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Official page of #ScienceSunday and #ScienceEveryday (brought to you by Robby Bowles, Allison Sekuler, Rajini Rao, Chad Haney, Buddhini Samarasinghe, Aubrey Francisco, and Carissa Braun)
You have found the official page for #ScienceSunday (co-curated by Robby Bowles, Allison Sekuler, Rajini Rao, Chad HaneyBuddhini SamarasingheAubrey Francisco, and +Carissa Braun)! Post anything related to science and tag it with #ScienceSunday, +ScienceSunday, and each curator to ensure we see your post. If you are a photographer, post a science related image and explain why it relates to science to you - doesn't need to be too specific (science is all around us!). If you are not a photographer, simply post anything related to science - drawings, movies, songs, and text are all welcome. Regardless of the type of post, feel free to add your 2 cents into a discussion in the comments. We always have some great posts with amazing images, great science information, and a lot of interesting conversations, and we're looking forward to even more in the weeks to come. If you miss the "Sunday" in #ScienceSunday, feel free to tag with #ScienceEveryday - we try to monitor those posts as well.



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Terra-fic Tiling

+Richard Green explains how Penrose or hexagonal tiles can be used in cartography. 

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Penrose Land Cover by Daniel P. Huffman

This land cover map of the continental United States was produced by Daniel P. Huffman using Penrose tiles.

A more traditional way to do this would be to use the technique of hexagonal binning, which achieves a similar result by using hexagonal cells, as in a honeycomb. It is possible to tile the entire plane using either identical hexagonal tiles or Penrose tiles. A key difference between the two is that the hexagons will produce a tiling with full translational symmetry, whereas the Penrose tiles will not.

Daniel Huffman recently remarked that “Penrose tilings are the new hexbins”. You can find this map, and some others of the same type, on Huffman's Twitter page:

Wikipedia seems not to have a good description of hexbins, but cartographer Zachary Forest Johnson wrote a nice blog post about them a few years ago, which you can find here:

Wikipedia has more on Penrose tilings here:

I have posted about Penrose tilings and related tilings several times, for example here:

#mathematics #cartography #sciencesunday

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Science Hangout on Air

Join +Science on Google+ on a fascinating HOA coming up this week.

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Please join us on 5/5 for a +Science on Google+ HOA with Dr.+Miguel Nicolelis, Professor of Neurobiology, Biomedical Engineering, Psychology, and Neuroscience at +Duke University, and founder of Duke's Center for Neuroengineering. Dr. Nicolelis is a pioneer in neuronal population coding (simultaneously recording from hundreds to thousands of neurons), Brain Machine Interface (controlling robotic or avatar limbs with thoughts), neuroprosthetics (prosthetic limbs that directly communicate with sensory and motor cortices), and Brain to Brain Interface (tactile or visual information encoded by rat 1 is decoded by rat 2). Dr. Nicolelis has published over 200 peer-reviewed articles, with many of these publications appearing in high impact journals such as Nature, Science, and Proceedings of the National Academy of Sciences (see below for a short list of publications). More recently, Dr. Nicolelis’ research made it possible for a quadriplegic child to use his mind to control a bionic exoskeleton and kickoff the opening game at the 2014 FIFA World Cup.

RSVP “yes” to add this event to your calendar. We will open up the Q & A app so feel free to post your questions on this event post or by using the app during the hangout.

Relevant Links:
Faculty page: 
Lab page: 
2012 Ted Talk: 
2014 Ted Talk: 

Relevant Readings (see for a more exhaustive list):

Schwarz D, Lebedev MA, Tate A, Hanson T, Lehew G, Melloy J, Dimitrov D, Nicolelis MAL. Chronic, Wireless Recordings of Large Scale Brain Activity in Freely Moving Rhesus Monkeys. Nat. Methods doi:10.1038/nmeth.2936, 2014.

Thomson EE, Carra R, Nicolelis MAL. Perceiving Invisible Light through a Somatosensory Cortical Prosthesis. Nat. Commun.10.1038/ncomms2497, 2013.

Ifft P, Shokur S, Li Z, Lebedev MA, Nicolelis MAL. A Brain-Machine Interface Enables Bimanual Arm Movements in Monkeys. Sci. Transl. Med. 5: 210, DOI:10.1126/scitranslmed.3006159, 2013.

