Hello everyone, my apologies for the absence of posts in December. For anyone even tangentially involved in education, December poses an energy sapping nemesis. I intend to start this year off strong, and with that goal, I am readying a post for publication later this week. In the interim I would like to share some statistics that WordPress (the host and tool I use to maintain this blog) has provided me.

This blog is extremely young; it is only five months old, yet it has done very well. I hope it is not too vain to post these results. I put a lot of effort into this blog, and I am glad people have read it and enjoyed it. I did not expect such a good reception and so many readers, especially not within only five months. Thank you, and happy New Year!

————Below this Line is an automated post prepared by WordPress—————–

The stats helper monkeys at WordPress.com mulled over how this blog did in 2010, and here’s a high level summary of its overall blog health:

The Blog-Health-o-Meter™ reads Wow.

Crunchy numbers

A Boeing 747-400 passenger jet can hold 416 passengers. This blog was viewed about 1,400 times in 2010. That’s about 3 full 747s.

In 2010, there were 14 new posts, not bad for the first year! There were 92 pictures uploaded, taking up a total of 59mb. That’s about 2 pictures per week.

The busiest day of the year was November 22nd with 298 views. The most popular post that day was Fibonacci Findings.

Where did they come from?

The top referring sites in 2010 were reddit.com, facebook.com, stumbleupon.com, twitter.com, and alphainventions.com.

Some visitors came searching, mostly for hypatia, fibonacci number and symbol pattern, john barth frame tale, masolino, and bunnicula.

Attractions in 2010

These are the posts and pages that got the most views in 2010.

Whoever wants to understand much must play much. –Gottfried Benn

This post has been a long time coming. For almost a month, my spare time has been filled by the brainstorming and experimentation that would become this post. It is probably the least organized post I have composed, yet it is also the most mathematically rigorous. What follows are several interrelated proofs annotated with the motivations for them and the logic behind them. This is a log of my mathematical play, and as such it is the closest this site has yet come to being a blog proper. The mathematics in this post can become dense and requires a good knowledge of algebra for full appreciation, however I have endeavored to explain things fully and have included full derivations that would normally be left absent in mathematical papers due to triviality. Feel free to ask me for clarification or enhancement on anything in this post.

The impetus for this stems from a result I found surprising while preparing the post that became Zombi(Nacci). In that post I investigate the rate of growth of a zombie population and find that it follows the Fibonacci sequence. This was not a result I knew beforehand. When creating the thought experiment and meditating on it, I was surprised to find the Fibonacci sequence naturally emerge from it. There is a peculiar joy to arriving at such a famous and simple solution unexpectedly, and it fueled my curiosity about the Fibonacci sequence and its properties. As such, my free time has been spent in its investigation.

The sequence is widely known even outside of scientific and mathematical circles, for, as a friend put it, it is simple and easily understood. It is. For reference here is its definition (this post relies heavily on mathematical symbols, and rather than struggle with the html to format them correctly, I have instead prepared images whenever they are necessary):

The final line states symbolically that each term of the Fibonacci sequence is the sum of the preceding two terms. The first ten terms are: 0,1,1,2,3,5,8,13,21,34. As stated, the first two values, 0 and 1, are called seed values and govern all the terms to come. You can apply the recurrence equation to any pair of seed values and arrive at a variety of different sequences. For instance, if the seed values are 1 and 3, you arrive at the Lucas Numbers named after the French mathematician Édouard Lucas. The first ten Lucas Numbers are 1, 3, 4, 7, 11, 18, 29, 47, 76, 123.

Interestingly, you can also use the definition to extend the sequence backwards, that is to extend it to negative terms. To do this you begin with the second Fibonacci number, 1, and ask yourself the question: “What number would I add to the preceding Fibonacci number, 0, to obtain the current number, 1?” Of course, the answer is 1; 0+1=1. Then you repeat the question, this time with 0: “What number would I add to the preceding Fibonacci number, 1, to obtain the current number, 0?” This time the answer is -1. Once again you attempt to answer the question: “What number would I add to the preceding Fibonacci number, -1, to obtain the current number 1?” The answer is of course, 2. So far the extended sequence looks like this …2,-1,1,0,1,1,2… Notice that it appears that the sequence is identical with the positive sequence except that its terms have alternating signs. This is indeed the case. These are often called the negaFibonacci numbers. The first ten negaFibonacci numbers are: 34,-21,13,-8,5,-3,2,-1,1,0.

