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Daniel Estrada
Robot. Made of robots.
Robot. Made of robots.


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Kepler, the Hydrogen Atom and the Fourth Dimension

Kepler loved Platonic solids. He even made a theory of the planets' orbits where they moved on spheres between a nested octahedron, icosahedron, dodecahedron, tetrahedron, and cube. It fit the data pretty well... but not perfectly. He eventually realized that the orbits aren't circles: they're ellipses!

We now know that the force of gravity drops off as the inverse square of the distance. That's why planets move in elliptical orbits. Platonic solids have nothing to do with it.

But last week, +Greg Egan, +Layra Idarani and I found a solid mathematical connection between the Platonic solids and the inverse square force law. It involves quantum mechanics, and a detour into the 4th dimension!

The force between the electron and proton in a hydrogen atom also obeys an inverse square law... but we need to use quantum mechanics to understand it. Instead of having a definite position, the electron has a wavefunction saying how likely it is that you'll find it in any location.

Amazingly, the wavefunction for the electron in a hydrogen atom can also be described as a function on a 3-sphere: a sphere in 4 dimensions. We can rotate a sphere in 4 dimensions and turn one wavefunction into another with the same energy.

This is a 'hidden symmetry' of the hydrogen atom. It's sort of obvious how 3-dimensional rotations act on a hydrogen atom. The amazing part is that you can do 4-dimensional rotations.

This picture, made by Egan, shows a wavefunction on the 3-sphere. It's positive in the blue regions, negative in the yellow regions, and almost zero where it's black. We’re seeing a moving slice of the 3-sphere, which is an ordinary sphere. When the image fades to black, our moving slice is passing through a sphere where the function is zero.

This particular function describes a very symmetrical state of a hydrogen atom, where it has a definite energy and 7200 symmetries, coming from the symmetries of the group of symmetries of a dodecahedron.

That's quite a mouthful! Luckily, I've already written a nice gentle explanation of this stuff with lots of pictures, including nice pictures of Kepler's original Platonic solid theory, start here:

Make sure to follow the links to Egan's explanation:

We worked out a lot of this math with +Layra Idarani, so if you're a mathematician you'll also enjoy his work:

It generalizes a lot of this stuff to even higher dimensions!

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From the icosahedron to E8

I love the icosahedron. One reason is that in math, every sufficiently beautiful thing is connected to all other beautiful things. So, the icosahedron sits in an intricate shimmering web of connections — a web that begs to be explored!

For example: the icosahedron has 60 rotational symmetries. You can describe rotations using quaternions. But a quaternion and its negative give same rotation. So, the 60 rotational symmetries of the icosahedron correspond to 120 quaternions.

And these 120 quaternions are the vertices of the 4-dimensional shape shown here!

Of course we had to project it down to 3d, and then 2d, to fit it onto your screen. So you're seeing a squashed version - but it's still pretty. The real thing has 600 faces that are all tetrahedra, so it's called the 600-cell. It's very symmetrical - it's a 4-dimensional Platonic solid.

Yes: the symmetries of a 3d Platonic solid, the icosahedron, can give the vertices of a 4d Platonic solid. How cool is that?

But that's just the start. I just wrote an article that explains how to get from this 4d shape to the E8 lattice - a beautiful lattice in 8 dimensions:

In fact I explain two ways to do this. I don't know how these two ways are related. But I feel they must be. So that's a good puzzle for anyone out there who feels up to it. If you figure it out, please let me know.

Here are some easier puzzles.

We can take the vertices of the 600-cell to be

(±½, ±½, ±½, ±½)

(±1, 0, 0, 0)

½(±Φ, ±1, ±1/Φ, 0)

and even permutations thereof, where Φ = 1.618... is the golden ratio. There are

2⁴= 16

points of the first kind,

2 × 4 = 8

points of the second kind, and

2³ × 4! / 2 = 96

points of the third kind, for a total of

16 + 8 + 96 = 120

points — which is good, since the 600-cell has 120 vertices.

The 16 points of the first kind are the vertices of a 4-dimensional orthoplex, the 4d analogue of an octahedron. The 8 points of the second kind are the vertices of a 4-dimensional hypercube, the 4d analogue of a cube. Taking these together, we get the 16 + 8 = 24 vertices of a 24-cell, another regular polytope in 4 dimensions.

