Absurd Geometry by Roger Pink

Just a few more examples of how not using actual numbers in mathematics causes problems with how things fit into reality.

Unfortunately the described method for making a Menger sponge will never yield zero volume, even with an infinite growth in surface area.

Continually subtracting any given percentage ratio apart from 100% can never result in zero- just a proportionately lessening convergence.

I think you can argue the exact opposite as well...that you will never get an infinite area with a proportionately expanding divergence.

Jon.

True but it will take forever to prove it!!!

Looks like Fractal Geometry.

Which Benoît Mandelbrot is a pioneer. Of which we own allot to for entertainment such as animated graphics among other things.

Do these things really have infinite surface area or circumference?

When the phrase ad infinitum is used to describe the construction it takes it away from reality. It's not possible to do any of the steps to infinity from a time perspective. Furthermore, one can not do it from a physical perspective because at some point you are at the atomic level and you begin to split atoms and deal with fractions of atomic size.

In the end, you do end up with something finite.

There's fields of mathematics that can take these to their convergence and prove what Roger is presenting.

Once you get to the atomic level you transfer to the mathematicians' universe where they all walk around with long capes and hoodies and mutter about partial derivatives and triple integrals and the limits as N approaches infinity....

Jon.

And that's kind of my point. It's only a mythical world where it exists. But it's presented to us and others in the non-mathematical world as something real (horn or box or star or whatever) with finite this but infinite that. It's only a mental construct, not something real.

Don't get me wrong, I think these things are cool, interesting and have a place.

But the real world is annoyingly populated with theoretical building blocks.

How long is a coastline? Putting aside the shear impossibility of two "points" in the real world, the answer depends on the resolution of your measurement. If you trace the outlines of all the sand grains at the water's edge (assuming an artificially static interface), you'll get a number much higher than taking a straight-line path.

You would have to put an imaginary sea level in there first. Last I looked the ocean moved around a fair bit!

It's only a mythical world where it exists.

If its a fractal, I have to disagree, What I understand Fractals are in nature......

And to answer Lynn W. here is Fractal Dimension of Coastlines with the accuracy is determined by the number of iterations in this case (∞) infinity number of iterations would do it.

I disagree.

Even if it's fractal, it's still finite in that as you zoom in on the coastline you will get to the point that your scale is on the atomic level and you will now be dealing with discrete points as opposed to distances that can be zoomed in on. The practical implementation is that we are not able to measure any smaller than a certain value. Are there things smaller than the smallest thing we know of? Possibly, but we don't know. Is it possible to find such a thing? Maybe, be again we do not know.

My point is that even in nature, the leaf or coast line still has a finite number of atoms which make it up.

True

Than not use infinity amount of iterations, and use the among of iterations necessary to match our capacity to be able to measure.

And as our ability to measure to a smaller value, add to the iterations,..... Because we do have an infinity amount.

;)

Ow...that made my head hurt.

Then zoom in further, inside the electron cloud and the nucleus. Then to the subatomic level. I may be wrong, but I think the latest LHC results indicate that we still can't say we've measured the smallest thing(s) in existence.

Reality is an approximation. (c)

The fact that ''Gabriels' Horn'' has finite volume but infinite surface area merely points to the fact that taking the limit of something out to infinity demonstrates that there is a flaw in the logic of doing so. Otherwise, they would both be equal quantities.

Starting with the fact that ''infinity'' is that which can never really be reached...

(except by Buzz Lightyear who does reach it, but keeps coming back, so he says...)