The opportunities to answer questions for potential publication can come from various sources. Help a Reporter, known as HARO, matches journalists with sources for their stories. Quora allows users to answer any question posed by their community. And professional membership organizations offer business leaders the opportunity to share their insights in industry publications.

The opportunities to add your own insights are out there. But why would you, as a business owner, devote your time to answering these questions? What are the benefits?

Here are five ways that answering questions online can help promote your business.

Connections
Search engine optimization (SEO)
Brand
Website traffic
PR opportunities

A small, overlooked federal agency is shouldering a massive relief effort for the nation's small businesses and their workers left reeling by the novel coronavirus. The U.S. Small Business Administration has committed to auditing every sizable emergency loan it approves, yet nearly two months since the $660 billion Paycheck Protection Program was launched, the agency has yet to make public the recipients of taxpayer aid.

A price strategy is every bit as important as what you have got to sell. After all, they dictate what you will charge for the goods and services you have on the market. In the end, getting these pricing strategies right is a bit of a balancing act. You will need to find that sweet spot between market demand and maximum profits.

1. Competition Based Pricing Strategy
If you are selling things that are similar, this is the pricing strategy your small business should be looking at. Here’s how competitive pricing works. First off, this works best for products rather than services. It works when the price for those products has reached a balancing point between your business and others.

This pricing strategy starts out using the going market rate to set a price.

2. Value-Based Pricing
Basically this is pricing based on what customers are willing to pay.

There are several steps to this pricing model. You need to do a bit of detective work here to find a competitive product. Next, you will need to list all the ways your product is different. Stress the financial value on these differences. This is a reasonable way to defend your pricing strategy.

Remote work was already on the rise before Covid-19 forced many employees to transition to working from home. Today, as many organizations continue to operate remotely due to the pandemic, how can business leaders address the array of security challenges their companies face?

Network security should be top of mind for businesses of all sizes across a variety of industries. While the list of companies experiencing major security breaches in recent years includes some well-known enterprises, addressing security threats is an especially hefty challenge for small businesses—particularly if those businesses lack the resources to implement strong controls and educate staff.

Ninety-six percent of new entrepreneurs say the pandemic is motivating or giving them the motivation they needed to start their own business. This positive statistic comes from Azlo, a banking platform for small business owners, freelancers, and entrepreneurs.

In mid-March 2020, Azlo witnessed an uptick in new accounts opening. Wanting to understand the reasons behind the boost in business, Azlo conducted a survey. ‘The COVID Economy’ report interviewed 1,000 of Azlos newest customers across the United States.

Businesses now can negotiate for perks and concessions that were unheard of in the commercial real estate market just a few months ago.

Asking rent prices have yet to fall, which is typical in a down cycle as landlords try to hold out as long as possible, says CBRE chief economist Richard Barkham. At the same time, Barkham says, landlords are eager to fill space, so they're willing to offer a bevy of concessions to the right tenants, including rent-free periods, build-out expenses, and flexible lease terms.

Whether your cell phone has a broken screen, is no longer the latest technology, or you are simply itching for the latest cell phone model, it is important that you take appropriate care in disposing of your old cell phone.

Retain and Protect Your Data
Cell phones are a treasure trove of information; think of the panic that strikes each time you mistakenly misplace your cell phone. The information contained in our cell phones is meaningful and it is important that you backup any data on the phone that you wish to retain. The data you backup would include: any music, photos, messages etc...  If you are moving to a new cell phone you can often transfer the data you wish to retain to the new cell device.

Protect Your Personal Information
Cell phones often contain sensitive personal data, including passwords, bank information, payment methods, messages, contacts, videos, account numbers and much more. After transferring the data to a new device, information is still retained on your original cell phone.  Providing this type of sensitive information to strangers could put you at risk for identity theft, other other types of fraud, or simply leave you very uncomfortable.

How to Dispose of Old Cell Phones; Protect Your Data and Privacy

A business owner who spent nearly $46 million over the years on Facebook ads said he got booted from the platform without warning.

Jordan Nabigon, the CEO of the Ottawa, Ontario, content-curation site Shared, said Facebook deleted his companys main Facebook page without warning in October, and without providing an explanation. He shared a Medium post detailing his experience, which has received more than 400 claps from readers.