Now, note that

120/16 = 7½

Puzzle 1. Can we partition the 120 vertices of the 600-cell into the vertices of 7 orthoplexes and one hypercube?

Puzzle 2. If so, how many ways can we do this?

On the other hand,

120/8 = 15

Puzzle 3. Can we partition the 120 vertices of the 600-cell into the vertices of 15 hypercubes?

Puzzle 4. If so, how many ways can we do this?

On the third hand,

120/24 = 5

Puzzle 5. Can we partition the 120 vertices of the 600-cell into the vertices of five 24-cells?

Puzzle 6. If so, how many ways can we do this?

By the way, the picture here was drawn using Robert Webb’s Stella software:

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In certain crystals you can knock an electron out of its favorite place and leave a hole: a place with a missing electron. Sometimes these holes can move around like particles. And naturally these holes attract electrons, since all they are is places an electron would want to be.

Since an electron and a hole attract each other, they can orbit each other. An orbiting electron-hole pair is a bit like a hydrogen atom, where an electron orbits a proton.

An orbiting electron-hole pair is called an exciton, because it's really just a special kind of 'excited' electron - an electron with extra energy, not in its lowest energy state where it wants to be.

An exciton usually doesn't last long: the orbiting electron and hole spiral towards each other, the electron finds the hole it's been seeking, and it settles down.

But excitons can last long enough to do interesting things. In 1978 the Russian physicist Abrikosov wrote a short and very creative paper in which he raised the possibility that excitons could form a crystal in their own right! He called this new state of matter excitonium.

In fact his reasoning was very simple.

Just as electrons have a mass, so do holes. That sounds odd, since a hole is just a vacant spot where an electron would like to be. But such a hole can move around, and it takes force to accelerate it, so it acts just like it has a mass! The precise mass of a hole depends on the nature of the substance we're dealing with.

Now imagine a substance with very heavy holes.

When a hole is much heavier than an electron, it will stand almost still when an electron orbits it. So, an exciton will be very similar to a hydrogen atom, where we have an electron orbiting a much heavier proton.

Hydrogen comes in different forms: gas, liquid, solid... and at extreme pressures, like in the core of Jupiter, hydrogen becomes metallic. So, we should expect that excitons can come in all these different forms too!

We should be able to create an exciton gas... an exciton liquid... an exciton solid.... and under certain circumstances, a metallic crystal of excitons. Abrikosov called this metallic excitonium.

People have been trying to create this stuff for a long time. A lot of people claim to have succeeded. A new paper claims to have found something else: a Bose-Einstein condensate of excitons. There's a pretty good simplified explanation at the University of Illinois website:

However, the picture here shows domain walls moving through crystallized excitonium - I think that's different than a Bose-Einstein condensate!

I urge you to look at Abrikosov's paper. It's two pages long and beautiful:

• A. A. Abrikosov, A possible mechanism of high temperature superconductivity,

He points out that previous authors had the idea of metallic excitonium. Maybe his new idea was that this might be a superconductor - and that this might explain high-temperature superconductivity. The reason for his guess is that metallic hydrogen, too, is widely suspected to be a superconductor.

Later Abrikosov won the Nobel prize for some other ideas about superconductors. I think I should read more of his papers.

Puzzle 1. If a crystal of excitons conducts electricity, what is actually going on? That is, which electrons are moving around, and how?

This is a fun puzzle because an exciton crystal is a kind of abstract crystal created by the motion of electrons in another, ordinary, crystal.

Puzzle 2. Is it possible to create a hole in excitonium? If so, it possible to create an exciton in excitonium? If so, is it possible to create meta-excitonium: an crystal of excitons in excitonium?

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The California effect

Before California started its cap-and-trade program, it was the 8th-largest economy in the world and the 12th-largest emitter of greenhouse gases. Now it's the 6th-largest economy and the 19th-largest emitter.

Here's how it works. Each year California sets a limit on greenhouse gas emissions - that's the cap. It allocates or auctions off permits to polluting companies. Companies that reduce their emissions below their allowed level can sell their permits to other companies - that's the trade. This gives them an incentive to do better.