For the remainder of this essay, we will be confined to the non-negative Fibonacci numbers, however the negaFibonaccis are an interesting extension of the sequence and I would be remiss not to mention them.

The next part of my investigation was to compare proceeding pairs of Fibonacci numbers. That is, to compare the current number to the previous number. I wanted to see if the numbers would eventually differ from one another by a common factor. To do this I divided each Fibonacci number by the previous number for fifty terms. Here are my results:

As you can see, the number 1.61803389 quickly emerges as the common quotient, but is it exact, and is it significant? What I’m attempting to do is find out if F(n)/F(n-1) is equal to some constant number for extremely large ns. Mathematically speaking this is known as looking for the “limit” of the sequence. If a mathematical statement has a limit, it means that as the terms become arbitrarily large the relationship tends towards a constant value. In the case of the Fibonacci numbers that value appears to be 1.61803389, but the evidence given by the preceding chart is insufficient to declare a mathematical truth. All evidence is circumstantial in mathematics and something is only true if it can be proven. To that end, I attempted to derive the limit of F(n)/F(n-1). To accomplish this, I first assumed that the limit did exist (sometimes they do not), and called it L. Note that limit is abbreviated lim, and that if the limit exists, then it is the same for F(n)/F(n-1) as it is for F(n-1)/F(n-2), that is limF(n)/F(n-1)=limF(n-1)/F(n-2)=L. Here then, is my derivation:

That final statement, (1±√5)/2, indicates two values. One value is (1+√5)/2, which is approximately the value we found before, 1.6180339… The other value is (1-√5)/2≈-0.6180339… Interestingly, (1-√5)/2=1-(1+√5)/2. However, more interesting is the fact that this is the answer to a well-known mathematical problem, that of the Golden Mean. The definition and derivation of the Golden Mean follow. Notice that the derivation is similar to the derivation of the Fibonacci limit:

The result obtained is often split into two ratios, the positive one being called the Golden Ratio, and often represented by the Greek letter phi(φ). Here are the explicit definitions and approximations for φ and 1- φ:

This establishes a relationship between the Fibonacci sequence and the Golden Ratio. It indicates that given any Fibonacci number, you can approximate the next Fibonacci number by multiplying the current one by (1+√5)/2, that is F(n+1)≈φF(n). Leonardo of Pisa discussed the Fibonacci sequence in relation to rabbit populations, and the ancient Greeks found the Golden Ratio fascinating due to its frequent occurrence in geometry; it is amazing that two pursuits so different from each other can have such an intimate connection. Math is full of these connections, and their presence and meaning are constantly fascinating to me. In this case, the connection goes even deeper.

The Golden Ratio has an interesting property: Its square is only one more than itself. Symbolically, φ^2=φ+1. This is a direct result of its derivation, however it can also be arrived at by squaring (1+√5)/2:

This property indicates that any power may be expressed in terms of an integer multiple of φ. Examining successively larger powers, a pattern emerges:

The numbers at the bottom of each column are the coefficient of (number in front of) φ and the number being added to it, respectively. I set them aside to draw attention to them. Notice the progression of φ’s coefficients, 2,3,5, and the second terms, 1,2,3. They are both Fibonacci numbers. The number being added to the multiple of φ is one Fibonacci number behind the number being multiplied by φ. In general it seems that the pattern is this: φ^n=F(n) φ+F(n-1). As with the limit of the Fibonacci numbers, the results found in these trials are not conclusive and our conjecture that φ^n=F(n) φ+F(n-1) requires a proof. Unfortunately, this proof requires methods that are more complex than the direct algebraic derivation of the previous proof. The proof I found relies on a proof strategy called mathematical induction, and to understand its success, you must first be introduced to this method.

Mathematical induction is a powerful proof strategy. Despite its name, it has no relationship to inductive reasoning, which is not considered rigorous in mathematics-mathematics striving to be a totally deductive discipline. Mathematical induction is often used to show that a given statement is true for all natural numbers (non-negative integers), and since we are dealing with the non-negative Fibonacci numbers it is perfect for our case. It can be intuitively understood like this: If it can be shown that a given statement is true for the simplest case and that if it is true for one number, it must be true for the following number, then you have proven it to be true for all natural numbers. Dominoes are often used as a metaphor for mathematical induction. You set up the first domino, and explain how if it topples, it will cause every successive domino to fall.