Nabigon spent $45,870,181 on Facebook advertising between 2006 and 2020 for Shared and his other company Freebies, according to expense reports reviewed by Business Insider. Shared employees three people full-time and 12 contract writers, Nabigon said.

Facebook increased its use of artificial intelligence to oversee advertising and other content during the COVID-19 pandemic, and Nabigon is among hundreds of business owners who said they suffered from Facebook's crackdown on ad policies.

The pandemic is accelerating existing trends.

Covid-19 has initiated some trends and altered the direction of others, but its most enduring impact will be as an accelerant. Take any trend--social, business, or personal--and fast-forward 10 years. Even if your company isn't living in the year 2030 yet, the pandemic has spurred changes in consumer behavior and markets. This is clear in the rapid increase in online shopping, in the shift toward remote delivery of health care, and in the spectacular increase in valuation among the biggest tech firms.

The more disruptive the crisis, the greater the opportunities--and the risks.

Here are five things in technology that happened this past week and how they affect your business. Did you miss them?

1 — Google plans to scrap third-party cookies by 2022.

Google announced this past week that it plans to stop the use of tracking cookies on Chrome by next year and— instead— will replace cookies with a profiling system

2 —Recruiting startup SeekOut raised $65M to take on LinkedIn and other talent acquisition companies.

3 —Small business owners adopted new software in 2020 and increased tech budgets in 2021.

Millions of businesses across the country are struggling, yet many are not taking the latest version of government aid: a second round of Paycheck Protection Program (PPP) loans. This is not happening because businesses are better off than they were last year; it is because the PPP still contains structural blockers that are stopping businesses from obtaining the aid they urgently need.

A recent survey by the Federal Reserve Bank found that 30% of U.S. small businesses — totaling 9 million — fear they will not make it through 2021 without more government assistance. And yet, many are not applying for aid. The Small Business Administration (SBA) reports that seven weeks after round two of PPP began, nearly half the funds remain, and only 31% of 2020 PPP loans have been forgiven to date.

Title: Flu Shots Lag in States With Low COVID Vaccine Uptake
Category: Health News
Created: 6/16/2022 12:00:00 AM
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Title: Could the Flu Shot Help Prevent Alzheimer's?
Category: Health News
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Title: Alpha-Fetoprotein Blood Test
Category: Procedures and Tests
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I mentioned that making Good Omens two is half of what I've been working on, and will be working on for next eighteen months, and I said I'd tell you soon enough what the other secret project I've been working on is.

It's this.

And I cannot tell you how happy I am to be making it, and making it in the way that we're making it.

Anansi Boys started in about 1996. I was working on the original Neverwhere TV series for Lenny Henry's film company, Crucial Films.

I loved a lot of what we were doing in Neverwhere. 25 years ago, it felt like we were doing something ahead of its time. 

Lenny and I went for a walk. Lenny grumbled about horror films. “You'll never get people who look like me starring in horror films,” he said. “We're the hero's friend who dies third.”

And I thought and blinked. He was right. “I'll write you a horror movie you could star in,” I told him.

I plotted one. I tried writing the first half-dozen pages of the movie, but it didn't seem to be right as a movie. And I was beginning to suspect that the story I was imagining, about two brothers whose father had been a God, wasn't really horror, either.

I borrowed Mr Nancy from the story I had not yet told and I put him, or a version of him, into AMERICAN GODS. 

In 2002 I was having lunch with my editor, and I told her the story of Anansi Boys, and said it was probably a novella. She waved her fork at me. “That is a novel,” she said, very certain. I was impressed enough with her certainty that I wrote the novel.

The creation and publishing of the novel is documented here on this very blog. Here's a useful bit, explaining its relationship to American Gods, and also explaining what Anansi Boys is:

https://journal.neilgaiman.com/2005/05/anansi-boys-question-of-day.asp

(For those of you who don't want to click, I talked about describing it thus:

My new novel is a scary, funny sort of story, which isn't exactly a thriller, and isn't really horror, and doesn't quite qualify as a ghost story (although it has at least one ghost in it), or a romantic comedy (although there are several romances in there, and it's certainly a comedy, except for the scary bits). If you have to classify it, it's probably a magical-horror-thriller-ghost-romantic-comedy-family-epic, although that leaves out the detective bits and much of the food. 

Which, oddly enough, is still a pretty good description.)