Each year, the state reduces the cap by 3%. This will gradually make the permits more expensive. By 2030, the goal is for greenhouse gas emissions to be 40% less than than in 1990.

This has lots of effects, some of which spread across the US.

Energy companies can reduce their emissions by switching from coal to natural gas and investing in clean energy like wind and solar. Manufacturers can improve their energy efficiency. And all businesses can offset up to 8% of their emissions by buying credits from organizations that grow forests and do other things to soak up CO2.

The last part is especially interesting, because these organizations don't need to be in California. This map shows you where they are.

Across the US, more than 300 projects in 30 states are coming online thanks to the California effect. They've already sucked up 69 million tonnes of CO2.

You can read about some examples here:

One is the Clinch Valley Conservation Forestry Program in Virginia. It was started by Stuart Land & Cattle, one of the largest cattle farms east of the Mississippi River, which owns a lot of land. It now protects 22,000 acres of forest.

So: states don't need to wait for the US federal government to catch up. They can start doing good things now.

Much of my post here was paraphrased from the above article, which was written by Brendan Borrell for the magazine of the Nature Conservancy, of which I'm a member.

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Physics and number theory - the math of Minhyong Kim

I met Minhyong in 1987 when I was a postdoc at Yale and he was a grad student there. He came to my course on nonlinear wave equations. He and his roommates started a tradition of "ice cream math socials". We had a huge amount of fun talking about math, physics and everything else. We've been doing it ever since, whenever we can. Now he's at Oxford, and I last saw him in his room at Merton College, which is about the closest thing to Hogwarts Academy that really exists. (They've got secret passages, and a library with medieval books in Latin chained to the desks.)

As a grad student he started out working with Greg Zuckerman on math connected to string theory, but then - much to my shock - he switched to working with Serge Lang on number theory. Ever since, he's done number theory influenced by physics.

If the connection sounds fantastical it’s because it is, even to mathematicians. And for that reason, Kim long kept it to himself. “I was hiding it because for many years I was somewhat embarrassed by the physics connection,” he said. “Number theorists are a pretty tough-minded group of people, and influences from physics sometimes make them more skeptical of the mathematics.”

But now Kim says he’s ready to make his vision known. “The change is, I suppose, simply a symptom of growing old!”

This is a quote from the Quanta article below. Read the whole thing!

The link is not as fantastical as the article makes it sound. For over a century, the most exciting parts of number theory are those that connect it to geometry. If you're looking for integer solutions to an equation like

x² + y² = z²

you might as well divide by z² and look for points with rational coordinates on the unit circle, so you're studying a curve "defined over the rationals". This idea is the tip of an iceberg called arithmetic geometry, where you accent the "e" in "arithmetic" so it becomes an adjective.

Also, for over a century the most exciting parts of fundamental physics are the parts that connect physics to geometry. Einstein realized that gravity is just the curvature of spacetime. Now we know the other forces are also various kinds of curvature, more abstract but still geometrical. By now, almost all the most exciting parts of geometry are inspired by physics. In particular, while string theory has been a dud when it comes to making experimental predictions, it's been an enormous boon for geometers.

So it makes some basic sense to connect number theory to the math coming from physics. But it's not so easy to make this work in practice! Especially if you're trying to impress number theorists.

The article says:

So far, Kim has made no mention of physics in his papers. Instead, he’s written about objects called Selmer varieties, and he’s considered relationships between Selmer varieties in the space of all Selmer varieties. These are recognizable terms to number theorists. But to Kim, they’ve always been another name for certain kinds of objects in physics.

Unfortunately this means I can't read his papers on Selmer varieties for information on how they're connected to physics. I'll have to ask him sometime.

But in some other papers, Minhyong is very explicit about the link between prime numbers, knots and a physical theory called Chern-Simons theory. It turns out that prime numbers are a lot like knots. They can get linked with each other... and you can use Chern-Simons theory to measure how linked they are!

This stuff is enormously fun. Especially because I remember Zuckerman shushing Minhyong and me when we were whispering at a talk by Stasheff on Chern-Simons theory!

