# Logical argument using infinity

#### arfa brane

##### call me arf
Valued Senior Member
The Euclidean plane is infinite in extent, so you can have a circle with a radius that gets as close to infinite as you like.

If you construct a pair of perpendicular lines, they intersect such that a pair of circles with a finite radius exist where each circle is tangent at two points to either line. Any circle tangent to both lines has a centre lying on the line bisecting one of the right angles.

As the circle radius increases (and the centre gets farther from the point of intersection of the pair of lines) the perpendicular lines are tangent to the circle 90 degrees apart, and as the radius approaches infinity, the points of tangency remain at zero degrees to the circle. But a circle with infinite radius has a straight line between any two points on it, so at infinity the two tangent lines are parallel and 180 degrees apart.

To restore a symmetry, you need another line parallel to one of the first two. A circle tangent at two points must lie between the parallel lines, and have the same radius everywhere in the plane.

However, the two parallel lines are paired with two perpendicular lines which are 180 degrees apart at infinity (divergent), so the locally parallel lines have to be perpendicular in the same place (convergent). So the tangent circle of constant radius shrinks to a point, or prosaically, a constant distance doesn't exist there.

So, where's the logical argument?!
EB

So, where's the logical argument?!
EB
Are you asking what geometry has to do with logic?
Do you mean what logic can be used with parallel or perpendicular lines? Or with circles tangent to them?

Aristotle said to Zeno "If tyrants apply your paradox to a taxation system we would need infinite money to pay for it" and Zeno responded "My Bad."

I don't think I entirely get the point from the word description. Perhaps a diagram would help.

I don't think I entirely get the point from the word description. Perhaps a diagram would help.
Ha ha. Like anyone can draw a circle with an infinite radius!

Aristotle said to Zeno "If tyrants apply your paradox to a taxation system we would need infinite money to pay for it" and Zeno responded "My Bad."

Replace "need infinite money to pay for it" with "have infinite transactions."

But a circle with infinite radius has a straight line between any two points on it, so at infinity the two tangent lines are parallel and 180 degrees apart.
This is the part I don't understand.

A circle of infinite radius is still a circle. It doesn't turn into a straight line.

This is the part I don't understand.

A circle of infinite radius is still a circle. It doesn't turn into a straight line.
But is a circle not a ideal object which needs no existence in reality.

Like Plato's solids?

Would the question be different if we posited the dimensional geometric properties of an infinite cube?

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A circle of infinite radius is still a circle. It doesn't turn into a straight line.
Yes it does, according to my 1st year calculus lecturer (a professor with a Ph.D.).

For a circle with a large radius, the curvature for any arc of the circle is close to zero, and as the radius increases without limit the curvature along any arc goes to zero. Therefore for a circle with infinite radius, any arc between any two points is a straight line.

The other way to look at it is as a boundary of the plane. At infinite distance from any point, there is an infinite horizon, a circle with infinite radius that any projected line intersects. Since the circle is infinite, it has no curvature.

The curvature of a circle with radius R is 1/R. When R = ∞, 1/R = 0. q.e.d.

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R is 1/R. When R = ∞, 1/R = 0. q.e.d.
When R = ∞, 1/R = 0 ?

What happened to the universe?

What happens if you draw a perfect circle or a sphere and "inflate" the boundaries infinitely large? Parallell lines anywhere?

I always thought that a circle is a logical object, not restricted to physics.

I heard somewhere that any equation that rest on infinity is unsolvable?

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It seems that: Just because it's logical doesn't mean it's rational.
Is infinity any more real/rational than leprechauns or unicorns?

I heard somewhere that any equation that rest on infinity is unsolvable?
Not if infinity is assumed to be like any other number (except, paradoxically, infinity isn't a number).

But that doesn't stop you defining a function with an infinite point in it. Like say, the Mobius transformation whose domain is the complex plane with such a point, at infinity. You can also assume that 1/R is well-defined when R is infinite, or that division by infinity is well-defined.
It seems that: Just because it's logical doesn't mean it's rational.
Is infinity any more real/rational than leprechauns or unicorns?
Indeed, logic doesn't have to relate rational arguments to rational arguments, it just has to be consistent.

