We’re Number…Huh?

If I gave you five guesses, could you identify the most famous equation in mathematics?  Perhaps this would be your first guess:

I must admit that is not a bad choice.  I think that one might be my personal favorite, but I do believe there is a general consensus about the most beautiful and elegant equation math has to offer.  It is known as Euler’s Identity, it looks like this:

If we do just a little manipulation we get this:

That is the version I prefer.  Within one simple equation, we get the five most important numbers in mathematics.  And as an added bonus, we get to see exponents.  Pretty cool.

There are lots of excellent videos on the internet on the topic of Euler’s Identity.  The equation can be derived in lots of different ways, each more fascinating than the next.  There is no need for me to include even a single derivation here.  I am going to do something much more interesting.

If there are any sports fans out there, I have a suggestion for you.  I don’t believe I have ever seen such a sign, but the world is undoubtedly in need of one.  The next time you head out to a game, you might want to take the time to get some poster board and markers, clear off a desktop, and get to work.  It won’t take long.

I suggest the following text:

If you really are proud and excited about your sign, feel free to add an exclamation point at the end, after all, 1! does indeed equal 1.  If a cameraman catches you, you just might make it onto ESPN.  If that happens, you can bask in the thought that 5 or 10 people across the nation chuckled when they saw your handiwork.

 

 

0.20787…

Check this out…

That looks familiar, right?  I think most of us have come across both of those definitions at some point.  What you probably didn’t learn is this…

Take a close look at that…and then look at this…

Truly astonishing, an imaginary number raised to the power of an imaginary number gives us a never-ending decimal.  As always, if you are interested in deriving the answer, do a Google search.  You will be amazed at what you will find.

 

 

How About a New Punctuation Mark?

Do you have any idea what this is?

How about a hint?  Oh yeah, that is a really good idea  

Can I have another one, that last one was so helpful  

OK, you can have a couple more.

You look great   Where are you going, gravedigging  

Nice ketchup stain, it really pulls your entire ensemble together  

Yes, I present the mighty Sarcmark.  Douglas Sak, a man in need of a punctuation mark to better denote sarcasm in his emails, founded Sarcasm, Inc. in 2006.  Shortly thereafter, the Sarcmark was offered for sale.  Not really a bad idea, is it?

I am always on board when someone is offering up ways to better clarify ideas, especially those of the written variety.  The website claims that the symbol is gaining traction but I must admit I have not seen it being used anywhere.  Except here.  Except now.  Where else on the entire internet could you get such valuable information this easily?  

If you really feel you need this Punctuation Mark in your grammatical arsenal, then I urge you to get over to the Sarcmark website ASAP.  When I came across the site years ago, the Sarcmark cost real money, now it is free.  I suggest you head on over, I would hate for you to send an email to a friend only to have them not realize you were insulting them.  

Inspired by a Yahoo: Inspiration is in Short Supply, we all need to take it Wherever and Whenever we can get it.

I am bursting at the seams.  Writing about Edwin Goodwin (the clown who “discovered” that π = 3.2) has elevated me; my ambitions now include worldwide fame.  I want riches, texts from supermodels (OK, Danica Patrick would do just fine), and a driver named Jeeves.  How am I going to get all these things?  Easy, I stayed up late last night and came up with the following rock-solid mathematical proof that 2 = 1.  Read on.

Why would I give you only one proof when I have two in the bag?  Behold the following:

I am basking in my own genius.  I am going to call my congressman and get this thing written into law ASAP.  I wonder which supermodel will text me first?  My mind is racing, good grief I might even hear from Athena (you will be hearing much more about her later this year).  Wow, I’ll probably have to set up some kind of a rotating schedule so that all these women don’t show up at my house at the same time.  I don’t want them fighting over little old me.  So, do you think this will be a big problem or have I missed something?  Do I need to buy some new socks and shirts or should I just chill?

 

Bill #246: Strange Doings in Indiana

‘we think it something on which the members of both houses can unite without distinction of party.'”

James Garfield, commenting on his proof of The Pythagorean Theorem (as discussed in a previous essay).

Does it strike you at all as strange that then-Congressman James Garfield would make such a comment? My initial thought was that he was having a little fun, that his tongue was firmly planted in his cheek as he made that statement about his neat little proof. Then I remembered something I learned years ago, I recalled an attempt to legislate mathematical truth by what can only be described as a group of yahoos in Indiana. Hold your breath and take a look at this.