Mathematical induction can be split into three sections. The first section, The Basis Step, shows that the relation holds for the number. The next step, The Induction Hypothesis, is to assume that the relationship is true for an arbitrary number k. The final section, The Inductive Step, is to show that if the relationship holds for k, then it must hold for k+1. However, mathematical induction is probably best understood through example. Here is a simple proof using mathematical induction. The statement being proven is this: The square of an odd number is always odd. Notice that an odd number is always one more than an even number and therefor can be written as 2n+1. Using this notation, the conjecture is (2n+1)^2=2m+1. Here is the proof:

The Square of an Odd Number is Odd

The small square at the end is known as a mathematician’s tombstone, or, less romantically, a Halmos symbol named after the amazingly prolific mathematician Paul Halmos who popularized its usage. It is used to indicate the end of a proof as are the initials QED that stand for Quod Erat Demonstrandum, which is Latin for “what was to be demonstrated.”

Using induction, we can prove the conjecture made before, φ^n=F(n) φ+F(n-1):

Result 1

Notice that the final statement, φ^(n+1)=F(n+1) φ+F(n) is exactly what we were looking for, because it is merely φ^n=F(n) φ+F(n-1) with 1 added to each n. This proves another connection between the Golden Ratio and the Fibonacci numbers. However, we have neglected half of the Golden ration that is approximately, -0.6180339, 1- φ. Does this value have a connection to the Fibonacci numbers? Some experimentation reveals that it may:

Again, the coefficients of 1- φ and the numbers being added to it follow the Fibonacci sequence. The conjecture is this (1- φ)^n=F(n)(1- φ)+F(n-1). This relationship can also be proven by induction:

Result 2

I had previously seen a function relating the Fibonacci numbers to the Golden Ratio, and had not understood it. Upon proving these two relationships, however, I am now able to derive it. It is known as Binet’s Formula after the French mathematician, Jacques Philippe Marie Binet. What is remarkable about it is that it takes a discrete relationship, the Fibonacci sequence, and finds a continuous analogue. Informally, in mathematics discrete means that a mathematical structure has no “intermediary” values. For instance there is no one third Fibonacci number, or a 12.5 Fibonacci number etc. Continuous is the opposite. This entire post has been my derivation of Binet’s Formula, and what follows is the final piece. I’m certain that there are more efficient ways to do it, and that I’m probably not the first to derive it this way, but this was how I proved it and understood it:

It is extremely satisfying to finish a proof and write QED at the end. They rival fin for finality, and perhaps disenthrone it. With those letters I affirm that I have truly understood something, and I know that its truth will far outlive me. They are denouement.

Here I compare the growth rate of a zombie population to that of vampires and werewolves. I had found that zombies grew according to the Fibonacci numbers and that vampires grew exponentially. However, I faked the graph. At the time I didn’t know of a continuous way to represent the Fibonacci numbers and therefor couldn’t adequately compare their growth to that of an exponential function, so I merely drew a line at the time when zombies would surpass the human population. However with Binet’s formula this can be done. To illustrate the differences in growth, this comparison assumes that vampires feed at the same rate as zombies. That is, I compare Binet’s formula directly with 2^x. As you can see, 2^x increases much quicker than do the Fibonacci numbers.

Notice the slight undulation between the first and second Fibonacci numbers. I do not know what that means. Neither do I understand the significance of the intermediate values that Binet’s formula returns. Binet’s Formula also occasionally gives complex results because (1-√5)^x can be imaginary. This indicates a relationship between the Fibonacci numbers, the Golden Ratio and the imaginary number i. I don’t understand any of these things now, but I’m looking forward to finding them out. Thank you for joining me in these investigations. QED

We could hardly ask any one, even did we wish to, to accept these as proofs of so wild a story. –Bram Stoker, Dracula

Though now it is almost a month since Halloween, and in internet time, that’s a length almost inconceivably distant, it is my intention to revisit that holiday once more. For me, the question of this post’s seasonableness is a nonentity for I am of that breed that anticipates Halloween for half a year and spends the other six months in reminiscence of the previous ‘eve. I had intended to write this earlier and publish it sooner so that it may have been timelier, but other responsibilities took precedence. The four posts I wrote for Halloween, Vampire Numbers, Infernal Integers, Zombi(nacci), and The Sum of All Fears, were wildly popular. The number of hits on Halloween weekend exceeded the site traffic received during any other similar period, and the residual popularity of those posts have made November my most successful month.

