Month: March, 2017

Zeno’s Paradox and the Planck Length

 

Zeno was an ancient philosopher/cult member who had some interesting thought experiments regarding infinity.
LINKS LINKS LINKS:

VSauce’s video on Supertasks:

https://www.learner.org/courses/mathilluminated/units/3/textbook/04.php
https://www.quora.com/Does-the-planck-length-explain-resolve-Zenos-Paradox
https://en.wikipedia.org/wiki/Planck_length
https://en.wikipedia.org/wiki/Reductio_ad_absurdum

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TRANSCRIPT:

Zeno was in a cult.

Back in ancient Greece, hero cults were a thing, and kind-of a big thing. A Hero in Greek culture was a man who had died and was now revered as not quite godlike, but close.

They were somewhere between man and God.

And followers would build temples in their names and establish churches around them, and devout followers would pledge their lives to their ideals.

But Zeno was a part of the first known hero cult based around a philosopher. A philosopher named Ameinias of Elea, who created the Eleatic School of Philosophy.

The Eleatics, led by the Philosopher Parmenides, believed that the senses cannot be trusted to reach truth, that only by thinking and logic can you arrive at truth.

They also believed that change and even motion was nothing but an illusion brought about by our senses.

To prove this idea, Zeno posited a series of paradoxes, the most famous being the Arrow paradox and the Achilles and the Tortoise Paradox.

Much like my spinning example earlier, the arrow paradox says that for an arrow to get from the bow to a target, it must pass through an infinite number of halfway points to get there, and the Achilles and the Tortoise paradox takes the use of movement a bit further by saying that if Achilles were racing a tortoise and gave the tortoise a head start that every time he got halfway to the tortoise, the tortoise would have moved forward a small amount, therefore even though he’s going much faster, could he ever actually catch the tortoise?

Now logic says that of course the arrow reaches the target, there’s not some invisible field keeping it from touching, and of course Achilles beats the tortoise.

These are some of the first examples of a rhetorical device called reductio ad absurdum, where you disprove a statement by showing that it inevitably ends with an absurd result.

Other versions of this that came later include Gabriel’s cake, which says if you slice a piece of cake in half, then slice one of those halves in half, and again and so on into infinity, and then stack each of those layers on top of each other, the cake would stretch to infinity.

Now these are all space paradoxes, but there’s also one that divides time called Thompson’s Lamp.

It says you take one minute and turn a lamp on and off at every halfway division, so turn it on at 30 seconds, turn it off at 15 seconds, turn it back on at 7.5 seconds, on at 3.75, and on and on until it’s blinking so fast we wouldn’t be able to see the difference. When the minute passes, is the lamp on or off?

Now, there have been countless resolutions of Zeno’s paradox through the years, some philosophers going so far as to reasoning that space and time don’t exist… which is kind-of what Zeno was going for…

But mathematical constructs worked to prove that the sum of infinitely decreasing quantities could result in a finite number.

And the idea for a limiting value to an infinite process is central to calculus, which relies on infinitesimals in order to ascribe a finite number to an infinite number of pieces.

Thanks, Sir Isaac Newton!

But maybe the easiest answer is simply that you can’t divide time and space forever. There might be a real, physical limit to smallness. Enter Max Planck.

Max Planck was one of the most substantial physicists of the early 20th century who proposed an elemental size to the fabric of spacetime, which he called a Planck Length.

The Planck Length is insanely small. It’s 1.6×10-35 meters.

To give you an idea of how small that is, This (graphic – 15 zeroes) is the length of a proton. To get to Planck’s length, you have to add not 5 (graphics change for each), not 10, not 15, but 20 zeroes. It’s one hundred quintillionth the length of a proton.

He came to this by combining three fundamental constants, Gravity, the speed of light, and his own Planck’s constant.

And from there, he created Planck Time, the time it takes for light to travel one planck length.

So if there truly is a smallest indivisible length of time and space, Zeno’s paradox is solved. But many still aren’t so sure.

 

The Toughest Ceramic Is Made From Mother-Of-Pearl Mimic

scientist

Biomimicry, technological innovation inspired by nature is one of the hottest ideas in science but has yet to yield many practical advances.

Scientists with the U.S. Department of Energy’s Lawrence Berkeley National Laboratory (Berkeley Lab) have mimicked the structure of mother of pearl to create what may well be the toughest ceramic ever produced.

Through the controlled freezing of suspensions in water of an aluminum oxide and the addition of a well known polymer, polymethylmethacrylate (PMMA), a team of researchers has produced ceramics that are 300 times tougher than their constituent components.




The team was led by Robert Ritchie, who holds joint appointments with Berkeley Lab’s Materials Sciences Division and the Materials Science and Engineering Department at the University of California, Berkeley.

Mother of pearl, or nacre, the inner lining of the shells of abalone, mussels and certain other mollusks, is renowned for both its iridescent beauty and its amazing toughness.

Nacre is 95-percent aragonite, a hard but brittle calcium carbonate mineral, with the rest of it made up of soft organic molecules. Yet nacre can be 3,000 times (in energy terms) more resistant to fracture than aragonite.

ceramic

No human-synthesized composite outperforms its constituent materials by such a wide margin. The problem has been that nacre’s remarkable strength is derived from a structural architecture that varies over lengths of scale ranging from nanometers to micrometers.

Two years ago, however, Berkeley Lab researchers Tomsia and Saiz found a way to improve the strength of bone substitutes through a processing technique that involved the freezing of seawater.

This process yielded a ceramic that was four times stronger than artificial bone. When seawater freezes, ice crystals form a scaffolding of thin layers.

ceramic

These layers are pure ice because during their formation impurities, such as salt and microorganisms, are expelled and entrapped in the space between the layers. The resulting architecture roughly resembles that of nacre.

In this latest research, Ritchie, working with Tomsia and Saiz, refined the freeze-casting technique and applied it to alumina/PMMA hybrid materials to create large porous ceramic scaffolds that much more closely mirrored the complex hierarchical microstructure of nacre.

To do this, they first employed directional freezing to promote the formation of thin layers (lamellae) of ice that served as templates for the creation of the layered alumina scaffolds.

ceramic

After the ice was removed, spaces between the alumina lamellae were filled with polymer.

For ceramic materials that are even tougher in the future, Ritchie says he and his colleagues need to improve the proportion of ceramic to polymer in their composites.

The alumina/PMMA hybrid was only 85-percent alumina. They want to boost ceramic content and thin the layers even further. They also want to replace the PMMA with a better polymer and eventually replace the polymer content altogether with metal.

ceramic

Such future composite materials would be lightweight and strong as well as tough, he says, and could find important applications in energy and transportation.

This research was supported by DOE’s Office of Science, through the Division of Materials Sciences and Engineering in the Basic Energy Sciences office.

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