Infinity within the finite

This is another of those "certain point of view" examples.

When i was first exposed to it, my first reaction was that cant be right, but from a maths pov it is.

It seems counter intuitive that infinity could be found within the finite.
How could the contents of the box be bigger than the box its inside.

A cupcake is finite, once you eat it its gone.

But Maths has a different take.

The number 1 has a finite value, one cupcake is one cupcake

But you can take one and divide it in half, discard half and take the remainder and divide that in half, and so on and so on.

As long as you have half, it can be divided in half.

Its sometimes expressed as a Sierpiński triangle



https://ibmathsresources.com/2020/09/26/sierpinski-triangle-a-picture-of-infinity/

Oh, yes, infinity is something many people struggle with at first. And with good reason, too, there are subtleties to be careful with, and far too often one is tempted to try and treat it like one would any other number. There is, for instance, and in keeping with roughly the topic you bring up here, in measure theory a very common assignment given to undergraduate students:

"Construct an open set that contains all rational numbers between and including 0 and 1, but whose Lebesque measure is less than 1."

Rational numbers here means any number that can be expressed as a fraction of integers, a ratio (hence the name), and only those.
An open set is a set where for any element inside one can also identify a neighborhood of that element that is still completely inside. So in other words, if you pick some point in it, there is a minimal distance you can go in any which direction away from that point without ever leaving the set.
Finally, the Lebesque measure is a function that maps subsets of a real space onto non-negative real numbers (or infinity, as the case may be), expressing the input set's "size". So in this context it would be the total length of the set we are looking for, ignoring any self-overlaps we may incur with our construction.

And, interestingly enough, eventhough the rationals are a dense set, i.e. there is no lower bound on the distance two can have - and consequently if one were to pick a finite but non-zero length interval, there would be infinitely many rationals in it, no matter how small a length we pick - the exercise is still not only soluble, but in fact one can use what ever construction solves it to produce a set that satisfies all those conditions and has an arbitrarily small non-zero measure.

Neither of us can know the calibre of the minds of our audience.

And i for one reading the posts and journals would never underestimate them, ive seen brilliance in them that should not surprise me, but in (delighted) honesty does.

None the less, lets take no risks. We should leave no one behind , lets keep this as simple as we can, this audience imo would be poorly served by any post that has the appearance of "baffling them with bullshit"

Consider it your test, your challenge if you will

Convey a complex idea in its simplest explanation, we do not patronize our audience in doing so, its a personal exercise in best practice communication

WoodfordJedi wrote: Neither of us can know the calibre of the minds of our audience.

And i for one reading the posts and journals would never underestimate them, ive seen brilliance in them that should not surprise me, but in (delighted) honesty does.

None the less, lets take no risks. We should leave no one behind , lets keep this as simple as we can, this audience imo would be poorly served by any post that has the appearance of "baffling them with bullshit"

Consider it your test, your challenge if you will

Convey a complex idea in its simplest explanation, we do not patronize our audience in doing so, its a personal exercise in best practice communication

It's entirely possible, likely even, that my comprehension is subpar at best. But, to be fair, I've been pretty baffled by a lot of your posts as well and have had to just acknowledge that I have so little understanding of the point you're trying to make sometimes that I wouldn't even know where to start asking questions.

Edit: To be clear, this initial post isn't one of them. I get the "travelling halfway forever and never arriving" thing. Just, ya know, what's good for the goose is good for the gander, and practice what you preach, and all that jazz.

What is "the simplest explanation" is a highly subjective thing to begin with, not to mention there is only so far one can go before further simplification starts amounting to mischaracterization, unless one is happy to accept ridiculous post lengths even by my own standards. I try and keep things reasonably simple. You notice, after presenting the task I explained key terms used in it that any student faced with that task would be already familiar with. I appreciate that some of what I say will, no matter how much I try, still be out of reach, if slightly, to some readers at first, and that's absolutely not a problem for any of us. I'm happy to elaborate on, or explain, or clarify anything that wasn't quite entirely clear at first. If anything, I expect an approach like that to be more effective, as this way I can address the specific knowledge gaps or misunderstandings my actual readers really face, rather than wasting resources in attempts at foreseeing imperfections in the communication, only to end up looking patronizing or condescending in the end, for over-explaining things some might have never had any trouble grasping as it stood.

This is all to say, in short: If something is here left unclear, and you, dear reader, are curious to clear it up, by all means, ask, and feel no shame in having to.

The human potential is amazing and difficult to express, and even more of an idea to practice. In it, the potential is there in so many different variations it can blow your mind and often does. When I read "from a certain point of view," honestly, I read that someone does this or it could be a possibility. One of the biggest things for me in my own Modern-day Jedi path is the NOW. What can I do now?
Thank you for posting the math problem so I could see it. I ain't good at math but I will tell you this, If I turn it into money - I can understand it.
Do you know of anyone who uses this equation for anything or anyone? In any way? Got a idea attached to it? Do you use it for anything?