In the late 1800’s something happened in the state legislature of Indiana that is inconceivable. Let me begin by telling you that I keep an unofficial list of the dumbest, most inexplicable things I have ever heard. The following short story is always at or near the top of that fluid archive.

In 1897 the Indiana General Assembly took up Bill #246, generally known today as The Indiana π Bill. It is going to be hard for me to finish off this essay because I have trouble typing while my head is shaking violently back and forth. You see, every time I am reminded of this story, I lose a little more faith in humanity. The story of what happened in Indiana is easily one of the dumbest things that have ever occurred in the legislative history of this country. You know what? Let me qualify that last statement and say that it is one of the dumbest things that has ever happened anywhere at any time in the history of humanity.

Our road to perdition begins with Edwin Goodwin, a physician and amateur mathematician (let me stress amateur) who decided he had figured out how to square a circle using only a compass and a straightedge. The big problem with that is that in 1882 a real mathematician named Lindemann had proved that such a thing was impossible. Such nasty little facts never, ever get in the way of a crackpot on a mission, and Goodwin certainly was a goofball with an agenda. As unbelievable as it may sound, he found many willing accomplices in the representatives of the people of Indiana.

The details of the mathematics are not necessary, his paper is so bad that I would not feel right telling you about it. Sometimes it is good to set up a straw man just to show how bad an argument is but not in this case. I think the story of what happened with this atrocious bit of mathematics is the interesting part. For our purposes, there is only one thing you need to know, namely that Goodwin came up with 3.2 as the value of π. As you probably know, the real value is 3.1415927… The decimal just keeps going, never repeating on its way to infinity and beyond.

Goodwin actually had his paper published by the American Mathematical Monthly, a journal founded in 1894 and still around today. The thing is, the people responsible for publishing the journal let Goodwin pay for the privilege of having his “genius” exposed to the world at large. The paper was printed with a disclaimer indicating that it had not been peer-reviewed and that it was published at the request of the author. Do you think anyone took note of these facts? Nope.

Goodwin took his paper to a state representative named Taylor Record, and in one of the worst decisions ever made by any person anywhere at any time, Record introduced a bill, Bill #246, to have Goodwin’s claims written into law. The bill was put to the vote, and as you might have guessed, the vote was unanimous. What you might not have supposed is that the bill passed!

Goodwin had figured the value of π to be 3.2 and the Indiana General Assembly readily agreed. Can you believe that? They all agreed, without a single dissenting vote, that the value of π was to be 3.2 and that would be the law of the land. What were they thinking? It is my contention that they were not thinking at all. They were apparently charmed by Goodwin and stood in awe of his “genius.”  It didn’t help that Goodwin told the politicians that the people of Indiana could use his result without paying royalties.  Of course, the rest of the world would have to pay up.

Fortunately, the bill had to then go on to the Indiana Senate. It failed in a close vote, a very close vote. Even though it was apparent to everyone but the politicians that they all had lost their minds, they still nearly passed a law indicating that anyone using a value of π different than 3.2 was guilty of somehow breaking the law. Not only that, but in 1985, a scholar went through Goodwin’s paper and found that there were seven different values of π implied by Goodwin’s ridiculous mathematics. Can you actually believe any of this nonsense?

This story is relevant today as the purveyors of Intelligent Design try, time, and time again, to discount evolution by attempting to legislate what science is and what it is not. It is also a remarkable story as the science of Climate Change has become politicized to the point where the actual science, and the rock-solid mathematics that is its foundation, doesn’t seem to matter at all. The story of The Indiana π Bill is a cautionary tale, a spook story, one where the stakes are as high as you and I can imagine. We all must remain vigilant to ensure that the crackpots and the self-described geniuses remain hidden in the dark underbrush where they indeed belong.

 

 

1,000,000 Digits Isn’t Nearly Enough: A Few Thoughts on π

A few years ago, I sat down at my computer intending to write a short essay on one of my favorite subjects, π.  I thought I would start by computing π to as many digits as possible.  It was at that point that I realized that I had never computed π, and I had no idea how to do it.  I knew about the old methods, like those of people like Archimedes, but I had no idea how to compute the digits using modern methods and a computer.  As odd as it sounds, it simply had never come up.  I was truly stunned as I sat motionless in my chair, unsure of what I was supposed to do.  It never occurred to me that if I were tasked with computing π, I would have no idea where to start.  It really was a strange and confusing feeling.