If you're into math, check out this one:

• Minhyong Kim, Arithmetic Chern-Simons Theory I,

Abstract. In this paper, we apply ideas of Dijkgraaf and Witten on 2+1 dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number fields. In the first three sections, we define classical Chern-Simons functionals on spaces of Galois representations. In the highly speculative section 5, we consider the far-fetched possibility of using Chern-Simons theory to construct L-functions.

You can see his self-deprecating sense of humor in that last sentence. I guess he wants to keep the skeptical number theorists at bay.

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A universal snake-like continuum

It sounds like jargon from a bad episode of Star Trek. But it's a real thing. It's also called the pseudo-arc. It's a monstrous object that's impossible to draw.

The pseudo-arc is the limit of a sequence of curves like those shown here, which get more and more wiggly. The pseudo-arc is infinitely wiggly — so wiggly that any picture of it is useless.

In fact Wayne Lewis and Piotr Minic wrote a paper about this, called 'Drawing the Pseudo-Arc'. That's where I got these pictures, which show stages 5, 6, 8 and 10 of drawing the pseudo-arc. The paper also shows stage 200 - and it's a big fat ugly black blob!

But the pseudo-arc is beautiful if you see through the pictures to the concepts, because it's a universal snake-like continuum. Let me explain. This takes some math.

A continuum is a nonempty compact connected metric space. When I think of this I think of a curve, or a ball, or a sphere. Or maybe something bigger like the Hilbert cube — a countably infinite product of closed intervals. Or maybe something full of holes, like the Sierpinski carpet or the Menger sponge. Or maybe something weird like a solenoid.

Very roughly, a continuum is snake-like if it's long and skinny. It's a bit like being 1-dimensional. But the precise definition is a bit harder:

We say that an open cover 𝒰 of a space X refines an open cover 𝒱 if each element of 𝒰 is contained in an element of 𝒱. We call a continuum X snake-like if each open cover of X can be refined by an open cover U₁, ..., Uₙ such that for any i, j the intersection of Uᵢ and Uⱼ is nonempty iff i and j are right next to each other.

Such a cover is called a chain, so a snake-like continuum is also called chainable. But 'snake-like' is so much cooler: we should take advantage of any opportunity to bring snakes into mathematics!

And here's what Mioduszewski proved in 1962: the pseudo-arc is a 'universal' snake-like continuum. That is: it's a snake-like continuum, and it has a continuous map onto every chainable continuum!

This is a way of saying that the pseudo-arc is the most complicated snake-like continuum possible. A bit more precisely: it bends back on itself as much as possible while still going somewhere! You can see this from the pictures below, or from the construction on Wikipedia:

I like the idea that there's a subset of the plane with this 'universal' property, which is so complicated that it's impossible to draw.

Here's the paper where these pictures came from:

Wayne Lewis and Piotr Minic, Drawing the pseudo-arc, Houston J. Math. 36 (2010), 905-934. Available at

For a bigger, better article on the pseudo-arc by me, try this:

Lots of nice pictures, including that big fat ugly black blob what Lewis and Minic got when they tried to accurately draw the pseudo-arc!

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An eternal golden braid

Any braid can be used to stir a liquid, as shown here. We can use this to define the entropy of a braid. The basic idea is that a braid with a lot of entropy mixes up the liquid a lot. But the cool part is that we can compute the entropy of any braid and get a specific number.

We can thus ask: which braid has the most entropy per 'switch'? The braid shown here has 4 switches, since reading it from bottom to top

first we switch aqua and blue,
then we switch aqua and yellow,
then we switch aqua and red,
then we switch yellow and red.

So: we can compute the entropy of a braid divided by the number of switches, and see which braids make this as big as possible!

For braids with 3 strands, all the winners have entropy per switch equal to the logarithm of the golden ratio!

The simplest winning braid is called the golden braid. Douglas Hofstadter should be eternally grateful.

For details, including a description of the winner, read my blog post:

At the end I sketch how the golden ratio gets into the game. But I also explain the entropy of a braid! It's an example of something called topological entropy.

I learned about this connection between entropy, braids and the golden ratio here:

• Jean-Luc Thiffeault and Matthew D. Finn, Topology, braids, and mixing in fluids,

and that's where I got the picture!

