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The Three Laws of Recursion

Like the robots of Asimov, all recursive algorithms must obey three important laws:

  • A recursive algorithm must have a base case.
  • A recursive algorithm must change its state and move toward the base case.
  • A recursive algorithm must call itself, recursively.

Recursion is the process of defining a problem (or the solution to a problem) in terms of (a simpler version of) itself. For example, we can define the operation “find your way home” as: If you are at home, stop moving. Take one step toward home.

Let’s begin our discussion of recursion by examining the first appearance of fractals in modern mathematics. In 1883, German mathematician George Cantor developed simple rules to generate an infinite set:

Cantor’s rule for an infinite set

There is a feedback loop at work here. Take a single line and break it into two. Then return to those two lines and apply the same rule, breaking each line into two, and now we’re left with four. Then return to those four lines and apply the rule. Now you’ve got eight. This process is known as recursion: the repeated application of a rule to successive results. Cantor was interested in what happens when you apply these rules an infinite number of times.

George Cantor

Dichotomy paradox – Zeno’s

“That which is in locomotion must arrive at the half-way stage before it arrives at the goal.”

— as recounted by Aristotle, Physics VI:9, 239b10

Suppose Atalanta wishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on.

Zeno’s paradox was recursive by cutting the distance in half each time to the infinitesimal. This is also how the Tortoise beat the Hair by questioning time over distance.

Recursive Function Calls

The tortoise and the Hair – the paradox of time
int factorial(int n) 
{ if (n == 1) { return 1; }
else { return n * factorial(n-1); } }

A function that does call others is called a nonleaf function. … The factorial function can be rewritten recursively as factorial(n) = n × factorial(n – 1). The factorial of 1 is simply 1. The image shows an object trace of the factorial function written as a recursive function. Each call goes in the run time stack until the base case is reached, and the the stack is popped as the result is passed to each function on the stack.

Five Factorial (5!) in recursion

What Is a Fractal?

The term fractal (from the Latin fractus, meaning “broken”) was coined by the mathematician Benoit Mandelbrot in 1975. In his seminal work “The Fractal Geometry of Nature,” he defines a fractal as “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole.”

Recursion in Nature

Looking closely at a given section of the tree, we find that the shape of this branch resembles the tree itself. This is known as self-similarity; as Mandelbrot stated, each part is a “reduced-size copy of the whole.”

The Three Laws of Robotics

Isaac Asimov was an American writer and professor of biochemistry at Boston University. During his lifetime, Asimov was considered one of the “Big Three” science fiction writers, along with Robert A. Heinlein and Arthur C. Clarke. A prolific writer, he wrote or edited more than 500 books.

  • A robot may not injure a human being or, through inaction, allow a human being to come to harm
  • A robot must obey the orders given it by human beings except where such orders would conflict with the First Law
  • A robot must protect its own existence as long as such protection does not conflict with the First or Second Laws
Partial sources: https://natureofcode.com/book/chapter-8-fractals/, Wikipedia, Google 
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Noise by Daniel Kahneman | 3 Distinctions

The Michael Shermer Show with Daniel Kahneman – Noise: A Flaw in Human Judgment

DESCRIPTION

Imagine that two doctors in the same city give
different diagnoses to identical patients. Now
imagine that the same doctor making a different
decision depending on whether it is morning or
afternoon, or Monday rather than Wednesday.
This is an example of noise: variability in
judgments that should be identical.

Shermer speaks with Nobel Prize winning
psychologist and economist Daniel Kahneman
about the detrimental effects of noise and what
we can do to reduce both noise and bias, and
make better decisions in: medicine, law, economic
forecasting, forensic science, bail, child
protection, strategy, performance reviews, and
personnel selection.

Video clip – 3 minutes

Noise by Daniel Kahneman | 3 Distinctions

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Full Video – Noise by Daniel Kahneman

Full video at https://youtu.be/5CFjERpwFys

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The Earth’s seasons

The Earth’s seasons are caused by three factors:

  • The Earth orbits the Sun once a year in a nearly circular orbit.
  • The Earth’s axis of rotation (the straight line through the center of the Earth between the north and south poles) is not perpendicular to the plane of the Earth’s orbit. The Earth’s axis is tilted by about 23.4° from the the direction perpendiular to the orbital plane.
  • The orientation of the Earth’s axis in space remains nearly constant even as the Earth revolves around the Sun. It always points in the general direction of the star Polaris.
Sun track

sun track

The result is that when the Earth is on one side of its orbit, the south pole is tilted toward the Sun (by as much as 23.4°) and the southern hemisphere experiences summer. Six months later, when the Earth is on the opposite side of its orbit, the north pole is tilted toward the Sun (by as much as 23.4°) and the northern hemisphere experiences summer. (Views of the Sun’s illumination on the Earth on any date are available here.) What we see from our viewpoint in the Earth’s northern hemisphere is that the Sun’s apparent daily track across the sky is much higher (that is, more northerly) in summer, and lower (more southerly) in winter. From horizon to horizon, the Sun’s track is longer in summer and shorter in winter; so that in summer, sunrises are much earlier and sunsets are much later than in winter. See, for example, the graphic above, or this photograph of the Sun’s paths through the sky at different times of the year.