The book came out and was my first New York Times Number One Bestseller. 

https://journal.neilgaiman.com/2005/09/theres-first-time-for-everything.html 

(This is the Ukranian cover.)

A top Hollywood director wanted to buy the rights to Anansi Boys, but when he told me that he planned to make all the characters white, I declined to sell it. It was going to be done properly or not at all.

And then, about ten years ago, two things happened at the same time. Hilary Bevan Jones, a producer who had made a short film I had directed (called Statuesque) mentioned she'd love to make Anansi Boys as a TV series, and a man named Richard Fee, who worked for a company called RED, spotted me eating noodles in a London noodle bar, waited outside so he didn't seem like a stalker, and told me how much he loved Anansi Boys and that he'd love to make it into television.

I loved the TV that RED had made, loved Hilary and her team at Endor, and, unable to decide between them, suggested that they might be willing to work together. They both thought this was a good idea. 

Work started. Somewhere around 2016 I agreed to work on it to help it get made, but we all knew that we would have to be patient as I was writing and making Good Omens. And when Good Omens was in post production we began to move forward.  Amazon had loved making Good Omens, and were blown away by the viewing figures and reaction to it, and wanted to make more things with me, so Endor and Red now had a place to make it for. We put together a fabulous team of writers -- Kara Smith and Racheal Ofori and Arvind Ethan David, not to mention Sir Lenny Henry, who came on board both as a writer and as an Executive Producer to make sure that the soul stayed in it. (I'm writing the first and the last episode). 

Douglas Mackinnon agreed to co-showrun it with me, because I knew I never wanted to be the sole showrunner of anything again and after the Good Omens experience I would trust Douglas with my life and (which actually may be more important) with my stories. We planned to shoot it all around the world...

Paul Frift had been the producer of Good Omens during the South African leg of the shoot, and was indomitable, so we were very happy when he agreed to come on board as our producer.

And then in 2020 Covid happened. The Prime Directive of making Big Budget International television suddenly became “Don't Travel and Especially Don't Travel All Around The World. We Mean It.”

Douglas came up with a Plan to bring Anansi Boys to the screen that was audacious, creative and brilliant. All we needed to make it work was the Biggest Studio in Europe and access to an awful lot of cutting edge technology. 

The biggest Studio in Europe happens to be in Leith, outside Edinburgh. 

Before Covid, the plan had been first to make Anansi Boys, then immediately to make Good Omens 2. (Good Omens 2 was going to be shot in Bathgate, outside Glasgow.) That was the plan we were working on through most of 2020. Then, in September 2020, Douglas and I got a call from Amazon. “We've got good news and complicated news for you,” they said. “The good news is we are greenlighting both Good Omens and Anansi Boys. The complicated news is... well, how do you feel about making them both at the same time?”

So...

Anansi Boys is coming.

Hang on. I want to do that again in a bigger font.

Anansi Boys is coming.

I'd loved the pilot episode of Star Trek Picard, and talked to Michael Chabon about the director, Hanelle M. Culpepper, and he gave her a rave recommendation as someone who could tell a story and stay in control of the technology. We reached out to her, sent her the scripts and the novel, and she loved the project. Hanelle is going to be our lead director, and will direct two episodes.

Hanelle, Sir Lenny Henry, Hilary Bevan Jones and Richard Fee are executive producers, as are Douglas and I.  Hanelle,  Jermain Julien and Azhur Saleem are our three directors.

We will start to announce the cast soon (it's thrilling).  (The crew are, to me, just as thrilling.)

(But I'll give you one clue: one of our cast members was on a public event with me at some point in the last five years. The first thing she said when we met backstage was that her favourite book was the audiobook of Anansi Boys, read by Lenny Henry. And when I told her that there was a part in the book I'd originally written with her in mind, she was overjoyed. So when it became a reality, she was the first person I asked, and the first to agree.)

(The Anansi Boys image above is by Michael Ralph, our amazing production designer.)

Just a note to say that this blog has become rather dusty and abandoned over the last two or three years. But I think it's time for me to use it a lot more. At least until Google notices that they still own Blogger, and close the whole thing down.

Hullo. Welcome back.

This is a good place, on the whole, this blog. I started it in February 2001, for American Gods. This was the first entry.)

Here is an Edward Gorey drawing called The Happy Ending, to celebrate the New Beginning.