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This is not an animated gif

This is Akiyoshi Kitaoka messing with your brain. He's a professor of psychology at Ritsumeikan University in Kyoto. He's spent a long time collecting and perfecting illusions. You can see them on his website:

where he writes

Should you feel dizzy, you had better leave this page immediately.

You can also see them on Twitter, where there is no such warning. And he has a book, The Oxford Compendium of Illusions.

I saw this picture in a post by +Alok Tiwari, who shared them privately to his circle of people who don't get seasick. So, I bequeath all the +1s on this post to him!

#geometry #illusions
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The Lakes of Wada

Wada lived on a white island in a red sea. On the island there was a blue lake and a green lake.

Wada became bored, and decided to dig canals.

On the first day, Wada dug a canal from the red sea so that every piece of land was within 1 mile of some red water.

In the next 1/2 day, Wada dug a canal from the blue lake so that every piece of land was within 1/2 mile of some blue water. The picture here shows what his island looked like then.

In the next 1/4 day, Wada dug a canal from the green lake so that every piece of land was within 1/4 mile of some green water. Can you draw it?

Wada continued this way, digging more and more canals. They were thinner and thinner, so there was always plenty of land left.

By the end of the second day, every piece of land touched the red sea, the blue lake and the green lake! He had built the famous Lakes of Wada.

You can create the Lakes of Wada effect by looking at the reflections in three mirrored spheres that touch each other.

You can also create this effect by applying Newton's method to a cubic polynomial with 3 distinct roots in the complex plane, such as z^3 − 1:

The 3 basins of attraction are not connected, and each one touches both others at every point of its boundary!

I got the picture from here:

and clicking the link at the bottom of this page will take you to more information on the Lakes of Wada. See also the Wikipedia article:

The Lakes of Wada were actually discovered by the Japanese mathematician
Takeo Wada (和田健雄) who lived from 1882 to 1944 and worked on analysis and topology at Kyoto University.

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Don't say you weren't warned

15,000 scientists signed this letter. They're warning you yet again: it's time to stop fucking around and get serious.

World Scientists’ Warning to Humanity: A Second Notice

Twenty-five years ago, the Union of Concerned Scientists and more than 1700 independent scientists, including the majority of living Nobel laureates in the sciences, penned the 1992 “World Scientists’ Warning to Humanity". These concerned professionals called on humankind to curtail environmental destruction and cautioned that “a great change in our stewardship of the Earth and the life on it is required, if vast human misery is to be avoided.” In their manifesto, they showed that humans were on a collision course with the natural world. They expressed concern about current, impending, or potential damage on planet Earth involving ozone depletion, freshwater availability, marine life depletion, ocean dead zones, forest loss, biodiversity destruction, climate change, and continued human population growth. They proclaimed that fundamental changes were urgently needed to avoid the consequences our present course would bring.

The authors of the 1992 declaration feared that humanity was pushing Earth's ecosystems beyond their capacities to support the web of life. They described how we are fast approaching many of the limits of what the ­biosphere can tolerate ­without ­substantial and irreversible harm. The scientists pleaded that we stabilize the human population, describing how our large numbers—swelled by another 2 billion people since 1992, a 35 percent increase—exert stresses on Earth that can overwhelm other efforts to realize a sustainable future. They implored that we cut greenhouse gas (GHG) emissions and phase out fossil fuels, reduce deforestation, and reverse the trend of collapsing biodiversity.

On the twenty-fifth anniversary of their call, we look back at their warning and evaluate the human response by exploring available time-series data. Since 1992, with the exception of stabilizing the stratospheric ozone layer, humanity has failed to make sufficient progress in generally solving these foreseen environmental challenges, and alarmingly, most of them are getting far worse. Especially troubling is the current trajectory of potentially catastrophic climate change due to rising GHGs from burning fossil fuels, deforestation, and agricultural production—particularly from farming ruminants for meat consumption. Moreover, we have unleashed a mass extinction event, the sixth in roughly 540 million years, wherein many current life forms could be annihilated or at least committed to extinction by the end of this century.