Shokur S, O’Doherty J.E., Winans J.A., Bleuler H., Lebedev M.A., Nicolelis M.A.L. Expanding the primate body schema in sensorimotor cortex by virtual touches of an avatar. Proc. Natl. Acad. Sci. 110: 15121-6, doi: 10.1073/pnas.1308459110, 2013.

O’Doherty JE, Lebedev MA, Ifft PJ, Zhuang KZ, Shokur S, Bleuler H, Nicolelis MAL. Active tactile exploration enabled by a brain-machine-brain interface. Nature 479: 228-231, 2011.

Fuentes R, Petersson P, Siesser WB, Caron MG, Nicolelis MAL. Spinal Cord Stimulation Restores Locomotion in Animal Models of Parkinson’s disease. Science 323: 1578-82, 2009.

Pereira A, Ribeiro S, Wiest M, Moore LC, Pantoja J, Lin S-C, Nicolelis MAL. Processing of tactile information by the  hippocampus. PNAS 104: 18286-18291 (Epub) November 2007.

Krupa DJ, Wiest, MC, Laubach M, Nicolelis MAL Layer specific somatosensory cortical activation during active tactile discrimination   Science 304: 1989-1992, 2004.

Nicolelis MAL, Dimitrov DF, Carmena J, Crist R, Lehew G, Kralik J, Wise S. Chronic, multi-site, multi-electrode recordings in macaque monkeys. PNAS 100: 11041-11046, 2003.

Nicolelis MAL. Actions from thoughts. Nature 409: 403-407, 2001.
This Hangout On Air is hosted by Rich Pollett. The live video broadcast will begin soon.
Brain-Machine and Brain-Brain Interface
Yesterday, May 5, 3:30 PM
Hangouts On Air - Broadcast for free

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Art or Science?

Can you distinguish the microscope image of basal cell carcinoma from the Van Gogh inspired quilt?

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Rajini Rao originally shared:
Cancer or Canvas?

★ Do you see cancer cells run amok or a beautiful rendition of Van Gogh's "Starry Night"?  In this addition to my Art or Science? collection, it's hard to pick out the microscope image from the artwork it inspired. The tiny biological details revealed by researchers at the University of Michigan Center for Organogenesis are captured in larger than life quilts by Fiber Artists @ Loose Ends who raise public awareness about the importance of the arts in healthcare settings.

★ On the Left is a cross-section of mouse skin with basal cell carcinoma, the most common form of human skin cancer. The top layer of skin is stained red, collagen fibers are stained blue and the deadly tumor cells appear in the red at the bottom. On the Right, artist Carole Nicholas renders the image with fabric and stitching to simulate the Van Gogh's brushwork in a quilt.

★ This type of common skin cancer arises exclusively from the base of the hair follicle, where a niche of stem cells reside. When the hair follicle is in its growth phase, these cells are temporarily activated by the hedgehog signaling pathway. In cancer, this pathway is permanently on overdrive, due to mutations in genes known as Patched (PTCH) or Smoothened (SMO). If you're curious about the origin of these amusing gene names, especially Sonic Hedgehog, Indian Hedgehog and Tiggywinkle Hedgehog, check out +Buddhini Samarasinghe's entertaining and informative post (! 

REF: Hutchin et al. Sustained Hedgehog signaling is required for basal cell carcinoma proliferation and survival: conditional skin tumorigenesis recapitulates the hair growth cycle.

Image Credits: Mark Hutchin, University of Michigan
Art Quilt by Carole Nicholas, Fiber Artists@Loose Ends

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Science Sunday Digest

Enjoy your weekly roundup of science and technology, curated by +Mark Bruce !

#ScienceSunday   #SciSunRR  
SciTech #ScienceSunday Digest - 18/2015.
Permalink here: 

Whole brain staining, Chemogenic neural switching, Neuronal secrets, Single electron radio, Remote immersive telepresence, Aural parsing machines, Novel materials scaleup, Self-guided bullets, Logistical automation robots, New visual prosthesis. 

