, or it's location, but determining one would result in not being able to determine the other. With this rule, let's assume these two scenarios are occuring in space.

1. The answer would be simple, wouldn't it? Maybe. Since we can only measure the speed or the location of a particle and not both, let's determine the location of these set points within each line segment, and the resulting ponits within each. Assume we're able to calculate the precise location of these points. We will come to conclude there are, in fact, more points within CD than there are in AB. Why? Because the points we're measuring do have mass, which limits the number of points within that line. If there is something that goes on into infinity, then it is not measurable.

2. Let us again assume these rectangles are in space. If both shapes are narrowing at the same time and same speed, it would be reasonable to assume that the shape with the narrower surface to begin with would be the taller object. At some point the two would have to become the same height, simply because each shape would be at it's maximum ability to narrow. The shapes wouldn't be rectangles without horizontal lines.

I realize I took some of your rules and threw them out the window, but I don't care. Defenestration is my specialty.
 
Here's my take.

1. There are actually different kinds of infinity, and even negative infinity. So, the infinity value in the line CD is greater than the infinity value in AB. Another way to explain this is CD has a greater infinity than AB's infinity. Thus, the line CD has more points in it. Here's a helpful video on infinity.

2. Two rectangles, one with the area of 4 cm2 (rectangle alpha), and one with the area of 5 cm2 (rectangle beta). I label them as such to remember which rectangle is which, making things simpler for us. So, calculating the perimeter of each rectangle, alpha's is 8cm and beta's is 8.944cm. With that, if we keep narrowing the rectangles, the height will be approximately half of the perimeter, which will make alpha's max height approximately 4cm and beta's max height 4.472cm. Making beta's height taller.

Hope you dont mind me using the SI units since you didnt provide a unit of measurement. It works the same with inches anyway.
 
Sounds like you would enjoy calculus immensely. This is one of the reasons analytic geometry is usually packaged with calculus/pre-calculus classes, as they go hand in hand in these very instances.
 
I'm going to invoke my sketchy knowledge of quantum physics here and state my belief that the Planck length makes it impossible for either line AB or CD to have an infinite number of points and that CD therefore has more points than AB.

With the rectangle, the lines can never merge as an area must be maintained, and the smallest possible width is the Planck length so at that point the rectangle with the largest area will have the greatest height.
 
Mr. Frog and Kimbal are correct, though the methods to determine the answers differ somewhat.

Simplified, CD has a larger infinite number of points than AB. The problem most people have is that we are used to dealing with hard numbers of a set unit. However, the problem is easy to understand if we realize that we are dealing with infinitely small units. There are simply more infinitely small units in the longer line.

Again, simplified, the question has a problem. The two lines cannot ever merge or the rectangles lose their second dimension and become lines. If this is allowed, then neither line is taller, since they both never end. If they cannot become lines, then the rectangle that started out with a greater area will have the greater height.

/thread
 
Just because a line consists of infinite points doesn't mean one infinity can't be larger than the other

Ap8Dg0w.png
 
ollee said:
Infinity + Infinity = Infinity.

Infinity^(2^x) = Infinity, where x > 1.
Click to expand...
You have to be careful when discussing infinity when you have no Real Number for reference. There are multiple types of infinity, and infinities of multiple sizes. In fact, there are infinite sizes of infinity, so your argument is invalid.
 
Casey said:
Sounds like you would enjoy calculus immensely. This is one of the reasons analytic geometry is usually packaged with calculus/pre-calculus classes, as they go hand in hand in these very instances.
Click to expand...

Calculus was a fun class for me especially since the analytic geometry part helped visualize fitting slopes to lines and adjoining numerous boxes under curves.

My answer:
1) In light of the foregoing discussion on different sizes of infinity, I would say that line CD has a greater infinity of points than line AB, but the question posed is bogus because infinity is not "countable."

