How to Derive a Taylor Series

Some people have wondered how I come up with such serial columns as General l’Hopital (v75, 1997; v85, 2001) and Quantum Loop (v77, 1998; v81, 1999). In fact, creating such Taylor Series’ is not that difficult, and I shall now reveal the appropriate steps in their creation to anyone who cares, or is actually interested in making their own Taylor Series derivative.

  1. Select a story arc. Something you can effectively babble about for six issues. If it’s related to math, all the better. You don’t have to stick to said arc the whole time but people will appreciate knowing that the column will eventually stop. This may deter them from coming after you with sharp objects (the operative word being ‘may’). Prior arcs of mine include “mathematical punctuation”, and “why chairs were chained in MC”.
  2. Write each column the same week it comes out in mathNEWS. Don’t prepare anything in advance. This way you can slip in any current events as well as give yourself a crash course in stress management. They usually only take between 2 and 4 hours to do anyway.
  3. When you write, have a math textbook on hand. Liberally insert terms into the column. As an example, don’t say “You did something wrong”, say “You didn’t log the major factors” or “What you-knit was not a-tribute”. There are actually two ways to do this.
    • Write the column, go back and change the wording. In my case, most often done with QLoop. Classic QLoop comments include “Lynn Kedlist cursed, then recursed”, “My processes have better threads than you, Admiral” and “the CPU will be arrested for making con currency to buy hash”.
    • Pick out some good catch phrases first, or words that you want to use together, and fill in plot elements around them. In my case, most often done with l’Hopital. Classic l’Hopital comments include a paragraph containing words all starting with “ex”, “Gran U. Larity is into cayenne distribution”, “back us now or form”, and the redoubtable “Gram Schmidt ortho gone, Ali Zay’s shun process”.
  4. If possible, include some mathematical trivia and words like paraskavedekatriaphobia (fear of Friday the 13th) to maintain reader interest. For QLoop this was required, for l’Hopital it seemed a moral at the end would suffice. Doing this means there’s a chance someone will actually learn something by reading the column. Of course, there’s also a chance they’ll simply be totally confused, but if they miss out on the enlightenment that’s their problem.
  5. Finally, make sure to carefully spell and grammar check your work. Whether you have available editors on a production night or not, you don’t want them correcting intentional “kneed” or “monic” errors anyway.

So there you have it. Note that if you have a topic you can only babble about for one issue, it may still be useful (witness Sine Field). I now leave my legacy in the hands of others… but remember, old mathNEWS editors never die. They just get written off.

-Greg “hologrami” Taylor

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[This appeared along with the last General l’Hopital, Volume 85, issue 6, March 30, 2001. By this point I had basically already graduated University, though I would contribute a couple columns later, as I did my first year of teaching in Sept 2001.

Did you find this informative? Recall there was also the behind the scenes notes for General l’Hopital Entry 1.5 & 1.6, if you like. Now, if you came here looking for information about Brook Taylor’s infinite sums of mathematical terms… sorry about that? Hope you’re still entertained, thanks for reading, feel free to drop a comment below.]

General l’Hopital: Entry 2d

A subsidiary of Quantum Loop Enterprises

Theorizing that one could time travel within their own lifetime, Doctor Sham Breakit stepped into the Quantum Loop accelerator… and vanished. He awoke to find himself trapped in the past, facing mere images that were not his own, and driven by unknown source code to arrange history for the better. His only guide on this journey is Hal, an observer at run-time, who appears in the form of a hologram that only Sham can see and hear. And so Dr. Breakit finds himself looping through life after life, striving to put right what once went wrong, and hoping each time that his next loop… will be the loop $home.

“Sham, we got problems,” Hal Calalilli asserted as he entered the imagine chamber.

“No kidding!” replied Sham. After dispensing with Elly by sending her on her rounds, Sham had spent the last hour looking about the l’Hopital. “Hal, everyone here is talking in math riddles or doublespeak!”

“Sounds like the joy of lex.”


“Sham, you’ve handled worse,” Hal said. “Now, back in Stall-Eons Gate, New Mexico we’ve got real trouble! Your parallel hybrid computer BigE is having a mental breakdown. Sushi and Xina can’t fix her, so it’s up to… er, Dr. Geeks.” Hal caught himself before mentioning Sham’s wife Drawna WeeBTree, or Sham’s daughter Shammy Pro Filer.