The great circle The great circle

So we are used to the fact that the length of daylight is significantly longer in summer than winter, and most of us know that the “longest day” (that is, the day when the Sun is above the horizon the longest) is the summer solstice, around June 21, when the Sun has reached its most northerly and longest track in our sky; and the “shortest day” is the winter solstice, around December 21, when the Sun has reached its most southerly and shortest track in our sky.

It would make sense, then, for the summer solstice to also be the date at which sunrise is earliest and sunset is latest; and for the winter solstice to be the date when sunrise is latest and sunset is earliest. However, that is not what happens! Nature sometimes defies our expectations.

The local meridian is a great circle passing through the celestial poles and through the zenith of an observer’s location on the planet. Image Credit: Daniel V. Schroeder

And that is because we have not talked about one other factor in sunrise and sunset times that is not at all obvious. It is that the Sun moves across the sky, in its apparent daily track, at slightly different rates at different parts of the year. Most of the Sun’s east-to-west apparent motion in the sky is caused, of course, by the rotation of the Earth, which is quite uniform (to milliseconds per day). But a small part of the Sun’s apparent daily motion depends on the position of the Earth in its orbit around the Sun. This component of the Sun’s apparent motion varies by a small amount over the course of a year due to the elliptical shape of the Earth’s orbit and to the tilt of the Earth’s axis.

Earth’s axial tilt

Earth’s axial tilt

Continue reading The Earth’s seasons

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Three types of Breathing

Change Your Breath, Change Your Life | Lucas Rockwood | TEDxBarcelona
Change Your Breath, Change Your Life | Lucas Rockwood | TEDxBarcelona

We do it as long as we live but mostly aren’t aware
of it: breathing. In his talk Lucas breaks down the
fundamentals of yoga breathing in a way that you
can easily remember and apply to your practice.
Lucas shows us how three breathing practices
water, whiskey, or coffee – can be used as a tool and help us to overcome any situation.

With a formal yoga training background in Hot
Yoga, Ashtanga Yoga, Gravity Yoga, and the Yoga
Trapeze®, Lucas has studied with some of the
most well-respected teachers on the planet. His
most influential teachers (all of whom he studied
with personally) include Sri K. Pattabhi Jois, Paul
Dallaghan, Alex Medin, Gabriel Cousens MD, and
SN Goenka.

Lucas founded Absolute Yoga Academy in 2006,
one of the top 10 yoga teacher training schools in
the world with 2,000 certified teachers (and
counting) and courses in Thailand, Holland, the
United Kingdom, and the Philippines.

In 2013, Lucas founded YOGABODY Fitness, a
revolutionary new yoga studio business model
that pays teachers a living wage and demystifies
yoga by making the mind-body healing benefits of
the practice accessible to everyone.

In search of nutritional products designed
specifically for achy yoga students’ bodies, Lucas
worked with senior nutritional formulator Paul
Gaylon and founded YOGABODY Naturals in the
back of his yoga studio a year later. The company
has gone from strength to strength and is now a
world-renowned nutrition, education, and
publishing organization serving 81 countries.

A foodie at heart, Lucas was a vegan chef and
owned and operated health food restaurants prior
to diving deep into the yoga world. He is also a
highly acclaimed writer, radio show host, TV
personality, business consultant, weight loss
expert, and health coach. This talk was given at a
TEDx event using the TED conference format but
independently organized by a local community.

Change Your Breath, Change Your Life | Lucas
Rockwood | TEDxBarcelona

Learn more at https://www.ted.com/tedx

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Old Harry Rocks

Old Harry Rocks are three chalk formations, including a stack and a stump, located at Handfast Point, on the Isle of Purbeck in Dorset, southern England. They mark the most eastern point of the Jurassic Coast, a UNESCO World Heritage Site. photography by @uwo

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Trifid Nebula

Trifid Nebula

In the Center of the Trifid Nebula Image Credit: Subaru Telescope (NAOJ), Hubble Space Telescope, Martin Pugh; Processing: Robert Gendler