Title: metronidazole antibiotic
Category: Medications
Created: 12/31/1997 12:00:00 AM
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Category: Diseases and Conditions
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Title: Toothache
Category: Diseases and Conditions
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(EMAILWIRE.COM, October 25, 2024 ) The global wheat protein market is expected to expand from $2.5 billion in 2023 to $3.2 billion by 2028, achieving a compound annual growth rate (CAGR) of 4.9%. This growth is driven by the rising popularity of meat-free diets, increasing obesity rates leading to...

(EMAILWIRE.COM, October 26, 2024 ) The global Photoacoustic Imaging Market, projecting growth from USD 80 million in 2024 to USD 105 million by 2029, with a CAGR of 5.5%. Key drivers include rising investments in imaging systems and strategic partnerships. Challenges include high system costs and...

(EMAILWIRE.COM, October 27, 2024 ) PET Bottle Market overview PET bottles are plastic containers crafted from polyethylene terephthalate (PET), which is a form of polyester resin. PET plastic is clear, durable, lightweight, and can be recycled. It is resistant to shattering, stable in extreme...

(EMAILWIRE.COM, October 27, 2024 ) Near Field Communication (NFC) India Market overview NFC technology is integrated with IoT devices for seamless interactions. Mobile payments are advancing with NFC for convenience. Security enhancements are prioritized for trust in NFC solutions. Near Field...

(EMAILWIRE.COM, October 28, 2024 ) The global IoT security market size is projected to grow from USD 24.2 billion in 2024 to USD 56.2 billion by 2029 at a Compound Annual Growth Rate (CAGR) of 18.4% during the forecast period. Organizations face significant financial losses and reputational damage...

(EMAILWIRE.COM, October 30, 2024 ) Growing vehicle manufacturing is driving the need for enhanced lighting systems worldwide. Increased visibility during nighttime decreases accidents, prompting more stringent regulations and greater consumer demand for these safety systems. Adaptive headlights are...

(EMAILWIRE.COM, October 31, 2024 ) Automotive Adaptive Front Lighting Market is expected to grow at a CAGR of 27.8% throughout the forecast period, to reach US$ 6.55 Bn. by 2030 The global automotive adaptive front lighting market is on the rise, driven by increasing safety concerns and consumer...

(EMAILWIRE.COM, November 01, 2024 ) The global Digital Instrument Cluster market is witnessing rapid growth, fueled by increasing demand for advanced automotive displays and enhanced in-car experiences. Driven by advancements in digital interfaces, these clusters provide customizable, high-resolution...

(EMAILWIRE.COM, November 01, 2024 ) Automotive Microcontrollers Market is expected to reach US$ 22.93 Bn. by 2030, at a CAGR of 8.2% during the forecast period. The automotive microcontroller market is experiencing significant growth, driven by the increasing integration of electronics in vehicles....

(EMAILWIRE.COM, November 04, 2024 ) The Aluminum Flat Rolled Products Market is set for expansion as demand rises in automotive, construction, and packaging industries. Aluminum’s lightweight and corrosion-resistant properties make it ideal for use in energy-efficient vehicles and sustainable buildings....

(EMAILWIRE.COM, November 06, 2024 ) The DDoS protection & mitigation security market is projected to grow from USD 5.17 billion in 2024 to USD 9.63 billion by 2029 at a Compound Annual Growth Rate (CAGR) of 13.2% during the forecast period. DDoS protection and mitigation security growth is driven...

(EMAILWIRE.COM, November 09, 2024 ) The global Al in Biotechnology Market is projected to reach USD 7.75 billion by 2029 from USD 3.23 billion in 2024, at a high CAGR of 19.1% during the forecast period. Increasing cross-industry collaborations and partnerships that foster innovation and resource...

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The National Company Law Tribunal on Tuesday granted relief to Ace Aviation Ltd., the potential buyer for Jet Airways Ltd.'s grounded aircraft. The court has allowed an application that allows them to purchase the debt-ridden airlines' aircraft, based on an agreement reached between Ace and the monitoring committee of Jet Airways.

New York Comic Con 2018: Constantine: City of Demons At the 2018 New York Comic Con, DVDTalk’s Francis Rizzo III sat down with the cast and crew of the new feature-length animated film Constantine: City if Demons, due out...