After I stopped rolling around on the floor (I couldn’t quite get into the fetal position), I started my research into how to compute π.  I quickly found that the process is not straightforward, and it certainly is not trivial.  I eventually found a freeware program called y-cruncher, which will compute π to n digits, depending entirely on the amount of RAM in your computer.  This was also a surprise to me.  I wasn’t quite sure what RAM had to do with the calculation then, but I have a better idea now.  All the previous numbers in π must be loaded into memory because they are used to derive the next digit, and the next, and the next.  Who knew?  I suspected that something like this might be the case, but I was surprised that hard drive space wasn’t used instead of RAM.

I loaded up the y-cruncher program on the killer computer I built a few weeks ago and looked over the settings.  After it scanned my system, the clever code told me that I could compute 5 billion digits of π with no problem.  Of course, I took the program up on its offer.  Before I knew what had happened, I had a file on my hard drive that contained 5 billion digits of π.  Amazing.  As I tried to load the 4.8GB file into a text editor, I ran into another unexpected problem.

Have you ever tried to open a large text document?  I mean a very large text document, say one with 5 billion digits?  It is not a simple thing.  Notepad won’t do it; Word can’t handle it, and on and on and on.  A developer wrote a program for Windows specifically for large files called Notepad++, but it wouldn’t open it either.  Fortunately, I am very familiar with a suite of programs called LibreOffice.  It mimics Microsoft Office and, best of all, the programs are free.  LibreOffice Writer has been my preferred text editor for years.  For now, the critical point is that LibreOffice claims not to have a file size limit.

I opened LibreOffice 6.2 and had a go at loading my giant file.  The file is so large that it crashed my computer, I mean the whole thing.  The program ran for hours, chugging along, trying its best to load the entire file.  It was taking so long that I went to the gym.  When I came back, the file was still loading.  I got something to eat, and when I came back to my computer room, I found that the system had crashed.

Not one to give up that easily, I tried loading it again.  This time it just crashed the LibreOffice program.  Progress!  Then it crashed again.  As of now, the file has processed about 140,000,000 characters out of the 5 billion total.  Will the file open?  I am not sure, but even if it does fully open, I doubt I will be able to search through it.  It is just too big.

At this moment, LibreOffice is using over 18 GBs of the 32 GBs of spiffy new RAM I installed in my new system.  The program is plugging along.  It has processed over 250,000,000 characters.  Sure, that is a lot of numbers, but it is only a small fraction of the total.

The program has been trying to open the file for about a day now.  After numerous crashes, it still has a long way to go.  I am writing this post while LibreOffice is working hard to fulfill my request.   It has over 625,000,000 characters open, we have reached a milestone, it is over 10% of the total.  The font is 10 point Liberation Mono, the page count is about 11,000.

Strangely, LibreOffice is splitting the digits into words.  Every 10,000 digits of π are counted as one word.  How odd.  I have no idea why it is doing that.  Apparently, no one else does either.  I have been doing some searching, and it appears I am the first person to ask such a question.  Everyone else probably instantly went to the proper program, the one known by scientists and mathematicians to be the go-to program for such nonsense.  I have been out of that loop for a long time, I stopped getting the flyers and emails years ago…

After a bunch of time and effort, I managed to get over 1 billion of the digits to appear in LibreOffice.  That was the limit, my system would not allow it to load more. Why?  Once again, we are back to the limitations my new computer has due to the amount of RAM I installed.  32 GBs, which is overkill for almost any system, simply falls short in this instance.  LibreOffice used it all up and wanted more.  The program crashed one final time, and then I gave up.

In the future, I will load a file with hundreds of millions of digits, or maybe even a billion, and see what I can find.  I am sure there is some interesting stuff in there.  Do we get 20 consecutive zeroes?  How about forty 9s in a row?  Is my phone number with area code in there?  How about my social security number?  In due time, we should have answers to these questions and more.

 

 

 

 

A Problem with a Post

A few days ago, I published a short essay entitled Multiplicative Persistence.  Now I want to tell the story behind the trouble I had trying to get the post, to well…post.  It was an ordeal.

As you might imagine, I go through lots of drafts when I write these things.  Actually, I go through a lot more than you might think.  It is not unusual to have 20 or 30 of them.  I am always tweaking a word here or there, and my good friend Grammarly always has something to add.