What’s happening at the center of the Trifid Nebula? Three prominent dust lanes that give the Trifid its name all come together. Mountains of opaque dust appear near the bottom, while other dark filaments of dust are visible threaded throughout the nebula. A single massive star visible near the center causes much of the Trifid’s glow. The Trifid, cataloged as M20, is only about 300,000 years old, making it among the youngest emission nebulas known. The star forming nebula lies about 9,000 light years away toward the constellation of the Archer (Sagittarius). The region pictured here spans about 10 light years. The featured image is a composite with luminance taken from an image by the 8.2-m ground-based Subaru Telescope, detail provided by the 2.4-m orbiting Hubble Space Telescope, color data provided by Martin Pugh and image

Trifid Nebula
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The comet, the whale and the hockey stick

Comet, whale, hockey stick

The Comet, the Whale, and the Hockey Stick

Image Credit & Copyright: Grand Mesa Observatory, Terry Hancock / Tom

Masterson

Closest to the Sun on March 1, and closest to planet Earth on April 23, this Comet ATLAS (C/2020 R4) shows a faint greenish coma and short tail in this pretty, telescopic field of view. Captured at its position on May 5, the comet was within the boundaries of northern constellation Canes Venatici (the Hunting Dogs), and near the line-of-sight to intriguing background galaxies popularly known as the Whale and the Hockey Stick. Cetacean in appearance but Milky Way sized, NGC 4631 is a spiral galaxy seen edge-on at the top right, some 25 million light-years away. NGC 4656/7 sports the bent-stick shape of interacting galaxies below and left of NGC 4631.

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Physics has a law that explains everything. And it’s brought to you by the number three

Jamie Seidel: News Corp Australia Network

The rule of three has become something akin to a social law of gravity — as if the number is behind everything.

Three groups of experimentalists have independently observed a strange state of matter that forms from three particles of any type and at any scale, from practically infinitesimal to infinite.

Forget pairs. They’re old pat. And 42? We still don’t know the question.

Three rings to rule them all, and in the darkness bind them ... apologies  Lord Of The Rings purists.
Three rings to rule them all, and in the darkness bind them … apologies to Lord Of The Rings purists.

Comedians insist three is the best pattern to exploit perceptions and deliver punchlines; three features prominently in titles, such as The Three Little Pigs, Three Musketeers, Goldilocks and the Three Bears; even the Romans believed three was the ultimate number: “Omne trium perfectum” was their mantra — everything that comes in threes is perfect.

Now, it seems Mother Nature may also think in threes. Especially at the very edge of physics — quantum mechanics.

A Soviet nuclear physicist first proposed the idea back in the 1970s — and was met with derision.

For 45 years number-crunchers around the world have been attempting to topple Vitaly Efimov’s idea and prove his equations wrong.

Shih-Kuang Tung of the University of Chicago
Shih-Kuang Tung of the University of Chicago holding Borromean rings, which have often been used as a symbol of unity. If one ring is removed, all three fall apart.

They’ve failed; and his “outlandish” theory is now on the point of being proven.

Most importantly, Efimov felt that sets of three particles could arrange themselves in an infinite, layered pattern. What form these layers take helps determine the makeup of matter itself.

Jump forward four decades, and technological advances now allow his groups of three quantum particles to be studied and manipulated.

The quantum condition — now known as Efimov’s state — is visible only under supremely cold conditions. Matter, when chilled to a few billionths of a degree above Absolute Zero, does strange things …

If you want the technical details, read Quanta Magazine’s article which examines the recent research papers.

Continue reading Physics has a law that explains everything. And it’s brought to you by the number three
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COVID -19: Fabric masks should have three layers of different material

10 JUN, 2020 – 04:06 

Roselyne Sachiti

Fearures, Health and Society Editor

The World Health Organisation (WHO) has published an updated guidance on the use of face masks for control of COVID-19.

WHO  wearing masks properly
WHO do’s and don’ts

In his opening remarks at a media briefing on Covid- 19 last Friday, WHO Director-General Dr Tedros Adhanom Ghebreyesus said the guidance is based on evolving evidence, and provides updated advice on who should wear a mask, when it should be worn and what it should be made of.

He said WHO has developed this guidance through a careful review of all available evidence, and extensive consultation with international experts and civil society groups.

“I wish to be very clear that the guidance we are publishing today is an update of what we have been saying for months: that masks should only ever be used as part of a comprehensive strategy in the fight against COVID.

“Masks on their own will not protect you from COVID-19,” he emphasised.

He said the updated guidance contains new information on the composition of fabric masks, based on academic research requested by WHO.

“Based on this new research, WHO advises that fabric masks should consist of at least three layers of different material. Details of which materials we recommend for each layer are in the guidelines.”

According to the latest guidelines, cloth masks should consist of at least three layers of different materials: an inner layer being an absorbent material like cotton, a middle layer of non-woven materials such as polypropylene (for the filter) and an outer layer, which is a non- absorbent material such as a polyester or a polyester blend.

