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Why the Golden Rule Must Be Practiced in Business

One of the most valuable possessions that you have is your reputation -- and it is important to guard it at all costs. Profits can be lost and regained, but rebuilding a damaged reputation -- that is far more difficult.

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Tips to Identify Your Customers

It is common knowledge that targeting customers will result in a higher sales conversion than non-targeted, random traffic that just happens upon your website. In order to effectively target your customers, you need to know who and where they are. Customers come in all shapes and sizes, so how do you discern which customers are the best to target?

The following are tips to identify your best customers...

Tips to Identify Your Customers




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10 Small Business Website Errors That Drive Customers Away

Visitors seldom leave a small business website with a neutral impression. For most prospects, your small business is an unknown quantity, so their first impression of your company will either make them more comfortable doing business with you or less comfortable. For this reason, you want their first experience with your website to inspire them to contact you—not your competitor.

These are the 10 biggest website problems that can drive potential new business away:

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10 Must-Read Business Books for 2017

Asking for that promotion or finally starting your own business is easier with an inspiring read in your back pocket. Plan to pick up one of these books in 2017 to boost your creativity, or learn leadership lessons from the best.

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Why Brick-and-Mortar Retail Technology Is One of the Best Industries for Starting a Business in 2017

Tech startups can help optimize retailers' in-store operations and customer experiences, enabling them to keep up with competition from e-commerce companies. This category includes hardware and software such as sensors that track foot traffic, interactive tablets that boost customer engagement, and mobile point-of-sale applications.

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Amazons Tepid Response to Counterfeiters Frustrates Sellers

On Tuesday, independent tech publisher No Starch Press found out that Amazon was selling counterfeit copies of Python for Kids, one of its popular programming books. A concerned customer alerted Bill Pollock, the founder and publisher of No Starch. He immediately knew what he was up against, because Pollock has dealt with this problem before.

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Outstanding Leaders Exhibit More Than Just Emotional Intelligence--They Have These 7 Traits, According to Neuroscience

The topic of emotional intelligence (EQ) continues to dominate leadership conversations. Rightly so. However, in a Harvard Business Review (HBR) article that highlighted research by Daniel Goleman and Richard Boyatzis (experts on the topic), EQ is only the beginning.

Whereas EQ has an emphasis on individual psychology, there is a more relationship-based version called social intelligence. Social Intelligence, as defined by Goleman and Boyatizis, is a set of interpersonal competencies built on specific neural circuits and responses that inspire others to be effective. In other words, based on neuroscience and biology, there are certain leadership behaviors that elicit positive emotional responses in your team members.

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10 Tips for Building a Customer Focused Business

For tips on keeping your business customer focused and more, check out this recent content from members of the online small business community.

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The 15 Most Profitable Small-Business Industries

Thinking about launching a new business? Here is where to look to for high profit margins.

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How AI Tech Is Changing Small Business Marketing And Customer Service

Whether you realize it or not, artificial intelligence has become a ubiquitous part of our lives. Companies large and small are embracing the opportunity to achieve more and set bigger goals by using AI. Solid proof of this embrace can be seen in CRM giant Salesforce, which embeds AI into its leading CRM platform. Driven by the idea of democratizing AI so companies of all sizes can deliver smarter, more personalized, and predictive customer experiences, Salesforce created Einstein, your personal AI data scientist to guide you through your day.

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How Clever Retargeting Persuades Customers Who Looked But Did not Buy

Did you know that nearly 96 percent of your first-time visitors are not ready to purchase from your website right away? That means in most cases, in order to make a sale, you need to encourage your visitors to return more often to your site.

Retargeting is the best way to target your first-time visitors on different marketing channels and encourage them to return to your site. Since retargeting helps you to target your existing prospects, it will likely boost more engagement than untargeted ads, making your entire campaign cheaper and cost-effective.

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Amazon takeover of Whole Foods may be good for small businesses in natural food industry

For entrepreneurs in the natural food and products industry like Constantine, Whole Foods has served as a launchpad of sorts for companies looking to get on mainstream store shelves. Companies and local farmers often would get into one or two locations and grow from there if their products took off. Like the big supermarkets and food manufacturers who are nervous about Amazon's size and power, many small suppliers can see benefits of the combination, but are also concerned what it could mean for them.

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How to Start a Sustainable Small Business

Roughly 60 percent of millennials want to increase sustainability efforts in business. Millennials make up the largest group of consumers besides baby boomers, and 72 percent are willing to pay more for products from businesses committed to sustainability.