1. Whole-Brain Staining for Whole-Brain Mapping
A complex new brain-staining method called BROPA is the first of its kind able to stain an entire brain including all neurons and synaptic connections This now raises the distinct possibility of using the now-standard block-face scanning electron microscope technique to image an entire brain, slice by slice, and stitch these images together into a complete brain connectome map. Until now brain-staining techniques have only been useful for small sections of brain tissue, which could be scanned to produce connectomes but piecing these together into whole-brain maps was infeasible. So far the technique has only been demonstrated for mouse brains and the group plan to produce a complete mouse connectome comprising 40 petabytes, but it is yet to be seen if the technique can scale to human level.

2. Chemogenic Switching of Neurons
A newly developed chemogenic technique allows neurons to be controllably switched on and off The chemogenic technique essentially represents an improved DREADD technology ( in which neural cell-wall receptors were modified and engineered to be sensitive to specific synthetic molecules or drugs. Subsequently, these receptors - and the firing of the neurons they adorn - could be activated or deactivated by adding or removing the synthetic molecule from the animal’s system, and in this case two different receptors were introduced to mature mice via viral-administered gene therapies. In different tests both voracious feeding and drug addiction behaviour could be switched on and off at will.

3. A Duo of Fundamental Neuron Function Discoveries
The first of these sheds new light on exactly how neurons form connections and memories at the molecular and cellular level; a specific signalling protein called Asef2 that actively promotes synapse formation by promoting outgrowths of actin from the neual cytoskeleton - its lack can lead to a range of disorders. The second shows that neurons make methylation alterations to their DNA on a regular and on-going basis; this is believed to be important for maintaining consistent synaptic signalling activity by modulating the activity of key signalling proteins. 

4. Detecting Radio Waves from a Single Electron
For the first time radio waves have been detected from a single spinning electron via new ultra-sensitive experimental techniques that involve trapping single electrons ejected by radioactive samples in cusomised wave-guides. As cool as this is the group ultimately hope to use the technique to perform the most accurate measurements to date for determining the mass of a neutrino. I wonder about the reverse: using a similar setup to influence and control a single electron . . . or a neutrino. 

5. Towards True Immersive Telepresence
An Oculus-linked robotic system is edging towards the first true immersive remote telepresence system in which a user can wear a VR headset and receive binocular video input from a distant, remote robotic system that mimics the movement of the users body and head The key to maintaining the feeling of immersion and presence in the remote location is minimal and unnoticeable lag between you moving your head, the robot moving its “head” in exactly the same way, and an updated video feed from the cameras arriving on the headset. 20ms lag is not perceptible and 60ms is considered an upper limit; the group currently have 70ms but hopes to drop this in the near future. 

6. Deep Learning and Aural Parsing
Software arising from deep neural networks has now been demonstrated able to separate human voices from background noise in a wide range of songs (often referred to as the cocktail party problem after the ease with which humans can do the same) The promise here isn’t just a next-gen karaoke machine able to remove the vocals from any and all songs of choice. This should also help make better hearing aids, bluetooth headsets, video transcripts, and other applications we haven’t thought of yet. 

7. Scaling-Up Novel Materials: Semiconductors & Metallic Glass
A couple of interesting scale-up advances this week. First, a new fabrication technique called metal organic chemical vapour deposition can successfully produce wafer-scale atomically-thin (3 atoms) films of molybdenum disulfide or tungsten diselenide for high-performance semiconductor applications Second, a new manufacturing process allows spinel (magnesium aluminate) to be produced in sheets up to 30 inches wide; spinel is a transparent mineral that is much tougher, stronger, and harder than glass - think display screens, camera lenses, building and car windows, etc. 

8. The Latest Self-Guided Bullets from DARPA
DARPA’s new EXACTO bullet is a self-guided 0.50 caliber round that can adjust its trajectory mid-flight In the demonstration video you can see the bullet not only move to allow a trained sniper to hit a moving target, and not only move to allow a novice shooter to hit a moving target but, indeed to move and swerve mid flight to hit a target that starts moving after the bullet has been fired. I’m just imagining swarms of military drones that shoot and never miss. 