“So I guess you don’t know why I’m here,” Sham sighed.

Hal tapped at his TI-85 calculator. “No, but we know the woman in the fating room is Dr. Waterson,” he offered.

“Doctor! Doctor Waterson!” called out Electra Lysis as she rejoined her associate. “Sir Cul wants to transcend dental work and eat pie. There’s also trouble with Jacob, Ian and May tricks. Oh, and Zeke wants help tracking down new classical records, but I just told him ‘Stop playing Haydn, Zeke’.”

Sham winced. “Stop speakin’ like that,” he pleaded.

“Like what?” Elly inquired. “I didn’t mean to go off on a tangent. Though I often strike a chord when not aligned with the story arc. But my maxim is, ‘If you’re cut, seek aunts’!”

“Sham, I was wrong,” Hal sympathized. “No matter what it takes, we’ll get you out of here.”


However, it was a few days before Hal could return with good news. “You won’t believe this, Sham,” he revealed at last. “The underlying situation here is a peculiar punctuation problem.”

“You positive?” Sham mused. “Because the Hopital logs show even their elementary operations involve calculations that are way off base.”

Hal flinched. “You’ve been here too long, Sham.”

“The trouble is I can’t project the point of origin for these errors!” Sham looked up. “Uh, Hal, can’t you wear proper ties?”

“Sham, what you need to do is make sure Dr. Waterson and Elly Lysis team back up with Max Value and his wife Ana. Elly’s presence is necessary to crack this missing punctuation case.”

“But what about the stats here? The error vectors on this scale are—”

“Stop it, you’re going dotty!” Hal interrupted. “Just get Elly to the Fourier Factory!”

“Wait, that’s it,” Sham realized. “Decimals and periods are both dots! There’s the connection, it even explains the missing cases of colon cancer. Brilliant, Hal!”

“Oh. Well, it was adjoint effort—” Hal stopped himself. “Anyway, just get that clique of people I mentioned back together so you can loop out of here.”

Sham quickly co-ordinated things. “But Hal,” he realized even as he left for the Fourier Factory. “I haven’t im-parted any mathematical information yet, like how there are 43,252,003,274,489,856,000 possible positions for a Rubik’s cube. How can I loop?”

Hal shrugged. “General l’Hopital is just less educational.”

Indeed, Sham did loop upon his arrival at the factory. However, the story did not end there.


Still going…

-Greg “hologrami” Taylor

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[Was this worth it, to hear about Shammy ProFiler? Or about Hal’s properties? Well, here’s something more random: Sham’s “stop speakin’ like that” line was a direct reference to FASS 2001, the February 2001 show for which I was CSW (chief scriptwriter). I’d written in a character who constantly spoke in music puns – a role I was ultimately cast into. In Act II scene iv, s/he was told to “stop speakin’ like that”. Weird that I still remember that 20 years later.

Anyway, that concludes Quantum Loop’s bonus episode of 2001. Two more parts remain in the Hopital!]

Quantum Loop: Entry 2f

“That went well,” Hal acknowledged. Sham was returning from the police office, where he had left a well documented statement. After all, once Sham looped, Ray Toobe would likely have no memory of the pivotal events.

“It’s remarkable that Millie came quietly,” Sham markedly remarked.

“I don’t think she really believed you could expose the truth,” Hal observed. “She seemed unphased.”

Sham shrugged. “I just figured that Millie’s encoding was going to have something to do with 357,686,312,646,216,567,629,137,” he pointed out. “No matter how you decapitate that number, you end up with another prime.”

“I don’t know how your swiss cheesed memory can recall stuff like that but forgets terms like the ‘lucky numbers’ or the ‘weird numbers’.”

“The weird numbers are abundant numbers without being the sum of any set of it’s own divisors. Like 70,” Sham noted.

Hal rolled his eyes. “If only you’d remembered that last loop…”

“So, what happens to Millie Niem anyway?” Sham continued.

“Oh… it looks like she gets off with a light sentence but they confiscate her viral device,” Hal answered, tapping at his calculator. “So everything’s back on track at the Project because of that… she doesn’t try it again. And hey, Sham, it looks like you accomplished your mission too!”

“What? I thought it was to provide room for terminals.”