New York Comic Con 2018: Batman: The Animated Series At the 2018 New York Comic Con, DVDTalk’s Francis Rizzo III sat down with Batman: The Animated Series creators Bruce Time and Eric Radomski, and actors Kevin Conroy, Loren Lester,...

Title: nicotine lozenge
Category: Medications
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Category: Diseases and Conditions
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This is a reasoned argument by Joanna (Why I Don't Vaccinate My Children) posted on Erwin Alber's VINE facebook page which was started in 2009, to help parents make an informed choice on behalf of their children. Image credit topnews.ae Joanna responds (below) to a lady who published an article saying that unvaccinated children are the cause of recent increased pertussis (whooping cough) outbreaks in areas where vaccination is actively pursued......

This is another installment of research into the biochemistry of HIV and Aids by Cal Crilly, an Australian who finds himself fascinated with the intricacies of biology. Crilly analyzes the seemingly unconnected studies that show the biochemical changes that accompany the presence of numerous retroviruses - one of them called HIV - in humans. The mechanism that makes retroviruses appear is hypomethylation, and it is the same mechanism that accompanies pregnancy and inflammation. Those retroviruses are produced in the course of normal biological activity and they are not infectious. There are many different types (ever heard of HIV 'mutating'?). As an aside, we declare pregnant mothers to be "HIV positive" as pregnancy causes the presence of retroviruses in the course of normal biological activity, and those harmless endogenous retroviruses react with what's generally called an "HIV" test. Certain basic nutrients - Selenium, Folate, B12, B6, Choline are the most important - counteract hypomethylation of the cells and thereby calm the production of human endogenous retroviruses. The toxic Aids drug AZT causes hypermethylation but it is so destructive of normal cell processes that most patients die. The 'life prolonging' effect of HAART, the drug cocktail that is prescribed to Aids patients today is due to a sharp decrease in the dosage of deadly AZT in the cocktail. Cal demonstrates those facts and more with reference to studies you can find as well, if you're interested in the details. Meanwhile we continue to treat immune compromised people with drugs that further compromise the immune system and - in many cases - kill the patient. When is medicine going to start treating those people by insisting on better eating and supplementation supplying the correct nutrients? How long will it take until the toxic drugs are phased out in favor of real prevention?...

In 2007, I posted two articles on this site suggesting that cancer had something to do with the energy mechanism of the human body. I am coming back to this through a comment on one of those articles by Viorel Bungau, a family medical doctor in Romania. The two previous articles are: Cancer and ATP: The Photon Energy Pathway Could cancer be a functional deviation of the cellular energy production mechanism that is open to correction by relatively simple means instead of a genetic mutation that is passed down through cell division? Heinrich Kremer, MD, says that this is indeed the case. His hypothesis of a photon-mediated cellular energy pathway may turn out to substantially add to our understanding of what cancer is, and why such natural substances as curcumin may be effective cancer fighting agents. Cancer: The ATP-Photon Hypothesis Could cancer be a result of the functional degradation of a cellular photon absorption pathway that is basic to the production of ATP, the cellular energy molecule? And could tumor cells merely be executing an age-old survival program they retained from the time of unicellular existence? An interesting new hypothesis of cancer causation spells out why some cells all of a sudden start dividing to form tumors, and how that urge to sustained growth could likely be 'switched off. Chromotherapy: The healing power of colors (image found here) Here (with some minor edits) is what Dr Bungau originally posted as a comment on the second article... CHROMOTHERAPY IN CANCER Dear Professor, Medical practitioner, I am passionate about cancer research. After years of study, we came to a hypothesis to be verified experimentally in collaboration with a field center. After many attempts (direct approach), no one has agreed to do this experiment. Please help make this research proposal possible. If proven true, it would change the medical approach to diagnosis and treatment of cancer and beyond. Yours sincerely, Dr.Viorel Bungau...

“Young man, in mathematics you don’t understand things, you just get used to them.” —John von Neumann¹

This, in a sense, is at the heart of why mathematics is so hard. Math is all about abstraction, about generalizing the stuff you can get a sense of to apply to crazy situations about which you otherwise have no insight whatsoever. Take, for example, one way of understanding the manifold structure on SO(3), the special orthogonal group on 3-space. In order to explain what I’m talking about, I’ll have to give several definitions and explanations and each, to a greater or lesser extent, illustrates both my point about abstraction and von Neumann’s point about getting used to things.