9. Another Industrial Automation Entrant
This week Fetch Robotics announced a duo of new robots called Fetch and Freight to tackle the logistics market The duo are intended to form a team in a warehouse, with the slower Fetch and its mobile manipulator shelf-picking arm confined to zones, and Freight a faster smaller unit zipping around between Fetches and a loading point. It’s good to see competition heating up in this space with the likes of Kiva and to a lesser extent ReThink for example. I also can’t resist a call-out to Stanford’s microtug robots able to 2,000 times their weight via novel controllable adhesive technology

10. A Vision Implant Powered by Light
A company called Pixium Vision is launching a new visual prosthetic that is powered by light and enables the blind to see The core of the system is based on a small chip that is implanted behind the retina and which includes pixels that have both a photodiode and retina-stimulating electrode; the person wears video glasses that capture the view in front of them and convert this into an infrared version that is beamed into the persons eyes which serves to both provide power and stimulate the retina. Tests in rats confirm restoration of 20/250 vision and they hope to soon achieve 10/120, below the limit of legal blindness. 

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What a teferric pun to kick off #ScienceSunday  ! 

Mz Maau originally shared:
I've not posted a #ScienceMeme in a while. This one goes around periodically.
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Iconic and ironic.
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Benefits of Daffodils

+Rajini Rao fills us in on how  exactly daffodils and snowdrops help contribute to improving memory loss and how extremely important it is to do our best to make sure research is properly funded!

#ScienceSunday   #SciSunCB  
Rajini Rao originally shared:
Daffodils and Dementia

✿ It's spring time in Maryland, and in the words of the poet Wordsworth, my heart dances with the daffodils. Through the long winter, I conjured up memories of these cheerful blooms in my mind:

For oft, when on my couch I lie
In vacant or in pensive mood,
They flash upon that inward eye
Which is the bliss of solitude;
And then my heart with pleasure fills,
And dances with the daffodils.

✿ But an estimated 44 million people world wide who suffer from Alzheimer's disease are robbed of their memories by a progressive dementia. As the 6th leading cause of death in the U.S., Alzheimer's cannot be cured or prevented. One of the handful of drugs available to improve memory loss in patients is galantamine, which is extracted from the leaves and bulbs of daffodils (Narcissus) and snowdrops (Galanthus). These extracts have been in use since ancient times. In Homer's Greek epic, Odysseus is said to have used snowdrops to clear his mind bewitched by Circe. In the 1950s, a pharmacologist observed inhabitants of a remote Bulgarian village rubbing the extracts on their forehead and shortly after, the drug was approved for medical use. Galantamine increases the action of the neurotransmitter acetylcholine in some parts of the brain, both by making the receptor more sensitive to its action and by slowing down its removal. The drug has other interesting properties: it is said to promote lucid dreaming, improve sleep quality, memory loss in brain damage, and some autistic symptoms (  

✿ No drug has yet stopped the inexorable progress of Alzheimer's. Early intervention is key to effective treatment: in my lab, for example, we are studying endosomal pathology which is the earliest sign of problems at the cellular level ( Yet lack of funding stifles productive research. As Newt Gingrich points out in his recent Op-Ed for New York Times, we spend only 0.8% of the estimated 154 billion dollars of annual medical costs related to Alzheimer's disease on research to cure or prevent it

News Story: Newt Gingrich: Double the NIH Budget. April 22, 2015 

Daffodil GIF:

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Celestial Mergers

While a Comcast-Time Warner merger may be huge, it does not generate short bursts of gamma rays! Read what happens when a black hole merges with a neutron star in +Jonah Miller's post for #ScienceSunday  .

Simulating Short Gamma Ray Bursts

The night sky is peppered with incredibly bright bursts of gamma rays. These so-called gamma ray bursts may be caused by a black hole eating a neutron star. Read on to learn more.

(To read this blog post in more detail, go here:

It was the mid 1960s. The United States and the Soviet Union had recently signed the Partial Nuclear Test Ban Treaty [1], which forbid the detonation of nuclear weapons except underground. Since neither nation trusted the other, each was carefully monitoring the other for non-compliance. In particular, the United States feared that the soviets might be, I kid you not, testing bombs behind the moon [2].