Humanity is now being given a second notice, as illustrated by these alarming trends. We are jeopardizing our future by not reining in our intense but geographically and demographically uneven material consumption and by not perceiving continued rapid population growth as a primary driver behind many ecological and even societal threats. By failing to adequately limit population growth, reassess the role of an economy rooted in growth, reduce greenhouse gases, incentivize renewable energy, protect habitat, restore ecosystems, curb pollution, halt defaunation, and constrain invasive alien species, humanity is not taking the urgent steps needed to safeguard our imperilled biosphere.

As most political leaders respond to pressure, scientists, media influencers, and lay citizens must insist that their governments take immediate action as a moral imperative to current and future generations of human and other life. With a groundswell of organized grassroots efforts, dogged opposition can be overcome and political leaders compelled to do the right thing. It is also time to re-examine and change our individual behaviors, including limiting our own reproduction (ideally to replacement level at most) and drastically diminishing our per capita ­consumption of fossil fuels, meat, and other resources.

The rapid global decline in ozone-depleting substances shows that we can make positive change when we act decisively. We have also made advancements in reducing extreme poverty and hunger. Other notable progress include the rapid decline in fertility rates in many regions attributable to investments in girls’ and women's education, the promising decline in the rate of deforestation in some regions, and the rapid growth in the renewable-energy sector. We have learned much since 1992, but the advancement of urgently needed changes in environmental policy, human behavior, and global inequities is still far from sufficient.

Sustainability transitions come about in diverse ways, and all require civil-society pressure and evidence-based advocacy, political leadership, and a solid understanding of policy instruments, markets, and other drivers. Examples of diverse and effective steps humanity can take to transition to sustainability include the following (not in order of importance or urgency):

(a) prioritizing the enactment of connected well-funded and well-managed reserves for a significant proportion of the world's terrestrial, marine, freshwater, and aerial habitats;

(b) maintaining nature's ecosystem services by halting the conversion of forests, grasslands, and other native habitats;

(c) restoring native plant communities at large scales, particularly forest landscapes;

(d) rewilding regions with native species, especially apex predators, to restore ecological processes and dynamics;

(e) developing and adopting adequate policy instruments to remedy defaunation, the poaching crisis, and the exploitation and trade of threatened species;

(f) reducing food waste through education and better infrastructure;

(g) promoting dietary shifts towards mostly plant-based foods;

(h) further reducing fertility rates by ensuring that women and men have access to education and voluntary family-planning services, especially where such resources are still lacking;

(i) increasing outdoor nature education for children, as well as the overall engagement of society in the appreciation of nature;

(j) divesting of monetary investments and purchases to encourage positive environmental change;

(k) devising and promoting new green technologies and massively adopting renewable energy sources while phasing out subsidies to energy production through fossil fuels;

(l) revising our economy to reduce wealth inequality and ensure that prices, taxation, and incentive systems take into account the real costs which consumption patterns impose on our environment; and

(m) estimating a scientifically defensible, sustainable human population size for the long term while rallying nations and leaders to support that vital goal.

To prevent widespread misery and catastrophic biodiversity loss, humanity must practice a more environmentally sustainable alternative to business as usual. This prescription was well articulated by the world's leading scientists 25 years ago, but in most respects, we have not heeded their warning. Soon it will be too late to shift course away from our failing trajectory, and time is running out. We must recognize, in our day-to-day lives and in our governing institutions, that Earth with all its life is our only home.


We have been overwhelmed with the support for our article and thank the more than 15,000 signatories from all ends of the Earth (see supplemental file S2 for list of signatories). As far as we know, this is the most scientists to ever co-sign and formally support a published journal article. In this paper, we have captured the environmental trends over the last 25 years, showed realistic concern, and suggested a few examples of possible remedies. Now, as an Alliance of World Scientists (­ and with the public at large, it is important to continue this work to ­document challenges, as well as improved ­situations, and to develop clear, trackable, and practical solutions while communicating trends and needs to world leaders. Working together while respecting the diversity of people and opinions and the need for social justice around the world, we can make great progress for the sake of humanity and the planet on which we depend.

Spanish, Portuguese, and French versions of this article can be found in file S1.


I have removed references above, to make the plea easier to read. You can find them here, along with the supplemental files:

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