Usually, I am sitting at my computer pounding away on the keyboard, but that is not how all the posts originate.  Lots of times, I am out at a bar or restaurant waiting for inspiration to strike.  I always have an open notebook and a few working pens by my side.  I have found that inspiration has its own timetable, and it doesn’t care much about what I want.   That said, I always try to be ready in case The Inspiration Gods decide to shower me with booty.  You will be learning much more about this process when I post the essays I wrote for a book called The Athena Chapters.  Those will be up soon enough.

This post is about the problems I had trying to get my essay on Multiplicative Persistence to save and then show up on my website.  The first few drafts were acting normal, but then the essay started to fight back.  It didn’t want to post, it didn’t even want to save any new drafts.  One moment I was typing and then, before I realized what was happening, I was engaged in battle.  This is my story…

When I clicked on Save Draft, the computer gave me a 502 Bad Gateway error, then it did it again, and again, and again.  I had never seen this error with WordPress or GoDaddy before, so I paused and tried to figure out what was going on.  I assumed that the problem was on their end, seems reasonable, right?  Nothing had changed on my system, and everything else I was working on seemed to be fine.

You know what I did, right?  I hit Save Draft another dozen times.  I got the 502 Bad Gateway error another dozen times.  After that enjoyable experience, I decided to hit the Troubleshooting Chat Button on GoDaddy’s website.  That is where the real fun began.

I can paraphrase the conversations…

Hello, what’s wrong?

502 Bad Gateway error when I try to save a draft of a post I am writing.

Ok, …give me a few minutes.

OK

I went to your site, and I was able to log in, no problem.

I do not have a problem logging in, I am having a problem trying to save a draft of a post I am writing.

Hold on…I am transferring you to a specialized team.

Hello, you are having a problem logging into your site?

No, I am getting an error when I try to save a draft of a post I am writing.

OK…give me a few minutes…

OK

We have put up a test post, and everything is fine.  Are you some kind of moron? (They didn’t come out and say that, but it was strongly implied.)

Here is a screenshot of the 502 Bad Gateway error I am receiving when I try to SAVE A DRAFT OF A POST I AM TRYING TO WRITE!

OK…thank you very much.  I am transferring you to a more specialized team.

All right.  Is Batman part of this group?  While he is not an actual superhero, he has world-class deductive skills. I am confident he could get me fixed up in no time.

No answer…

(I really love this next part.)

OK, sir, it is evident that you have no idea what you are doing.

Is this Batman?

No, sir, this is not Batman.

Can I speak to Batman?

There is no one here named Batman.

What should I do next? Hours are racing by, and my readers are clamoring for a new essay on multiplication.  If I don’t get the post up soon, I can only imagine the level of rioting in the streets.

At this point, all I saw was a blinking cursor on the chat screen.  I guess this person was trying to find Batman to tell him that I was out of my mind.  This poor representative, with limited imagination, couldn’t understand why any person would want to read an essay on multiplication.  The fact that he is pretty much right is beside the point.

Sir, please try another network and see what happens.  We are thinking that the problem is on your end.

I don’t have another home computer network.  My setup I not as sophisticated as that of someone like Batman.  I am sure he has redundancies built-in, don’t you think?

I really don’t know.  Again, there is no one here named Batman.  Could you reset your router?

If I do that, I will lose you, and I will have to go through all this nonsense again, correct?

No answer…

I will reset, but can I have the direct line to Batman to save me some time after I reconnect?

No answer…

I tried the reset, lost the guy helping me, and still got the error message when I tried to save a draft.  It was at this point that I started to utilize all those years of education I am rumored to have.

I asked myself a question: Is there anything different about that post?  Is there something unique about it?  The answer was yes.  It was those stupid right-pointing arrows that I included to show the progression of the numbers after the digits were multiplied together.  Those were the culprit, and, of course, there is a story behind their use.

The first few drafts of that post were behaving normally.  The Save Draft button was working fine.  I finally realized that the problems started when I decided to put those arrows in.  The arrows are specialized symbols, getting them into a program like WordPress is not the easiest thing to do.  You don’t tap the keyboard, you have to have a specialized script for mathematical symbols, or you need an equation writer.  For decades, scientists and mathematicians have used a program called LaTeX (pronounced lay-tech) to write their papers.  It is not very user-friendly, I use it when I have to, and in WordPress, I have to.

I went back to my draft of the essay, and I removed all the arrows.  I tried to save the draft, and it worked.  I really wanted to use arrows, so I didn’t give up on them. I then inserted a different kind of arrow (this one from a Special Symbol Editor, not an Equation Editor), and everything was fine.  I was instantly back on track, and the world is now a better place because a handful of people now know what Multiplicative Persistence is.