The United States solved this problem with the Vela satellites [3]. When a nuclear bomb goes off, it emits a short burst of gamma rays, which are rays of extremely high energy light. The Vela satellites were gamma ray detectors in space, orbiting the Earth 65,000 miles above the surface. Figure 2 shows one of these satellites in a clean room.

The Vela satellites did detect gamma rays all right, but they didn’t come from nuclear weapons… they didn’t even come from the solar system. The satellites repeatedly detected short, very intense bursts of gamma radiation that nevertheless took too long to be from nuclear weapons blasts.

Gamma Ray Bursts

For a long time, we didn’t know anything about these events or what caused them. So we gave them the enigmatic name gamma ray bursts [4], and made up many models for what could cause them.

This changed in the late nineties, when we were able to measure X-rays and visible light emitted from the same source after the burst, which we call an afterglow. We now know that there are many causes for gamma ray bursts. Some bursts take a relatively long time and we’ve linked them to supernovae in distant galaxies [5].

The relatively shorter gamma ray bursts (creatively called short gamma ray bursts ) are less common and less extensively studied. And we therefore know a lot less about them. One popular theory [6] is that they’re caused by the merger of a black hole and a neutron star.

A Quick Aside on Neutron Stars

Neutron stars are the densest stars we know of. Figure 3 shows that a neutron star twice as massive as our sun might have a radius smaller than Manhattan. Indeed, the only thing preventing a neutron star from forming a black hole is the Pauli exclusion principle (which I wrote about here:

Ordinary matter is made up of mostly empty space. The radius of an atomic nucleus is about a picometer, while the radius of an atom is about an angstrom. This means that, on average, 99.9999 % of matter is empty space. Not so with a neutron star. A neutron star is made up of neutrons packed as tightly as possible, like spheres [7]. This means that in a neutron star, only about 25 % of a neutron star is empty space. (Obviously take this analogy with a grain of salt. The properties of a neutron star depend heavily on quantum mechanics and nuclear physics… so the neutrons aren’t actually packed like spheres. They’re waves. See:

Anyway, neutron stars are incredible.

Black Hole-Neutron Star Mergers

When a black hole gets too close to a black hole, the black hole can eat it. But as I’ve discussed before, black holes are messy eaters. The matter in the neutron star gets distorted and forms an accretion disk around the black hole, which glows incredibly brightly. (see:

As the accreting matter falls the black hole, that matter can be accelerated to incredible velocities and launched out the poles, forming an ultrarelativistic “jet.” These jets are common in many circumstances, but we believe that the jet from a black hole-neutron star merger might be the source of short gamma ray bursts [8].

There’s a long and beautiful history of studies of accretion disks and the jets they drive. And we’ve known that black-hole neutron star mergers produce accretion disks of the right type. But there’s still a lot we don’t understand about jets, accretion physics, and neutron stars [9].

The Jet Emerges: A Piece of the Puzzle

Recently, Vasileios Paschalidis, Milton Ruiz, and Stuart L. Shapiro, of the University of Illinois at Urbana-Champaign numerical relativity group, helped add a bit to our understanding. For the first time, they simulated a black hole-neutron star merger, watched as the accretion disk formed, and the relativistic jet emerged. This provides additional evidence that black-hole neutron star mergers might be the progenitors of short gamma ray bursts. Figure 4 shows snapshots of the simulation as the black hole disrupts the star, accretes the matter, and finally drives the jet.

You can find a preprint of the article, which is currently undergoing peer-review, here:

Now, studies like this have been attempted before. Researchers routinely run many simulations of black hole-neutron star mergers to make predictions about gravitational waves [10]. And many groups around the world have run simulations of the jets driven by black holes. However, no previous simulation has successfully observed a jet after merger. All the previous jet simulations started with an accretion disk already in place.

Paschalidis, Ruiz, and Shapiro got their jet to emerge by correctly configuring the magnetic field of the neutron star before merger. Previously, all magnetic fields were assumed to be confined only within the star, and not exist outside it. Paschalidis, Ruiz, and Shapiro argue that this isn’t particularly realistic and, by including the exterior magnetic field, the jet emerges naturally.

This is a pretty cool piece of science!

Related Reading

If you found this interesting, you might enjoy my other posts on astrophysics.