First off, SO(3) has a purely algebraic definition as the set of all real (that is to say, the entries are real numbers) 3 × 3 matrices A with the property A^TA = I and the determinant of A is 1. That is, if you take A and flip rows and columns, you get the transpose of A, denoted A^T; if you then multiply this transpose by A, you get the identity matrix I. The determinant has its own complicated algebraic definition (the unique alternating, multilinear functional…), but it’s easy to compute for small matrices and can be intuitively understood as a measure of how much the matrix “stretches” vectors. Now, as with all algebraic definitions, this is a bit abstruse; also, as is unfortunately all too common in mathematics, I’ve presented all the material slightly backwards.

This is natural, because it seems obvious that the first thing to do in any explication is to define what you’re talking about, but, in reality, the best thing to do in almost every case is to first explain what the things you’re talking about (in this case, special orthogonal matrices) really are and why we should care about them, and only then give the technical definition. In this case, special orthogonal matrices are “really” the set of all rotations of plain ol’ 3 dimensional space that leave the origin fixed (another way to think of this is as the set of linear transformations that preserve length and orientation; if I apply a special orthogonal transformation to you, you’ll still be the same height and width and you won’t have been flipped into a “mirror image”). Obviously, this is a handy thing to have a grasp on and this is why we care about special orthogonal matrices. In order to deal with such things rigorously it’s important to have the algebraic definition, but as far as understanding goes, you need to have the picture of rotations of 3 space in your head.

Okay, so I’ve explained part of the sentence in the first paragraph where I started throwing around arcane terminology, but there’s a bit more to clear up; specifically, what the hell is a “manifold”, anyway? Well, in this case I’m talking about differentiable (as opposed to topological) manifolds, but I don’t imagine that explanation helps. In order to understand what a manifold is, it’s very important to have the right picture in your head, because the technical definition is about ten times worse than the special orthogonal definition, but the basic idea is probably even simpler. The intuitive picture is that of a smooth surface. For example, the surface of a sphere is a nice 2-dimensional manifold. So is the surface of a donut, or a saddle, or an idealized version of the rolling hills of your favorite pastoral scene. Slightly more abstractly, think of a rubber sheet stretched and twisted into any configuration you like so long as there are no holes, tears, creases, black holes or sharp corners.

In order to rigorize this idea, the important thing to notice about all these surfaces is that, if you’re a small enough ant living on one of these surfaces, it looks indistinguishable from a flat plane. This is something we can all immediately understand, given that we live on an oblate spheroid that, because it’s so much bigger than we are, looks flat to us. In fact, this is very nearly the precise definition of a manifold, which basically says that a manifold is a topological space (read: set of points with some important, but largely technical, properties) where, at any point in the space, there is some neighborhood that looks identical to “flat” euclidean space; a 2-dimensional manifold is one that looks locally like a plane, a 3-dimensional manifold is one that looks locally like normal 3-dimensional space, a 4-dimensional manifold is one that looks locally like normal 4-dimensional space, and so on.

In fact, these spaces look so much like normal space that we can do calculus on them, which is why the subject concerned with manifolds is called “differential geometry”. Again, the reason why we would want to do calculus on spaces that look a lot like normal space but aren’t is obvious: if we live on a sphere (as we basically do), we’d like to be able to figure out how to, e.g., minimize our distance travelled (and, thereby, fuel consumed and time spent in transit) when flying from Denver to London, which is the sort of thing for which calculus is an excellent tool that gives good answers; unfortunately, since the Earth isn’t flat, we can’t use regular old freshman calculus.² As it turns out, there are all kinds of applications of this stuff, from relatively simple engineering to theoretical physics.

So, anyway, the point is that manifolds look, at least locally, like plain vanilla euclidean space. Of course, even the notion of “plain vanilla euclidean space” is an abstraction beyond what we can really visualize for dimensions higher than three, but this is exactly the sort of thing von Neumann was talking about: you can’t really visualize 10 dimensional space, but you “know” that it looks pretty much like regular 3 dimensional space with 7 more axes thrown in at, to quote Douglas Adams, “right angles to reality”.