Isn’t it strange that a small piece of random code for a right-leaning arrow caused all these problems?  That little symbol led to a lot of issues and cost me a bunch of time and aggravation.  The mysterious ways in which some computer code can interact are not to be underestimated.

After I figured out what was going on I decided that it was best if I write the guy from the chat, I still had his email from when I sent the screenshot of the error message. I thought that maybe my story would help in the future when some other poor slob stumbled across something similar. I told him to let Batman know that I had solved the problem and that everything was fine.  Neither of them wrote me back.

 

 

 

 

 

 

Multiplicative Persistence

277,777,788,888,899 is an unusual and special number.  When it comes to Multiplicative Persistence, it is an unparalleled superstar.

Check out the following table.  Can you figure out what is going on?   The digits of any given number are multiplied together to get a new number, and then those digits are multiplied together, and so on.  The Multiplicative Persistence of a number is equal to the number of steps required to get to a single digit.  Really simple and pretty cool, isn’t it?

MP        n

0              0

1              10→0

2              25→10→0

3              39→27→14→4

4              77→49→36→18→8

5              679→378→168→48→32→6

6              6788→2688→768→336→54→20→0

7              68889→27648→2688→…→0

8              2677889→338688→27648→…→0

9              26888999→4478976→338688→…→0

10           3778888999→438939648→4478976→…→0

11           277777788888899→4996238671872→438939648→…→0

 

So, 77 has a Multiplicative Persistence of 4 because it takes four steps to get to a single digit, in this case, 8.  What about 11?  Why did we stop there?  Because 11 is the record, and 277777788888899 (commas aren’t necessary, right?) is the shortest number to share in that record.  Other numbers, with many more digits, tie our special number, but none beat it.  Did you get that?  Do you fully understand the strength of that statement? The conjecture is that any number, any single one you can think of, has a Multiplicative Persistence of 11 or smaller.  No number has been found that takes even 12 steps to get to a single digit.

This is quite extraordinary, don’t you think?  If you like, take out a computer and start coding.  Mathematical immortality awaits, but my guess is the search is futile, just like it is with the 10,958 problem I wrote about some time ago.  I think that a series of digits with a Multiplicative Persistence of 12 or greater, if it exists, would have been found a long time ago.

I am happy that I get to mention the great Paul Erdos before I close out this short post.  Erdos had a finger or two in this particular mathematical pie.  He suggested that we ignore all zeroes and just multiply together all the other digits.  After all, if you come across a zero, you are sunk.  This makes for a tasty mathematical stew.  There are people actively doing research in this area.  If you ignore zeroes, I have seen a Multiplicative Persistence as high as 22.  That said, I recently came across a paper on this very topic in French.  I tried to understand it as best I could.  The strange thing is that this is the first time I can remember that the specialized math did not lose me, I got lost in the language differences long before that could happen.  Those French, it’s like they have a different word for everything.   And yes, I tried to translate the page, but Google only decoded the numbers…

 

 

The Feynman Point

Richard Feynman was unusually intelligent.  He earned a Nobel Prize in physics.  He was one of those people who had a VIP pass to look at The Book virtually whenever he wanted. There are a bunch of biographies and autobiographies out there about him.  It wouldn’t be a waste of time to pick them up and read them.

This short post is about a little known aspect of .  There is a particular sequence in  that starts at what has been named The Feynman Point. The Feynman Point starts at decimal digit 762 and runs for six consecutive 9’s before it takes off, once again, on its random journey. How cool is that?

…2113499999983729…

That is not the only instance of consecutive digits, either. One of the most interesting sequences starts at decimal digit 1,699,927. You are not going to believe what happens at that point. There are six consecutive zeroes before it takes off again. I find that genuinely extraordinary.

…5105800000059277…

Wait, there is more.  How about 8 consecutive zeroes?

Starting at decimal digit 172,330,850 we get the first zero in this sequence:

…655810000000012202…

So, this brings up a question.  Since  is infinite, does that mean that, at some point in the sequence, we will get 1,000,000 zeroes in a row?  How about 10,000,000?  Furthermore, does contain every known, or possible, number string?  Will my DNA sequence show up at some point?  How about my Social Security number, that sequence is obviously a lot shorter.  I did check, and my Social Security number does not appear in the first 200,000,000 digits.  Neither does my phone number with area code.