“Yeah, well it looks like Millie’s son, Tita Niem, was going to prevent some initiative by the University of Mizuloo this year… that is, in the year 2000. But due to this disruption in her life, now Millie never has kids. And without the hinderance to the Mizuloo space program, it seems there is no longer a need for you to renovate the original MC design!”

“Oh.” Sham paused. “But then if I’ve fixed Y2K and accomplished what I was supposed to do here, why haven’t I looped yet? Am I supposed to prevent your alien invasion in the year 2000 too?”

“Oh… that turned out to be a hoax. It was a scam run by some cracked group of PCs,” Hal admitted. “And… it seems you’re still here because you need to render your art.”

“My art? Do I have to make a mosaic?”

“Hmmm… if ice in act, I’ve…” Hal smacked his calculator. “Sorry, you’re here to render your artifice inactive. BigE is playing with parsing…”

“My artifice?”

“You know, the thing you used to broadcast your mapping? It’s going to cause problems for some guy named Sine Field who ends up living on the sixth floor.”

Sham shrugged and detoured back to where he’d left his equipment. Arriving promptly at his destination, Sham Breakit switched off his device and began to dismantle it. “So, I guess that even though this wasn’t a routine loop, it at least has a good end,” Sham concluded.

“Uh oh… wait a minute,” Hal abruptly broke in. “It looks like there’s still a lot of unexplained Y2K problems in the future.”

“Pardon?” Sham mused, putting away his circuits and wires before looking back at his friend.

“Sham — someone else has a Y2K Bug!” Hal declared, shocked. “And… oh man, Sham, it’s…”

Sham looped.


–Greg “hologrami” Taylor

PreviousMISC INDEXBonus Next

[Thus ends “Quantum Loop” as a serial, in December 1999. I hope you enjoyed it. In the issue’s mastHEAD, I was pleased with myself for avoiding coffee to that point in my life. I’m pleased to say the trend has continued over 20 years later.

That said, there was ONE more entry, as part of the “General l’Hopital” serial in 2001. Once it’s published, I’ll switch the “Next” to link out to it. Beware, there’s no returning from that. Alternatively, you can look up the standalone “Sine Field” entries (a column about nothing), referenced above, of which there were four. Seems I loved to multiverse my stories even back then.]

Quantum Loop: Entry 2e

Hal looked up from his calculator. “The lucky numbers? They’re similar to the prime numbers. You start by striking out all the even numbers. Then after 1 you have a 3, so you strike out every third number. That gets rid of 5, 11, 17, etc. Now after 3 you have a 7, so take out every seventh number, and so on.”

“Oh, right,” Sham recalled. “13 is the fifth lucky number then, isn’t it?”

“Oddly enough, yeah,” Hal confirmed. “But you don’t have triskaidekaphobia, do you?”

Sham blinked. “That’s like paraskavedekatriaphobia, isn’t it?”

Hal shrugged. “Not quite. Look it up.” He punched at his calculator. “But it doesn’t look like that will work either. Your higher dimensions on the sixth floor are never used as extra space for terminals. You’ll have to try something else.”

Sham frowned. “Would the odds improve if I provided the mapping information directly?”

“We have zip on that,” Hal indicated.

“There’s that big room right in the middle – how about I just place some sort of device near that location which will activate in 1999?”

“Well, it would be difficult constructing it with the technology of this decade,” Hal said, nonplussed.

“It’s better than restarting,” Sham insisted. “Go get the Quantum Loop staff and BigE to work on it.”

“I think you’re using fuzzy logic.”

“Just open the imagine chamber door, Hal.”

“I’m sorry, Sham. I can’t do that,” Hal intoned ominously.


“It’s going blooie again…” Hal smacked his calculator and the door opened. “There we go. Wish we could thrash the originator of this Y2K bug.” Sham sighed, stacking some papers as Hal popped off.

Sometime later, Sham was prepared to put his plan into action. But Hal still didn’t have high hopes. “Sham, BigE has calculated a 90% probability that activating your device will only delay the 1999 reconstruction in the Red Room by 4 months. People will actually have LESS terminals to work on.”

Sham sighed. “Well, currently I have no alternative so let’s try the direct route. Here goes nothing.”

Sham switched on his abstract device. And Hal blinked. “Sham, in tracking your signal, BigE has picked up an analogous analog object!”

Sham blinked. “How do you account for that?”