Okay, so the claim is that SO(3), our set of special orthogonal matrices, is a 3-dimensional manifold. On the face of it, it might be surprising that the set of rotations of three space should itself look anything like three space. On the other hand, this sort of makes sense: consider a single vector (say of unit length, though it doesn’t really matter) based at the origin and then apply every possible rotation to it. This will give us a set of vectors based at the origin, all of length 1 and pointing any which way you please. In fact, if you look just at the heads of all the vectors, you’re just talking about a sphere of radius 1 centered at the origin. So, in a sense, the special orthognal matrices look like a sphere. This is both right and wrong; the special orthogonal matrices do look a lot like a sphere, but like a 3-sphere (that is, a sphere living in four dimensions), not a 2-sphere (i.e., what we usually call a “sphere”).

In fact, locally SO(3) looks almost exactly like a 3-sphere; globally, however, it’s a different story. In fact, SO(3) looks globally like , which requires one more excursion into the realm of abstraction. , or real projective 3-space, is an abstract space where we’ve taken regular 3-space and added a “plane at infinity”. This sounds slightly wacky, but it’s a generalization of what’s called the projective plane, which is basically the same thing but in a lower dimension. To get the projective plane, we add a “line at infinity” rather than a plane, and the space has this funny property that if you walk through the line at infinity, you get flipped into your mirror image; if you were right-handed, you come out the other side left-handed (and on the “other end” of the plane). But not to worry, if you walk across the infinity line again, you get flipped back to normal.

Okay, sounds interesting, but how do we visualize such a thing? Well, the “line at infinity” thing is good, but infinity is pretty hard to visualize, too. Instead we think about twisting the sphere in a funny way:

You can construct the projective plane as follows: take a sphere. Imagine taking a point on the sphere, and its antipodal point, and pulling them together to meet somewhere inside the sphere. Now do it with another pair of points, but make sure they meet somewhere else. Do this with every single point on the sphere, each point and its antipodal point meeting each other but meeting no other points. It’s a weird, collapsed sphere that can’t properly live in three dimensions, but I imagine it as looking a bit like a seashell, all curled up on itself. And pink.

This gives you the real projective plane, . If you do the same thing, but with a 3-sphere (again, remember that this is the sphere living in four dimensions), you get . Of course, you can’t even really visualize or, for that matter, a 3-sphere, so really visualizing is going to be out of the question, but we have a pretty good idea, at least by analogy, of what it is. This is, as von Neumann indicates, one of those things you “just get used to”.

Now, as it turns out, if you do the math, SO(3) and look the same in a very precise sense (specifically, they’re diffeomorphic). On the face of it, of course, this is patently absurd, but if you have the right picture in mind, this is the sort of thing you might have guessed. The basic idea behind the proof linked above is that we can visualize 3-space as living inside 4-space (where it makes sense to talk about multiplication); here, a rotation (remember, that’s all the special orthogonal matrices/transformations really are) is just like conjugating by a point on the sphere. And certainly conjugating by a point is the same as conjugating by its antipodal point, since the minus signs will cancel eachother in the latter case. But this is exactly how we visualized , as the points on the sphere with antipodal points identified!

I’m guessing that most of the above doesn’t make a whole lot of sense, but I would urge you to heed von Neumann’s advice: don’t necessarily try to “understand” it so much as just to “get used to it”; the understanding can only come after you’ve gotten used to the concepts and, most importantly, the pictures. Which was really, I suspect, von Neumann’s point, anyway: of course we can understand things in mathematics, but we can only understand them after we suspend our disbelief and allow ourselves to get used to them. And, of course, make good pictures.

² As a side note, calculus itself is a prime example of mathematical abstraction. The problem with the world is that most of the stuff in it isn’t straight. If it were, we could have basically stopped after the Greeks figured out a fair amount of geometry. And, even worse, not only is non-straight stuff (like, for example, a graph of the position of a falling rock plotted against time) all over the place, but it’s hard to get a handle on. So, instead of just giving up and going home, we approximate the curvy stuff in the world with straight lines, which we have a good grasp of. As long as we’re dealing with stuff that’s curvy (rather than, say, broken into pieces) this actually works out pretty well and, once you get used to it all, it’s easy to forget what the whole point was, anyway (this, I suspect, is the main reason calculus instruction is so uniformly bad; approximating curvy stuff with straight lines works so well that those who who are supposed to teach the process lose sight of what’s really going on).