The answer to the question I posed is unknown.  Mathematicians do not yet know if the universe lies within .  I don’t know what to think, the universe is a vast place but infinity is an awfully long “time.”

Give Me Some Space!

It is very difficult for old people to change their ways.
Buford Lister (personal communication)

A few years ago, I got an email from a friend of mine.  As I was reading it, I began to become irritated, then I became agitated, and then…well, I didn’t throw my computer monitor out the window, but I thought about it.  Why?  Was the content of the message that annoying and frustrating?  No, not at all.  I can’t even remember what the email was about.  What I do remember is that the author only put one space after each sentence, and I found that visual to be compact and quite disturbing.  Welcome to my world, a universe unto itself where the spacing between sentences is far more critical than the content of the text itself.

I am 57 years old, which means I grew up with typewriters; back in the day, personal computers were nothing more than a figment of somebody else’s imagination. I learned to type on an old mechanical device.  You had to push down hard on the keys to get them to strike with enough force to make an impression. Also, and this is the crucial point, everyone was taught to put two spaces between sentences.  That was how it was done, no questions asked.

Typewriters use monospaced fonts, which means that every character is given the same amount of space on the page.  An “I” and an “m” get the same area even though the “I” certainly doesn’t need or deserve it.  The use of monospacing led to a consensus that hitting the spacebar twice after a sentence was required to make it easier for the reader to see the end of one sentence and the beginning of the next.

We all know what happened, right?  Computers came along, and word processing programs started using proportional fonts, the type of fonts where an “I” gets less space on the screen, and the page, than an “m” or some other broader letter gets.  Before any of us knew what was happening, people were only hitting the spacebar once, dogs and cats were living together, and the ghost of Shakespeare was seen floating through English departments throughout college campuses worldwide.

The people who argue for one space after sentences hate, and I mean hate, to see two spaces being used anywhere.  They complain about rivers of white flowing through a passage of text.  It somehow offends them that there are still people walking the earth who prefer the two space method.   Sadly for them, I am a proud “Two-Spacer,” and I fully intend to die that way.  Hey, all you “One-Spacers,” do your worst, I am fully prepared for the onslaught.  Present the evidence in favor of your position, of which there is none.  Then sit back and behold the science supporting my position.

There was a study recently done; yes, you heard that right.  People take this stuff so seriously that someone is trying to further their academic career at a university somewhere by addressing this pressing issue.  The author of the study found that using two spaces after a sentence does increase reading speed as well as comprehension.  Take that!  Of course, the opponents say that the research must be flawed, how else could the wrong conclusion be reached.  So it goes…

I recently read a blog post somewhere about an older woman who was asking for advice about this issue.  She explained that she was too old to change, but she didn’t want her readers to think that her text was written by some sort of modern-day keyboard wielding buffoon.  So, what to do?  The reply was genius, shocking coming from a One-Spacer.  The One-Spacer said that the woman should type as she always does.  Keep right on tapping that spacebar twice, continue to do it out of habit, no worries.  When the document is complete, all she has to do is perform a search and replace.  Search for the two spaces and replace them with one space.  In one fell swoop, her document would then be acceptable for polite and sophisticated company the world over.  Not bad, right?

It is surprising (or maybe it isn’t) how worked up people get over this issue.  Lots of professional writers, as well as English professors and random commentators, take firm stances.  Their opinions are strong and unwavering.  While I much prefer two spaces, I am not going to take out a loan, purchase a tank, and go to war over it.  As for some of the others, I think they have already met with their credit unions.

Now for the big reveal, I have secretly left a trail of intrigue in this short essay.  I am conducting my own little, non-scientific study. I put two spaces after some of the sentences, and others got one space treatment.  Did you even notice?  Are you offended at this travesty?  My guess is no one noticed, but I bet you do in the future.  Once that genie escapes, they cannot be shoved back in the bottle.  Oh boy, I just used the word “they” to refer to a singular genie.  Not a bad segue to a future essay that I am finishing up now.  More on that soon enough.

 

Notes:  The article about spacing is entitled Are Two Spaces Better Than One? The Effect of Spacing Following Periods and Commas During Reading.  Rebecca Johnson, an associate professor at Skidmore College, led the team that conducted this outrageous and groundbreaking research.  Three cheers and a tiger for her and her colleagues, they are doing the world a service by putting those distrustful keyboard jockeys in their proper place.