Of the Halloween posts, Vampire Numbers received the most attention. This is almost certainly due to the attention given to it by Clifford Pickover, the mathematician who defined vampire numbers. Next up was Zombi(Nacci) followed by Infernal Integers. Disappointingly, the post that went least read was also the post that took the most effort to write: The Sum of All Fears. In retrospect, this is not surprising. It is written in a tone and style quite different than any of my previous posts. At ten pages long, it is of intimidating length as far as internet documents are concerned. Worse, the first five pages are oddly organized exposition. The heart of the post doesn’t begin until page five, and its mathematical content doesn’t begin in earnest until page six. I fear that this presented something unappealing to most readers and that many found this surface impregnable. I apologize for this, and would like to devote the rest of this post to explaining some of The Sum of All Fears’ oddities.

The Sum of All Fears is my attempt at writing a mathematical “weird-tale.” I’m a great fan and collector of horror literature, especially the eccentric, romantic, gothic literature of the nineteenth century and the tone of The Sum of All Fears is my parody of the lexicon and phraseology of those works. Its organization (as a collection of documents) and introduction is a parody of Dracula, which begins in a very similar fashion. In fact, it had initially been my intention for the final Halloween post to be merely a link to a short story by Bram Stoker,The Judge’s House, which has a math student as its protagonist. However, I felt that this was an unsatisfactory, anticlimactic way to end what had been a great creative week for me, and so I endeavored at the eleventh hour to cobble together my own horror story.

The post is loaded with allusions to the horror genre. To someone not wise to their inclusion, their presence probably makes the whole article even more opaque. I will try to provide some clarity by explaining the different references here. I apologize for the absence of mathematics in this post, but the post following this one is mathematically dense, and more than makes up for its absence here.

“The Miskatonic Messenger” is an invention of my own. It is a newspaper serving the communities residing near the Miskatonic River in Massachusetts. The Miskatonic River is a fiction of the prolific horror/fantasy/sci-fi author, H.P. Lovecraft. According to him, the river flows from north central Massachusetts to the fictional town of Kingsport just north of Salem, Massachusetts. He created an entire region surrounding the river and locates several fictitious towns around it. These include Arkham, and Dunwich, where my story takes place.

The seal following the introduction is a design created by other fervent Lovecraft fans for Lovecraft’s Miskatonic University. Several of his stories mention this institution and it is within its mysterious library that a copy of the dreaded and forbidden Necronomicon is held.

The date of the first newspaper article is June 6, 2006 or 6/6/06. Its awkward title, Dunwich’s Exciting, Adventurous Double, was merely an excuse to force in an acrostic for DEAD. Robert Kams, is an anagram of Bram Stoker. The name of professor mentioned in the second paragraph, Andre Delambre, is the name of the scientist in the 1958 version of The Fly. Larry Talbot is the doomed protagonist from the 1941 movie, The Wolf Man. The letter penned by “D” is of course written by Dracula, and he intends to stay at Hill House, Shirley Jackson’s famous haunted house from The Haunting of Hill House.

The next article is dated June 11, 2006, which was the day after the first full moon in June of that year. The first wolf howls were heard on Elm St., the eponymous street from Wes Craven’s A Nightmare on Elm St. The first victim of the werewolf is Frank Cotton, the first victim in Clive Barker’s Hellraiser.

Dr. Giancomo Rappaccini is the prototypical mad scientist from Nathaniel Hawthorne’s Rappaccini’s Daughter. The family he works for, the Torrances are the family from The Shining. Their daughter, Regan, shares the name of the possessed from The Exorcist. Finally, she goes to Bates High School, the high school of Carrie, which is itself an allusion to Norman Bates of Psycho.

Riget Hospital is the name of the hospital in Lars von Trier’s strange TV series, Riget, which inspired Stephen King’s Kingdom Hospital. The name of the first doctor mention, Dr. Stauf, is an anagram for Faust of legend. Dr. Brundel is the scientist in David Cronenberg’s The Fly (1986).

Regan dies on Wednesday, July 19 giving 43 days from the onset of her anemia as is the course of the malady that afflicts Lucy Westenra in Dracula. Her parents are Gomez and Morticia from The Adams Family. The poem, “The Death-Bed” by Thomas Hood, printed with her obituary was excerpted by Abraham van Helsing to eulogize Lucy in Dracula.