“Someone else has an assembly!” Hal addressed Sham. “And… it looks like the other one is actually responsible for Y2K?!?”


“I’m trying to get a fix… but it’s looking like the Y2K Bug that hit us in 2000 was actually a virus initiated by someone named Millie Niem here in the 60s. That’s why even Y2K approved computers were affected!”

“I knew Quantum Loop was compliant,” Sham muttered.

“Sham, you can repair our problems in the future if you catch the person in the field nearby!” Hal finally declared.

“Would a net work?” Sham wondered, hurrying towards the location.

But as it turned out, a net was unnecessary. The person was Millie Niem herself, and she stopped willingly when Sham confronted her. However, she was not very forthcoming (or even thirdcoming).

“A virus to truncate dates to two digits in the Year 2000?” Millie retorted, laughing. “How fantastically fantastic! I’d like to see you validate such a claim.”

“You can prove it if you crack the encrypting of her device,” Hal assured Sham. “Unfortunately we don’t have a starting point.”

Sizing up the situation, Sham reached down and grabbed a bit of paper that had been left on the ground. “‘Use largest known truncatable prime’?” Sham read. “I think this will provide enough information to substantiate my statements.”

“That won’t help you,” Millie stated defiantly.

“On the contrary — I think everything can be determined now,” Sham retorted in satisfaction.

What has Sham figured out? Will all this really solve the problems with Y2K? And what about Sham’s mission to provide Mizuloo’s terminals? Find out when the story concludes in issue six…

–Greg “hologrami” Taylor

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[This issue also had more “Cynic’s Corner”, and the results of my survey on Saskatchewan. I’d needed some filler in the prior episode. I was kind of clever in my youth. Maybe I still am.]

Quantum Loop: Entry 2d

Hal walked into the Central Processing Room of Project Quantum Loop. “Any news?” he asked.

Sushi, the head programmer, stood up from behind the centre dais. “Dr. Geeks was looking for you earlier,” he replied. “That Katho guy in the fating room asked for some punched cards to toy with, and she doesn’t think he’s playing with a full deck.”

Hal nodded. “Well, I’m still trying to convince the nozzles who want to shut us down that we could fix Y2K better than their aliens.”

“As long as I don’t have to interface with all the infected computers,” the omnipresent BigE reminded. “It would cause hard wear.”

Hal nodded. “I’m currently working on just the right angle for my next presentation.”

“Then will things get back to normal?” Sushi mused.

“It may result in some basis of normality,” Hal shrugged. “So, anything else I should know before I see Sham?”

“The only people still reading this column are the ones looking for the occasional mathematical number theory,” BigE offered.

Hal blinked. “BigE, sometimes your statements don’t make a heap of sense,” he accused.

“This from someone in a lime green tinted suit. My processes have better threads than you, Admiral.”

Hal frowned, while Sushi patted a console, consolingly. “There, there. We’ll get to the root of this,” he assured.

Just then, Drawna WeeBTree entered the room. “Hal! How’s Sham?” she inquired.

“Haven’t checked on him yet, but I’m sure he’s doing fine,” Hal said with confidence.

Drawna nodded. “Xina was telling me that this looked like an easier loop.”

“Awk!” muttered Sushi, having gone back to his calculations. “Well, debugging this Y2K isn’t easy – I’m starting to think someone planned all this two-digit-bug business from the start.”

“I’m sure you’ll pull through when the chips are down,” Drawna comforted.

“Anyway, I’m off to the imagine chamber,” Hal declared. He grabbed his TI-85 and headed up the appropriate slope. Hal arrived holographically back in the 1960s shortly after, where he found Sham computing an inverse in verse. “Hey, Sham, how go the Harshad numbers?” Hal intoned.

Sham paused in his singing. “I decided that a number divisible by the sum of its own digits wouldn’t work for the mapping,” Sham responded. “But I think everything is finally in place now.”

Hal peered down at the schematics. “Sham, this floor labelling makes no sense,” he protested. “You’ve even labelled the potential elevators on each floor. 1101, 2079, 3093, 4115, 5220, 6312… where’s the pattern in that?”

“They all sum to numbers divisible by 3, except 4115.”

Hal tapped at the calculator in his hands. “Sham, no one figures this encoding out. There’s still the space problem in Mizuloo. Maybe you need to apply a mapping using the lucky numbers.”