Is infinity real? Is there more of the universe than we can see? How much more?

Is infinity real? Is there more of the universe than we can see? How much more?
Well, if the universe is finite, then what could possibly be infinite? A permittive condition? Nothing at all?

For a circle with a large radius, the curvature for any arc of the circle is close to zero, and as the radius increases without limit the curvature along any arc goes to zero. Therefore for a circle with infinite radius, any arc between any two points is a straight line.
No. A straight line is the shortest distance between two points. Consider two points on opposite sides of a circle of radius $R$. The straight-line distance between them is $2R$, whereas the distance along the arc is $\pi R$. The distance along the arc is a factor of $\pi /2$ times larger than the straight-line distance, for a given circle of any radius, including a circle of infinite radius.

Of course, you might point out that if $R\rightarrow \infty$ then the straight-line distance and the distance along the arc are both the same (i.e. both $\infty$) and therefore even though the arc length is $\pi /2$ times larger than the straight-line distance, they are still both the same distance.

This is the kind of thing that happens when you start talking about infinities. It's probably what's causing your problem with the tangents to the circle that are at right angles.

Of course, you might point out that if R→∞ then the straight-line distance and the distance along the arc are both the same (i.e. both ) and therefore even though the arc length is π/2 times larger than the straight-line distance, they are still both the same distance.
That doesn't make sense; you can't distinguish the arc length and the straight-line distance like you're trying to do there.
It's probably what's causing your problem with the tangents to the circle that are at right angles.
It isn't a problem, nor is it "my" problem; the tangents at right angles exist locally, where circles have a finite radius.
The statement that a line is an infinitely large circle can ... be framed in terms of curvature. For each R > 0 let C be a circle of radius R; then it has curvature 1/R. Then in the limit as R → ∞, the curvature goes to 0. In some sense, at "R = ∞" we end up with curvature "1/∞ = 0," a straight line. The way this is interpreted more formally is that as R → ∞, small segments of the circle C become better and better approximations to straight lines. Armed with this interpretation, we can amend the above bolded statement: there is only one kind of curve of constant curvature: circles, including the infinite ones (lines).

Addendum: This is only one way to view the statement that lines are "infinite circles." Another common way to think about this is in the context of projective geometry, where lines can "become circle" by "closing them up" by adding "points at infinity." Also, you can think about them in the context of inversion geometry (geometry of the map ↦ 1/ in the complex plane) and more generally Mobius transformations, which overall tend to map lines and circles to other lines and circles; if one makes the convention that a line is an infinite circle, then Mobius transformations can be said to map circles to circles, which makes their description somewhat neater.

Sorry, a typo. that phrase about inversion geometry should read "geometry of the map z ↦ 1/z"
P.S. If I'm right then James R is wrong with
The distance along the arc is a factor of π/2 times larger than the straight-line distance, for a given circle of any radius, including a circle of infinite radius.

. . . since it isn't true for a circle of infinite radius; it doesn't hold because not only is a constant distance meaningless (the distance between parallel lines is zero), but the ratio of a circle to its diameter is also meaningless, there isn't any factor of π/2, at infinity.

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The kind of infinity in the OP is the kind that relies on the notion of projection. Or of a line (a radius R) or pair of lines being extended without limit.

So define a function, f(R) =1/R. Clearly f decreases as R increases and stays finite. But the limit of f is zero, as R "goes to" infinity, a place, not a number.
Then show that circles tangent to perpendicular lines, with finite radius r, are in the same place (infinity) when f(r) = 0, as a circle with infinite radius centred on the point of intersection of the two perpendicular lines (the infinite boundary of the plane).

I don't understand the OP either. As James R said, all circles have the same shape over a 360 degree arc, regardless of the radius. It seems to me that all Arf is doing when he lets the radius approach infinity, is taking into consideration an arc which spans a smaller and smaller number of degrees of the whole circle. If so, he could do the same with a circle of any finite radius, by simply considering an arc spanning fewer and degrees. This is one of the reasons flat-earthers exist, because at the local level, the curvature of the earth is small enough that it seems flat locally. Or am I missing something here?