The name of the author of “The Dunwich Problem, its Global Implications, and the Need for Immediate Action,” Philip Ward, is the name of a recurring character in Lovecraft’s fiction who may represent Lovecraft himself. The excessive amounts of post-nominal titles are a parody of van Helsing who appends his notes in Dracula in a similar fashion. Similarly, his arrogant, eccentric style is a parody of van Helsing’s.

There are a smattering of Latin phrases throughout his paper; their meaning is thus: Barba tenus sapientes– is a phrase attributable to Desiderius Erasmus that means something like “only superficially intelligent.” Ars magna– “The Great Art.” This was Gerolamo Cardano’s term for algebra. Dat deus incrementum– By god they increase. Finally, Pax Dei, Pax Americana, Pax Mundus- Peace of God, American Peace, World peace.

The chief editor of the Journal of Miskatonic Studies, Paul Sheldon is the writer from Stephen King’s Misery.

The towing service is named after Ripley from the Aliens franchise. The truck towed is owned by the Umbrella Corporation from the Resident Evil franchise. It is insured by Milton, Chadwick & Waters, the law firm from The Devil’s Advocate. The vehicle crashes into Crystal Lake, the lake from which Jason Voorhees emerges in Friday the 13^{th}. It carries the T-Virus, again from the Resident Evil series.

I hope this in some way makes the post more approachable and enjoyable. I had great fun writing it, and hope that people will enjoy it as well. However, I realize that its enjoyment is dependent on the reader having a great deal of specialized knowledge. If you haven’t already read it, I hope these notes allow you to understand its structure, and if you have read it, thank you, and I hope that with these explanations you will enjoy it more. While it was fun to write, it may have been too idiosyncratic a piece to include here. I’m already writing to a limited audience by creating a math-centric site, and when I insist that they also have an encyclopedic knowledge of the horror genre, I may be cutting my audience down to an extreme minority. Perhaps only one:

No snowflake in an avalanche ever feels responsible.

-(Disputed) Voltaire or Stanislaw J. Lec

Now that Halloween is behind us, it is time to move onto really frightening matters. Tuesday November 2^{nd} is Election Day in the United States, and democratic elections are very deeply mathematically problematic. The primacy of the democratic system has been so often repeated to twentieth and twenty-first century citizens that most people do not consider examining its structure. However, if its foundations are inspected, one does not find pillars resting on bedrock, but impossible Escherian structures that cannot exist, but do. At the base of democracy is voting, and voting is embroiled in many paradoxes.

Whoever gets the most votes wins, whoever is voted against the least wins, right? It would be an absurd for an election to conclude with the candidate that the majority voted against winning, wouldn’t it? Consider the following: Four candidates, Groucho, Harpo, Chico, and Zeppo are in a very tight political race. When November comes, and the votes are tallied these results are found:

Clearly, Zeppo has triumphed- he wins! However, the vast majority, 70%, did not vote for Zeppo! Sometimes in democratic elections, the will of the majority can be overlooked.

It would seem fundamental to a democracy that the will of the majority is accounted for, and numerous contingencies have been proposed in the event of a situation like this. The most common is the requirement of a run-off election between the top candidates. Does this alleviate the problem? Consider another election, this time there are three candidates: Moe, Larry, and Curly. There are also three types of voters:

Moe Voters: Moe Voters ideally prefer Moe to Larry, and prefer Larry to Curly.

Curly Voters: Curly voters prefer Curly to Larry, and prefer Larry to Moe.

Larry Voters: Larry voters ideally prefer Larry, but are split in their preference of the other candidates.

There is an election producing these results:

From the first round, no one holds a majority- no one obtained more than fifty percent of the votes. So there is a run-off election between Moe and Curly. In this election, the Larry Party is split with 8% voting for Moe and 4% voting for Curly. Here is the rundown:

Moe wins with 53%! However, it is questionable how well this result represents the will of the majority. If the contest was between Moe and Larry instead, the Curly voters would prefer Larry to Moe and give their votes to Larry. So would all the Larry voters. Therefore, when it is Moe v Larry, Larry wins with 55%. Likewise, in a contest between Larry and Curly, the Moe voters give their votes to Larry, and Larry again wins, this time with 57%. Notice no matter the outcome, it is hard to justify the statement that the will of the majority is being served.