Sham paused. “How do you define those again?” he inquired.

Can Sham still save the day? Are lucky numbers important? Do you feel lucky? Then keep reading when this series next continues…

–Greg “hologrami” Taylor

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[Happy New Year 2022! I still chuckle at ‘My processes have better threads than you, Admiral.’ Yes, we needed a scene set in the far future of 2000 to see the Drawna character. For those who don’t know, we now have the following Quantum Leap (Quantum Loop) name associations: SAM BECKETT (Sham Breakit); AL CALAVICCI (Al Calalilli); ZIGGY (BigE); TINA MARTINEZ (Xina); DR. “GOOSHIE” GUSHMAN (Sushi); DR. VERBENA BEEKS (Dr. Geeks); DR. DONNA ELEESE (Drawna WeeBTree). If you’re puzzled about Calalilli that’s a self-referential joke for hardcore Leap fans. Make sense?]

Quantum Loop: Entry 2c

Mr. Aba Cuss shook his head. “The fact that an abundant number is a number less than the sum of its factors (excluding itself) changes nothing. The first number of that form is 12 and I cannot present such a design to the architects. It’s too much.”

“I suppose such an abundant number of floors would be odd,” Sham admitted.

Aba stared. “An odd abundant number? There’s no way we could build a structure of 945 floors!”

“No, no! That is… well, can’t we add just one more floor?” Sham continued worriedly.

“Not a floor or a ceiling,” Ada responded. “Don’t you remember how our original design for the MC was that big castle structure…? Some people had real math news issues with that.”

Sham pondered for a moment, trying to determine what to do. “How about enlarging the floors we already have?” he proposed.

“Won’t do. Anyway, it’s all out of our hands now, our feat was merely working out the dimensions,” Aba reminded.

“And it’s too late to change them?” Sham verified. This loop was becoming infinitely more complex.

“Yes,” Aba declared firmly. “Though if you’re this concerned I can send along the schematics when they come in.”

Sham nodded. “Please do,” he requested.

A few days later, Sham found himself on a plane, observing cubism as he grappled with a tesseract. Indeed, he had managed to work out a diabolical plot on his graph paper. “You’re looking well co-ordinated,” Hal observed, stepping into the imagine chamber.

Sham jumped and fell, almost fracturing his spline. He spun to face the hologram. “Where have you BEEN?” he demanded. “And what are you WEARING?” he continued.

Hal’s pink tie was clashing with his plaid suit. “Aw, Sham, you don’t remember?” he chided. “I wore this tie last time you were in Mizuloo. Remember how my fifth wife graduated from here? …or maybe it was the fourth…”

“Nevermind,” Sham sighed.

“Anyway, sorry about the loss of contact. BigE is better but we’ve had our hands full in the year 2000 what with the aliens.” Sham stared. Hal continued. “See, they seem to be negotiating with the President about fixing the whole Y2K Bug using superior alien technology. But they’re asking for a lot of money. Thus the government is considering rerouting the funds currently going to Quantum Loop.”

“What on earth on you talking about?”

“No, they’re not from Earth, that’s the point. But anyway, what are you up to here?”

Sham decided to ignore Hal’s aliens for now. “I’m fixing the Math & Computing building’s problem,” he explained. “It seems like I ended up getting here too late to alter the three dimensional construction – so I’m adding a few extra dimensions to the sixth floor.”

Hal blinked. “How would anyone know to look in n-space to install extra computers??”

“There are mathematicians in the building, right? I figure I just have to introduce the proper labelling system for the doors and people will be able to work out a mapping from that. Access follows.”

Hal looked dubious. “Access at Mizuloo is not known for it’s reliability,” he stated. “What are you going to be basing this mapping on?”

“Probably something to do with Harshad numbers. What do you think?”

“I think I have no clue what you’re talking about,” Hal concluded.

What is Sham up to now? Is he as spaced out as the aliens in the future? Are you as lost as Hal? Then look for the continuation in two weeks time…

–Greg “hologrami” Taylor

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[A prior issue of mathNEWS had featured a castle structure as cover art. Access was the University’s unreliable software program for co-op. With the inside jokes out of the way, I’ll just mention that the previous Loop entry fit very nicely into one column. This entry, not so much. The joys of layout, when one is the editor.]