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4 High-Tech Industries (and a Really Low-Tech One) That Are Booming With No End In Sight

The beginning of the new year is when the worlds top investors make their predictions of industries to keep an eye on. Innovative technologies and ways of doing business are creating growth in certain industries. For those looking for something new to invest time or money in, take a look at these five industries that experts predict to have an increase in growth in 2018.

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12 Must-Have SaaS Tools For Small Businesses

In fact, some business leaders get so overwhelmed by the din of competing voices from service providers, they sit on the sidelines and fail to take advantage of some productivity and profitability enhancing systems.

I want to point out several good apps and SaaS tools, but before I get into specifics, I want to create a simple framework in which you should evaluate these tools. Further, keep in mind that one of the biggest advantages these tools give you is the ability to successfully manage virtual teams. However, you will find almost all of them appropriate for local use as well.

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10 Helpful Insights to Help You Make Use of Video and Better Communicate with Customers

Video is a medium that gives your small business lots of different options for communicating and sharing information with customers. But it can be sort of overwhelming for some small businesses. Arielle Kimbarovsky of CrowdSpring offers some suggestions for businesses looking to make the most of video in a recent post.

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What is PCI Compliance and Why MUST Small Business Owners Be Concerned?

The Payment Card Industry Data Security Standard (PCI DSS) is a set of security standards, designed to ensure businesses which accept and process credit and debit card information, do so in a safe and secure environment.

No matter what industry you operate in or what size business you have, if you accept card payments and process, transmit and store cardholder data, you must host your data securely with a hosting provider that is PCI compliant.

The PCI security Standards Council was formed in 2006 by the five major credit card brands — American Express, Visa, MasterCard, Japanese Credit Bureau (JCB) and Discover. While each credit card brand has its own compliance programs, the PCI standards are the foundation for all of them.

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The Never-Ending Struggle to Sustain a Small Business

Delissa Reynolds, an actress, ran a popular local bar for years, until the neighborhood changed.

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The 5 Most Important Negotiation Skills You Must Master

Intimidated by negotiations? Focus on these five simple negotiation skills first and get much better.

Think of all the times in your business week you negotiate:  with new hires and existing employees; with sales prospects and long-term clients; with vendors and suppliers. If you're a business owner or leader, you need to know how to negotiate. This is non-negotiable.

Here are the five most important negotiation skills you should focus on first. Each of these skills has proved to be worth millions to my clients and to me over the past 25 years. This all recently came to a head when I had 35 of my top business coaching clients join me for a day-and-a-half program on negotiation in my hometown of Jackson Hole, Wyoming. These were 35 of the most successful entrepreneurs and business owners in the United States.

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Three Small Business Trends That Many Businesses Just Do Not Need

The standards for small business success have unquestionably gone up. Todays small businesses must do a lot more to stay competitive than their predecessors. Myriad businesses, however, take this notion a bit too far. Once they discover yet another resource or strategy with the potential to elevate their success, they jump all over it. This is partially due to the many online articles urging business leaders to devote more effort to virtually every element of their businesses.

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Three Small Business Trends That Many Businesses Just Do Not Need

Here are three popular things that are actually not mandatory for all small businesses:

1. Aggressive Marketing

It is easy to go overboard with marketing. With so many tools available, your gut instinct is to try anything with the slightest potential. Even if you stand to lose money, you want the assurance of knowing you are doing everything you can to gain the interest of your target audience. And it is difficult to ignore a marketing tool that has worked marvelously for other small businesses. If they had to do it all over again, these businesses would likely spend 10% of their time developing their products or services and 90% marketing them.

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3 Ways Your Small Business Can Reach New Customers

Grow or die.

It sounds like the title of a rap song, but its actually the business idea that if your company does not grow, it is probably not going to make it. The reality is that you don't need to get bigger. What you need is a stream of customers to replace the ones that move, buy from you a little less, or have other life circumstances that change the level of business they do with you.

Yes, it is nice to add business and grow.

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13 Must-Have Words to Include In Your Resume

Diction or word choice is important when it comes to drafting your resume, not just to ensure that your resume is reviewed positively by software, but also because you want to wow recruiters with your skills, competencies and relevant credentials.

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Facebook Just Revealed It Is Doing the 1 Thing a Brand Should Absolutely Never Do

One thing brands should never do.

But it leads to something no brand should ever do: You should never assume that your customers feel the same way about your brand that you do. Your brand is how other people feel about your company, not how you feel about it.

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Small Businesses Must Take Advantage of the Democratization of Data

Learn how the easy availability of big data can help SMBs compete with big businesses.