This paradox is due to preference being an intransitive relation. A relation is transitive if when the relation holds between the first element and the second element, and it holds for the second element and third element, it also holds for the first element and the third element. For example, if Fred is bigger than Barney who is bigger than Wilma, we can conclude that Fred is bigger than Wilma. Size is a transitive relation. However, if Rhet loves Scarlet and Scarlet loves Ashley, we cannot conclude that Rhet loves Ashley.

Love is Intransitive

Love is an intransitive relation. Preference is intransitive like love, and it is because of this that the paradox emerges. This paradox, The Voting Paradox, was first recognized by the eighteenth century French mathematician and philosopher, the Marquis de Condorcet. It was latter revived by Duncan Black, a Canadian economist, and became part of Kenneth Arrow’s 1972 Nobel Prize winning work. Kenneth Arrow presented five conditions that are essential to democracy. They are listed and explained in Morton Davis’s 1980 book, Mathematically Speaking:

1.) The decision making procedure must yield a unique preference order. Whatever the preferences of society’s members, the procedure should come up with one and only one preference order for society.

2.) Society should be responsive to its members. The more individuals like an alternative, the more society should like it too.

3.) Society’s choice between two alternatives is based on its member’s choices between those two alternatives.

4.) The decision making procedure should not prejudge. For any two alternatives X and Y, there must be some possible individual preferences that allow society to prefer X to Y. Otherwise, Y is automatically preferred to X and the group preferences are unresponsive to those of its members.

5.) There is no prejudgement by an individual. Arrow assumes there is no dictator, that is, society’s choices are not identical to the choices of any single individual. If this condition didn’t have to be satisfied, it would be easy enough to find a voting mechanism, but Arrow wouldn’t consider it representative of the individuals in the whole group.

These five precepts have been almost universally hailed as reasonable conditions for a democracy. Arrow did more though, he proved that it is impossible to have a democratic system in which the will of the majority always wins and have it satisfy all five of his conditions. If Arrow’s 5 conditions are defining of democracy, then that means that for there to be a democracy the will of the majority must sometimes be overruled. These paradoxes are then inherent to our electoral system, if it is indeed democratic. I offer them here only as something to think about when you are waiting for the voting booth. That is, if I haven’t scared you away from it.

If voting could change anything, it would be illegal.

NOTE: This post is reference heavy. For those that feel that they may be missing something I have explained all the allusions here.

Welcome back, dear reader, to the finale of our Samhain Celebrations. I have been honored to be your humble guide, but my term is now up. For this final piece, I bow out and let the voices of history be heard:

AUTHOR’S PREFACE

I will make no remarks of how the documents comprising this manuscript have come into my keep. Neither will I comment on their inclusion, arrangement, or order; the reader of this story will very soon understand how the events outlined here constitute a logical whole. Out of respect for the dead, and in deference to those who wish to present these events to the eyes of the public, I have contained all editorializing to this prefatory note, in all other respects, I leave the manuscript unaltered. Without controversy, it can be asserted that there is no doubt to the veracity of the events documented within, nor is there any call to question the authority of the witnesses, however absurd their testimony may at first appear. I am quite convinced that the plot described herein will always remain to some extent incomprehensible, despite continuing advances in psychology and natural science, for as the bard remarked, “there are more things in heaven and earth/ then are dreamt of in your philosophy.”

Massachusetts,

October 2010,

W.B.

“The Miskatonic Messenger”

June 6, 2006

Dunwich’s Exciting, Adventurous Double

Robert Kams

Editor in Chief

DUNWICH- The oft-forgotten people of Dunwich are clamoring with excitement over two new arrivals to their town whose patronage could put them on the map. You may be forgiven if you don’t know of Dunwich, its insignificance was all but officialized when a recent congressional candidate made the remark, “that’s not a real town, not in Massachusetts; it doesn’t exist,” but real it is. Located in north central Massachusetts just beyond Dean’s Corner, Dunwich is a community that seems to have never recovered from the Great Depression, and the recent economic slump has done it no favors. A casual inspection by a visitor will reveal that most of the houses are deserted and falling to ruin, and that the broken-steepled church now harbours the one slovenly mercantile establishment of the hamlet. The houses that are inhabited are home to some of the poorest and least accounted for citizens of the northeast. On The Hill, it has long been widely accepted that the unspoken policy concerning Dunwich is one of planned negligence. “Let the lonely, curious country reclaim it,” one legislator is rumored to have said.