I don't think this particular equation works for the way I use things. Half of something is usually never enough in the real world. I do much self-reflection and I can say, the "half-sies" in my path was more of a hack,or slice, in half, blindly.
I will tell you this, in ministry I do use one practice... 1 for 1. For every discouraging thing I see or hear, I try to GIVE one encouraging thing back. I will find someone. I try. What happens when you turn the triangle upside-down? What happens when you add, instead of subtract? What happens when you multiply, instead of divide?

River wrote: . But, to be fair, I've been pretty baffled by a lot of your posts as well and have had to just acknowledge that I have so little understanding of the point you're trying to make sometimes that I wouldn't even know where to start asking questions.

Edit: To be clear, this initial post isn't one of them. I get the "traveling halfway forever and never arriving" thing. Just, ya know, what's good for the goose is good for the gander, and practice what you preach, and all that jazz.

I shall endeavor to do better, Thank You

Carlos.Martinez3 wrote: .
Do you know of anyone who uses this equation for anything or anyone? In any way? Got a idea attached to it? Do you use it for anything?

For me its an exercise in perspective, that what we consider reality sometimes has hidden dimensions, many of them counter intuitive.

The Sierpiński triangle is a fractal.

The whole universe is fractal, and so there is something joyfully quintessential about Mandelbrot's insights.

Fractal mathematics has many practical uses, too - for example, in producing stunning and realistic computer graphics, in computer file compression systems, in the architecture of the networks that make up the internet and even in diagnosing some diseases.

Fractal geometry can also provide a way to understand complexity in "systems" as well as just in shapes. The timing and sizes of earthquakes and the variation in a person's heartbeat and the prevalence of diseases are just three cases in which fractal geometry can describe the unpredictable.

Another is in the financial markets, where Mandelbrot first gained insight into the mathematics of complexity while working as a researcher for IBM during the 1960s.

Mandelbrot tried using fractal mathematics to describe the market - in terms of profits and losses traders made over time, and found it worked well.

https://www.bbc.com/news/magazine-11564766

I think strictly speaking a lot of physical applications (at least the ones I've seen) of fractals are really more akin to space filling curves. The whole "the universe is fractal because your neuron looks like a galaxy" thing really is just a cosmetic similarity in my opinion. The finance application seemed a little counterintuitive, so I looked it up and it's not really related to fractals but rather a "buy low sell high" concept couched in post-hoc analysis of local maximums/minimums with enough caveats for it to seem not particularly useful.

Math once you get past calculus is fun like this.

Rex wrote: I think strictly speaking a lot of physical applications (at least the ones I've seen) of fractals are really more akin to space filling curves. The whole "the universe is fractal because your neuron looks like a galaxy" thing really is just a cosmetic similarity in my opinion. The finance application seemed a little counterintuitive, so I looked it up and it's not really related to fractals but rather a "buy low sell high" concept couched in post-hoc analysis of local maximums/minimums with enough caveats for it to seem not particularly useful.

Math once you get past calculus is fun like this.

Some Practical fractal applications

Applying Fractals to Trading

Most charting platforms now provide fractals as a trading indicator. This means traders don't need to hunt for the pattern. Apply the indicator to the chart, and the software will highlight all the patterns. Upon doing this, traders will notice an immediate problem: this pattern occurs frequently.

Fractals are best used in conjunction with other indicators or forms of analysis.

https://www.investopedia.com/articles/trading/06/fractals.asp

With respect to markets, one can see that stock prices move in fractals. Due to this characteristic, technical analysis is possible: in the same way that the patterns of fractals repeat themselves along all time frames, stock prices also appear to move in replicating geometric patterns through time.

https://www.investopedia.com/terms/f/fractal-markets-hypothesis-fmh.asp

Fractal geometry has become very useful in the understanding of many phenomena in various fields such as astrophysics, economy or agriculture and recently in medicine. After a brief intuitive introduction to the basis of fractal geometry, the clue is made about the correlation between Df and the complexity or the irregularity of a structure. However, fractal analysis must be applied with certain caution in natural objects such as bio-medical ones. The cardio-vascular system remains one of the most important fields of application of these kinds of approach. Spectral analysis of the R-R interval, morphology of the distal coronary arteries constitute two examples. Other very interesting applications are founded in bacteriology, medical imaging or ophthalmology. In our institution, we apply fractal analysis in order to quantitate angiogenesis and other vascular processes.

https://pubmed.ncbi.nlm.nih.gov/10783467/