- Data is now easier to collect and cheaper to host, making it accessible to nearly every business.
- The availability of data helps small businesses compete against big corporations, but they may still struggle to use that data in practical ways.
- Three ways small businesses can use big data to grow include checking on credit data about your business, monitoring supply chain data, and keeping an eye on artificial intelligence and automation innovations.

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Banking industry has concerns about the small business bailout program

Many small businesses have shut their doors but the bills are still piling up. The Small Business Administration is launching a program to assist, but many lenders are voicing concerns about it, calling the S.B.A.s expectations unrealistic.

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FinTechs Must Be Involved In Paycheck Protection Program Lending For Small Business

Small businesses drive the economy and create the lions share of jobs in the private sector economy. However, right now they are struggling mightily. Although it is recommended that companies have at least six months’ worth of revenue in the bank to weather economic disruptions such as the coronavirus, the reality is that most small companies don’t have enough cash to operate more than a couple of weeks.

For the service industry: restaurants, nail salons, haircutters, landscapers, athletic trainers, and others, the loss of weekly revenue is devastating for the owners and staff of small businesses. Most service workers live paycheck to paycheck and cannot go very long without being paid. We have already seen an enormous spike in unemployment claims, ending an era when the economy has basically been at full employment.

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10 Must-Have Content Marketing Tools For Small Business Owners

Customers go through a journey before they commit to a purchase. As a small business owner, it is your duty to engage and interact with them until they do so. And when they do, you continue to nurture them so they become your brand’s advocates.

With content marketing, you can convert random online searchers to website visitors, make them a part of your tribe and drive more sales by consistently providing them with valuable information.




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6 Cybersecurity Must-Haves for Your Business

In the current environment, cybersecurity is essential for businesses of all sizes. Many small or medium-sized businesses find themselves without adequate cybersecurity, either as a result of believing that they don’t need it or simply overlooking it among the many demands that come with running a business.

1. Use protection against ransomware
2. Invest in employee security training
3. Adopt multifactor authentication
4. Use a Security Information and Event Management system
5. Implement effective systems for protecting and monitoring data
6. Have a plan for mobile device security




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3 Promising Industries for Starting a Business Right Now

Entrepreneurs have seized the opportunity to start new companies in a wide variety of industries during the Covid-19 pandemic. While not all of these ventures will be successful, businesses that help alleviate some of the new challenges created by the pandemic are poised for long-term growth. Here are three industries that hold promising opportunities for entrepreneurs looking to start new businesses.

1. Contactless Tech
2. Telehealth
3. Education Tech




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12 Resources and Communities Entrepreneurs Should Follow for Industry Insight and Tips

Staying tuned in to the pulse of your industry is key to becoming a successful entrepreneur. Public groups, online forums and the like are among the most valuable resources for gathering and contributing industry information. But if someone is looking for in-depth insight into their business niche, locating the right groups where this discussion occurs is the first step. T




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How just a few days cost some small businesses thousands on their PPP forgivable loans

For some of the smallest businesses that applied for forgivable loans through the Paycheck Protection Program, waiting just a few days or weeks would’ve gotten them thousands of dollars more.

But they had no way of knowing what was coming.

The Biden administration in late February announced a slew of changes to the loan program, which offered forgivable loans in return for keeping employees on a company’s payroll, after it reopened in January with $284 billion in funding. Those amendments included an adjusted loan formula that would mean larger amounts for sole proprietors as well as expanded eligibility for small business owners with certain criminal records, were delinquent on student loan debt or were non-citizens.




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Just a note to say...

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.






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Wheat Protein Market Growth: Key Trends and Opportunities for Industry Leaders

(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...




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Industrial Cybersecurity Market to Reach $135.11 Billion by 2029 at a CAGR of 9.8%

(EMAILWIRE.COM, October 26, 2024 ) The industrial cybersecurity market was USD 84.54 billion in 2024 and is expected to reach up to USD 135.11 billion by 2029, growing at a CAGR of 9.8 %. Rising government and private investments to create safe and secure industrial environment, deployment of AI/Gen...




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Photoacoustic Imaging Industry Worth $105 million by 2029, with a CAGR of 5.5%

(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...




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Digital Instrument Cluster Market Growth Accelerates with Rising Demand in Automotive Sector, as per Maximize Market Research.

(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...




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AI in Fashion Market Grows as Industry Embraces Digital Transformation, as per Maximize Market Research

(EMAILWIRE.COM, November 05, 2024 ) The Global AI in Fashion Market is expanding rapidly, with brands leveraging AI to enhance design, personalization, and inventory management. From predictive analytics to virtual try-ons, AI is revolutionizing consumer engagement and operational efficiency. Fashion...