“Fabric cloths (e.g., nylon blends and 100 percent polyester) when folded into two layers, provides 2-5 times increased filtration efficiency compared to a single layer of the same cloth, and filtration efficiency increases 2-7 times if it is folded into 4 layers. Masks made of cotton handkerchiefs alone should consist of at least 4 layers, but have achieved only 13 percent filtration efficiency. Very porous materials, such as gauze, even with multiple layers will not provide sufficient

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These Three Things Need to Happen Before Stocks Bottom Out, Credit Suisse Says

By Callum KeownUpdated March 16, 2020 10:29 am ET / Original March 16, 2020 9:58 am ET

When will stocks reach the low and what will the recovery look like?

Workers wearing protective gear in South Korea.ASSOCIATED PRESS
Workers wearing protective gear in South Korea.ASSOCIATED PRESS

Credit Suisse said it needed to see three conditions required for a trough in global stocks:

1. Clear-cut fiscal easing in the U.S. — which happened late on Sunday;

2. A peak in daily infection rates

3. A trough in global purchasing managers indexes, which it said could happen in May.

In the severe acute respiratory syndrome crisis, markets bottomed out a week daily new infections hit a peak, the bank’s research analysts said.

“We expect a V-shaped recovery ultimately and would be buyers of equities on a one-year view; we believe markets will rise 15-20% over the next 12 months. 

“Historically when we look at exogenous supply-side shocks, markets tend to rise very rapidly from the trough (SARS, Kobe earthquake, Suez, 1987),” they said.

The analysts, led by Andrew Garthwaite, favored stocks in Asia (a commodity-importing region on top of the virus) relative to Europe. In a new realistic worst-case scenario, U.S. earnings would drop 20% and the S&P 500 would fall to 2,200 points, they added.

They also expected “massive” monetary and fiscal stimulus. “This should enable a V-shaped recovery that by the end of 2021 could make up for much of 2020’s lost growth,” they said.

Source: https://www.barrons.com/articles/these-three-things-need-to-happen-before-stocks-bottom-out-credit-suisse-says-51584367103

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Coronavirus can stay infectious for 3 days on surfaces. But it’s still okay to check your mail.

March 13, 2020 at 8:00 AM EDT

Excerpt: Scientists studying the novel coronavirus are quickly uncovering features that allow it to infect and sicken human beings. Every virus has a signature way of interacting with the world, and this one — SARS-CoV-2, which causes the disease covid-19 — is well-equipped to create a historic pandemic.

Corona virus covid-19
Corona virus covid-19

The coronavirus may take many days — up to 14 — before an infection flares into symptoms, and although most people recover without a serious illness, this is not a bug that comes and goes quickly. A serious case of covid-19 can last for weeks.

This coronavirus can establish itself in the upper respiratory tract, said Vincent Munster, chief of the Virus Ecology Section of Rocky Mountain Laboratories, a facility in Hamilton, Mont., that is part of the National Institute of Allergy and Infectious Diseases. That enables the virus to spread more easily through coughing and sneezing. Munster and his colleagues have been studying the novel coronavirus under laboratory conditions to better understand its viability outside a host organism — in the air and on surfaces.

Those experiments found that at least some coronavirus can potentially remain viable — capable of infecting a person — for up to 24 hours on cardboard and up to three days on plastic and stainless steel.

Source and to read more: The Washington Post at https://www.washingtonpost.com/health/coronavirus-can-stay-infectious-for-days-on-surfaces/2020/03/12/9b54a99e-6472-11ea-845d-e35b0234b136_story.html


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Up the duff

up the duff. (Britain, Australia, New Zealand, Newfoundland, colloquial, slang) Pregnant. The expression UP THE DUFF originated in Australia in the 1940s.

Duff. If you describe something as duff, you mean it is useless, broken, or of poor quality. “Sometimes you have to take a duff job when you need the money.” [British, informal, disapproval]

Up the Duff by Kaz Cooke

Up the duff
Up The Duff

First published in 1999, Kaz Cooke’s best-selling Up the Duff is firmly established as the most loved and trusted book for Australian and New Zealand women on pregnancy.

This 20th-anniversary edition has been fully revised and updated. Australia’s most trusted advisor on women’s health delivers the lowdown on pregnancy, birth and how to best prepare for life with a baby. There’s no bossy-boots advice – just lots of cartoons and the soundest, sanest, wittiest advice you’ll ever get.

Kaz Cooke
Kaz Cooke

Inside there’s the crucial week-by week info on what’s happening to you and the baby, coupled with the hilarious diary of Hermoine the (even more) Modern Girl, and everything you need to know about preparing for pregnancy and birth:

Read more Up the Duff by Kaz Cooke at the link below…

Source: https://www.penguin.com.au/books/up-the-duff-2020-edition-9780143795339