2) In the limit that the two vertical lines join, the rectangle with an area of 5 would have a greater height than the rectangle with an area of 4.
 
lacar1601 said:
2) In the limit that the two vertical lines join, the rectangle with an area of 5 would have a height than the rectangle with an area of 4.
Click to expand...
Wait. Does this mean that a line doesn't have a width of zero? Do they join when the distance between them is something other than zero?
 
lacar1601 said:
Calculus was a fun class for me especially since the analytic geometry part helped visualize fitting slopes to lines and adjoining numerous boxes under curves.

My answer:
1) In light of the foregoing discussion on different sizes of infinity, I would say that line CD has a greater infinity of points than line AB, but the question posed is bogus because infinity is not "countable."

2) In the limit that the two vertical lines join, the rectangle with an area of 5 would have a height than the rectangle with an area of 4.
Click to expand...
1. Not countable, but we aren't dealing with quantitative properties here, we're dealing with qualitative properties, so the question is perfectly answerable.
2. Here, we are dealing with 2D objects becoming 1D, so the question is mathematically impossible. If we assume that the rectangles become infinitely NEAR to 0 width, you're correct. Assuming the adjustment to the question is correct, it now deals with quantitative properties rather than qualitative, so the answer is that the height of the rectangle with the greater area will be greater than that of the rectangle with the lesser area. The question can be compared to the graph of y=1/x^2 (alternatively, x=sqrt(1/y) ), such that the graph will infinitely approach 0y and 0x, but never quite attain either (with theoretical mathematics and imaginary numbers [which can allow you to divide by zero], it's theoretically possible; but from a practical, real-number approach, it is not).
 
MiniCacti said:
Wait. Does this mean that a line doesn't have a width of zero? Do they join when the distance between them is something other than zero?
Click to expand...

The condition for number 2 is achieved when the object is simultaneously a line and a rectangle.

It's like asking someone what phase of matter H2O is in at 0 degrees Celsius (and pressurized to 1 atm). There are two possible answers that can be argued.
 
oozinator said:
The condition for number 2 is achieved when the object is simultaneously a line and a rectangle.

It's like asking someone what phase of matter H2O is in at 0 degrees Celsius (and pressurized to 1 atm). There are two possible answers that can be argued.
Click to expand...
Actually in that case, the answer is both :p Chemistry is amazing.

Aside from the fact that you're dealing with a perfect situation, which is never the case, any amount of water will have multiple molecules in both states, and in actual fact some will be gaseous as well. However, if you are dealing with a single molecule of dihydrogen monoxide - that answer is actually none of the above. A single molecule has no state at any moment, since states are based off of the interactions between multiple molecules and each other (and the environment).

However, assuming all molecules have the exact same amount of energy (meaning they are all one state), and you can balance on exactly zero degrees, the answer is solid - since at exactly zero degrees, molecules of water lose 1 of their 3 possible movement states (lateral, rotational, and vibratory). The lost movement is lateral movement - which means the water is solid.
fox_news said:
WHAT LANGUAGE ARE YOU ALL SPEAKING!?!!??!??
Click to expand...
English for smart people.
 
TheGurw said:
Actually in that case, the answer is both :p Chemistry is amazing.

Aside from the fact that you're dealing with a perfect situation, which is never the case, any amount of water will have multiple molecules in both states, and in actual fact some will be gaseous as well. However, if you are dealing with a single molecule of dihydrogen monoxide - that answer is actually none of the above, just solid, just gas, just liquid, just solid and liquid, just liquid and gas, just solid and gas, and all three.....and all of this is simultaneous. A single molecule has no state at any moment, since states are based off of the interactions between multiple molecules and each other (and the environment).

However, assuming all molecules have the exact same amount of energy (meaning they are all one state), and you can balance on exactly zero degrees, the answer is solid - since at exactly zero degrees, molecules of water lose 1 of their 3 possible movement states (lateral, rotational, and vibratory). The lost movement is lateral movement - which means the water is solid.

English for smart people.
Click to expand...
דצכ
לגלגל
דלכללהדגלגל גלגל כלכלה לפשלה
 
You must log in or register to reply here.
Back
Top