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Exoskeleton Market Poised for Growth with Demand in Healthcare and Industrial Sectors, as per Maximize Market Research

(EMAILWIRE.COM, November 05, 2024 ) The Exoskeleton Market is projected to grow rapidly, driven by applications in healthcare rehabilitation and industrial productivity. Exoskeletons enhance mobility for those with disabilities and improve worker efficiency and safety in industries such as manufacturing...




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Biomarkers Industry worth $93.8 billion in 2029, with a CAGR of 10.2%

(EMAILWIRE.COM, November 05, 2024 ) Biomarkers Market in terms of revenue was estimated to be worth $57.7 billion in 2024 and is poised to reach $93.8 billion by 2029, growing at a CAGR of 10.2% from 2024 to 2029 according to a new report by MarketsandMarkets™. The growing significance of biomarkers...




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Orthopedic Devices Industry worth $48.1 billion by 2028, with a CAGR of 4.8%

(EMAILWIRE.COM, November 08, 2024 ) Orthopedic Devices Market in terms of revenue was estimated to be worth $36.3 billion in 2022 and is poised to reach $48.1 billion by 2028, growing at a CAGR of 4.8% from 2022 to 2028 according to a latest report published by MarketsandMarkets™. Increasing cases...




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Industrial Utility Communication Market worth $4.2 billion by 2028

(EMAILWIRE.COM, November 11, 2024 ) The report "Industrial Utility Communication Market by Technology (Wired, Wireless), Component (Hardware, Software, Services), End-use Industry( Power Generation, , AC Transmission, Oil & Gas, Transportation), and Region - Global Forecast to 2028" The global Industrial...




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***** Reena Singh - Customer Service Manager - Prime Aviation ... (rank 30)

Reena Singh. Customer Service Manager at Prime Aviation Services Pvt. Ltd. Location South West Delhi, Delhi, India Industry Airlines/Aviation




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Dan Harmon and Justin Roiland

Rick and Morty: Dan Harmon’s Next Experiment After stunning the world (or at least the portion of the world that pays attention to what showrunners do) by reclaiming his series Community, after being removed for season four, Dan Harmon has...




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2010 Subaru Outback from Australia and New Zealand

In summary, quite a useful general purpose car without being too shouty




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2004 Volkswagen Beetle Convertible from Australia and New Zealand

Remarkably solid machine, and great fun to drive and own




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2018 Subaru Outlander from Australia and New Zealand

Great vehicle otherwise. Just this issue causing me disappointment




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Saw a doctor (good adjustments); drew a pickle (while talking to friends); crossed a river.

from Instagram https://instagr.am/p/DCIneGJpq6Z/ via IFTTT




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Quackbusters, Skeptics and the Web of Trust

What are 'quackbusters', you might ask. Well, Tim Bolen has the answer to that question. On his site (quackpotwatch.org) he explains: The "quackbuster" operation is a conspiracy. It is a propaganda enterprise, one part crackpot, two parts evil. It's sole purpose is to discredit, and suppress, in an "anything goes" attack mode, what is wrongfully named "Alternative Medicine." It has declared war on reality. The conspirators are acting in the interests of, and are being paid, directly and indirectly, by the "conventional" medical-industrial complex. These so-called quackbusters seem to be a branch of a larger movement, the "skeptics". Their website at www.skeptic.com/ shows who they are. Skeptics think of themselves as having opinions based on scientific 'truth'. They are very outspoken and very much "out there" to disabuse the rest of us of any idea that does not fit into their version of the scientific world view. While real scientific procedure requires there to be observation and experiment, formation and testing of hypotheses, and open discussion of both experiment and theories, the skeptics have firmly made up their mind on a number of issues. And they don't hesitate to tell us where we are going wrong... Mercury and fluoride for instance are not poisons for skeptics, and anyone who thinks they are must clearly be a conspiracy nut. Vaccination is good for you, as are chemotherapy and radiation cancer treatments offered by conventional medicine. If you oppose either of them you are simply a 'quack' or at the least you are an easy target for those who take advantage of your stupidity. The practices of alternative medicine, including "chiropractic, the placebo effect, homeopathy, acupuncture, and the questionable benefits of organic food, detoxification, and ‘natural’ remedies" are a favorite subject of the skeptics. They know that only mainstream medicine should be relied on and everyone who is into those practices really needs to have their head examined....




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"...you just get used to them"

“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 ATA = 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 AT; 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.


1 This, by the way, is my second-favorite math quote of the year, behind my complex analysis professor’s imprecation, right before discussing poles vs. essential singularities, to “distinguish problems that are real but not serious from those that are really serious.”

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).