1. In this post, I describe accretion disks in more detail:

2. In this post, I describe how planets form:

3. In this post, I describe the relativistic jets of active galactic nuclei:


#science   #physics   #astrophysics   #generalrelativity   #blackhole   #neutronstar   #space   #astronomy   #computers   #simulations   #supercomputer   #ScienceSunday   #ScienceEveryDay  
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Gator Gar!

Now that's a big fish for your #ScienceSunday  catch.

"The alligator gar is eight feet long, 254 pounds and about 44 inches around, and was caught on Lake Texoma in Marshal county last Thursday."

Alligator gars are the largest of the gar species and can live for many decades. While the 254 lb (115 kg) gar recently caught is a record for Oklahoma, the largest on record comes from Mississippi weighing 327 lbs (148 kg) with a length of 95 in (2.4 m). Texas comes in close with a record 302 lb (137 kg) alligator gar, and it is not unusual to reel in specimens over 200 lbs (91 kg). They have recently become popular for sports fishing which may contribute to their decline in the southeastern portion of its range, but some states are starting to impose limits to protect populations. While it gained its common name for the rows of large teeth, the alligator gar is not know to attack humans; however, the eggs are poisonous if ingested, and alligator gar can pose as an obstacle on water courses. Alligator gars have few predators, but the alligator is one of them.

A quick link of interest for your #ScienceSunday . Alligator gars are too cool to not make more widely known of their existence.

Sources and Further Reading
Alligator Gar (Atractosteus spatula) : TPWD
Alligator Gar (Atractosteus spatula) : NatGeo
OKLAHOMA-- An alligator gar caught on Lake Texoma is going down in the record books.
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Neuroscience Blog

We're delighted that +Brad Esau has shared his neuroscience blog with us, for #ScienceSunday  .

Okay, I thought #sciencesunday might make a good time to re-introduce my neuroscience blog.

As the name implies, A Layman's Guide to Your Inner Galaxies, my blog is to attempt to make the beauties and wonders of the human brain and the philosophical questions that arise in its study more accessible to the general public.

I was very enthused with this project when I launched it in late 2013 but life and health issues got in the way and I had to allow it to fall fallow for over a year.

But my enthusiasm never waned and I am now going to try keep it active with new posts. For now I'm just recycling some old material that I had originally written for Taming the Polar Bears (but which now falls outside of my direction there).

#consciousness #neuroscience #brain  
Consciousness Explained Part One Okay, one of my favourite subjects about the brain – consciousness! There are a couple of reasons I want to talk about consciousness. Firstly, and most importantly, I would like my de...
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Consciousness explained is the title of Daniel Dennett's book on the subject, a real good read I'd recommend to anyone.  I'd rather believe in what he says about having a multi-draft model of consciousness that's continuously redrafted.  We have psycho-neural identities apparently, look it up
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Perils of Climate Change

+Noah Diffenbaugh reports on the recent estimates that one in six species may go extinct as a result of climate change. 

#ScienceSunday   #SciSunRR  
"Accelerating extinction risk from climate change"

There is a new paper by Mark Urban in Science (from +AAAS - The American Association for the Advancement of Science ) analyzing the risk of extinction from climate change. Urban provides a "meta-analysis" of the previous literature on extinction rates. The image shows the risks by continental region. The magnitude of risk varies by region, and will also depend on the magnitude of climate change that occurs.

The peer-reviewed paper is here:

The Editor's Summary reads:
"There is great interest in understanding how species might respond to our changing climate, but predictions have varied greatly. Urban looked at over 130 studies to identify the level of risk that climate change poses to species and the specific traits and characteristics that contribute to risk (see the Perspective by Hille Ris Lambers). If climate changes proceed as expected, one in six species could face extinction. Several regions, including South America, Australia, and New Zealand, face the greatest risk. Understanding these patterns will help us to prepare for, and hopefully prevent, climate-related loss of biodiversity."

+Carl Zimmer  has an informative article in +The New York Times  (including quotes from other scientists):

The image is from the article, via +CityLab  from +The Atlantic :

#climatechange   #globalwarming   #biodiversity   #science   #sciencecommunication   #sciencesunday  
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That said, ocean acidification is still a nasty thing to have to deal with.
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Centrally Symmetric Pieces

Our resident mathematician +Richard Green gives us a wonderful explanation (and images) relating to cutting a regular octagon into convex and centrally symmetric polygon-shaped pieces.

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Tiling an octagon with centrally symmetric pieces

It is easy to cut an equilateral triangle into four smaller equilateral triangles, or to cut a square into four smaller squares, or to cut a regular hexagon into six equilateral triangles (think of a Trivial Pursuit playing piece). Less obviously, it is possible to cut a regular octagon into polygon-shaped pieces that are both centrally symmetric and convex. Some of these are illustrated in this picture.

The picture comes from the recent paper Decompositions of a polygon into centrally symmetric pieces by Júlia Frittmann and Zsolt Lángi ( The introduction of the paper gives a brief survey of some related problems. For example, it is trivial to cut a square into a set of triangles of the same area as each other, as anyone who has tried to cut a square piece of bread into triangles will know. (Note, however, that the resulting triangles will probably not be centrally symmetric.) Given this, it may be surprising to discover that that P. Monsky proved in 1970 that it is impossible to cut a square into an odd number of triangles, all of the same area.

Frittmann and Lángi's paper illustrates all 111 irreducible edge-to-edge decompositions of a regular octagon into convex polygonal pieces (tiles), where each tile is rotationally symmetric about its centre. The 18 decompositions shown in the picture are among the 111 irreducible decompositions.

In order for this classification to make complete sense, some terms need to be defined. A decomposition is called edge-to-edge if each edge of each tile either lies in the boundary of the surrounding polygon, or if it meets the edge of some other tile along the entire length of the edge. A shape is called convex if, whenever two distinct points are chosen within the shape, the straight line connecting the two points lies entirely within the shape. A decomposition with n tiles is called irreducible if, whenever at least 1 but at most n–2 tiles are removed from the decomposition, the remaining tiles form a non-convex shape. 
If A and B are tilings of the same polygon, we say that A is equivalent to B if there is a one to one correspondence between the set of tiles of A and the set of tiles of B, in such a way that two tiles of A touch each other in the same way that the corresponding two tiles of B touch each other. [Precise definition for mathematicians: two tilings are equivalent if the face lattices of the corresponding CW-decompositions are isomorphic.] There are infinitely many ways to tile an octagon under the constraints mentioned above, but there are only finitely many different ways up to equivalence. 

The main result of Frittmann and Lángi's paper generalizes this result to the case of a polygon with an even number, 2k, of sides, where k is at least 4. It turns out that in order for such a decomposition to be possible at all, the big polygon needs to be centrally symmetric. The authors show that up to equivalence, there will be a finite number of irreducible edge-to-edge decompositions of a centrally symmetric polygon into centrally symmetric, convex, polygonal parts. For an octagon, this number is 111, and G. Horváth proved in 1997 that the corresponding number for a hexagon is only 6. In their paper, Frittmann and Lángi give an upper bound for the number of decompositions for larger values of k.

It would be interesting to know what happens in the case where the big polygon has an odd number of sides. As I mentioned in the first paragraph, it is clearly possible to do something analogous for an equilateral triangle, although I don't know what would happen in the case of, say, a regular pentagon.

Relevant link
I mentioned CW decompositions above. These are associated to a CW complex, which is an important type of topological space. The “CW” is not somebody's initials, or an American TV channel, but rather stands for “closure-finite” and “weak topology”. The definition of a CW complex is rather technical, but it can be found here:

#mathematics #sciencesunday #spnetwork arXiv:1504.05418
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Making Science Accessible

What are quasicrystals? According to the well-known theorems of cystallography, only certain symmetries are allowed (square, rectangle, parallelogram triangle, or hexagon), but not others (pentagons). Quasicrystals are crystals that ignore these theorems and take on symmetries that are forbidden for crystals. The topic of quasicrystals can get complicated quickly, become difficult to explain, and cause someone to lose interest fast - unless you make it into a comic book. Thanks to +Peter Smalley for sharing this interesting piece!

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If you can't communicate your science, you're not really doing science. This researcher gets it.

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