Significant demographic data has not been collected from Dunwich since the 1965 census, but its current population has been estimated to be at most 400, which would make it one of the smallest, if not the smallest towns in Massachusetts. Furthermore, the population is almost certainly shrinking, though the causes of this are poorly understood. The trend is clearly not attributable to emigration from Dunwich, because truly no one can be said to have met a man from Dunwich. The “Dunwich Decline”, as it has become known, is an often discussed curiosity among the faculty of Miskatonic University’s School of Sociology. Former Professor Emeritus, Andre Delambre PhD., has become infamous for his remark concerning the flagging population: “[They] was no doubt eat up by what he had call’d out of ye Sky.”

Despite all this, the average Dunwichian remains quietly resolved to her isolated, rural existence without complaint. This stoicism has recently been broken as the town’s Main St. is decorated for the first time in decades. It can never be said that the atmosphere in Dunwich is electric, but this is as close as it ever has been. The cause of this commotion is the unexpected announcement that two persons of international interest and influence will be making Dunwich their place of residence, at least temporarily. Both are European nobleman of some distinction.

The first is a Mr. Larry Talbot of Wales. He explains that his visit to The States, and specifically Massachusetts is in the interests of his health. In Dunwich he has found just the quiet and solitude his doctors have proscribed. It seems our fortune is unfortunately at Mr. Talbots misfortune, for he has come to Dunwich convalesce after a recent animal attack. When questioned about the matter, Mr. Talbot merely responded that the animal was “a great wolf.” We here at “The Miskatonic Messenger”, and everybody in the Miskatonic Valley wishes Mr. Talbot the greatest expediency in his recovery.

Our second distinguished guest is far more enigmatic. Rumoured to be of Central European descent and of an ancient family, little else is known of him. His influence and finances must be considerable, however, for in the months prior to his arrival, he had purchased and renovated the admirable Hill House. Despite repeated attempts by this office, no interview could be arranged by the time of printing. However, we have received a letter from our new neighbor, and in the interest of the public we present it here in full:

My Dear Mr. Kams and company,

I regretfully inform you that I will be unable to provide any interviews, now or in the foreseeable future, and the foreseeable future is, for me, very far indeed. I keep unorthodox hours, and I’m afraid they are currently occupied with my hunger for knowing your country. Please do not take this too harshly. In Romania, our ways are not like your ways, and acquaintances grow slowly. I’m sure in time our bond will be as blood.

Your Friend,

D.

The promise of these two arrivals is unmistakable, and the enthusiasm in the small town of Dunwich is infectious. We at “The Miskatonic Messenger” wish the best to our eminent guests. It would seem that the darkness around Dunwich is coming to an end. The future now looks brighter than ever.

“The Miskatonic Messenger”

June 11, 2006, evening edition

Bevy of Animal Sightings Interrupt Summer Night

DUNWICH- The night of the tenth was a sleepless night for many as residents across the town of Dunwich and the surrounding countryside were awoken from their slumber, not by the cockcrow, but by the howling of an animal that has not been seen in the Miskatonic Valley for more than forty years. Around 10PM, residents near Elm St. reported hearing a wolf’s howl, and shortly after, reports were being made from every part of Dunwich. Several courageous citizens left their homes in search of the animal, leading to a smattering of sightings. Eye witnesses are consistent in their reports that the animal was a large canine. One witness, a Mr. Frank Cotton, claims to have been bitten by the beast, but his wounds are consistent with no known animal. Arkham’s Animal Control assisted the Dunwich Police Department in attempting to capture the animal, but in spite of their efforts it eluded them throughout the night. The search has continued throughout the day, but no trace of the animal has been found. Officials are urging people to contact the police if they see or suspect the presence of the creature. Under no circumstances, they say, should you approach it yourself.

Diary of Dr. Giancomo Rappaccini

Family Physician to the Torrance Family

June 18

Another sleepless night! I feel an unyielding shame stemming from my utter incapacity to offer any meaningful assistance to poor Regan. Her symptoms are all consistent with severe anemia, but I am unable to account for the lost blood. A thorough physical examination reveals no wounds large enough to account for her hypovolemia, though two anomalous puncture wounds were found on her neck. She claims no memory of how she came to have such wounds, and I am inclined to dismiss them as inconsequential to her current condition. I have only just finished reviewing the results of a bone marrow biopsy of her femur I had ordered, and it offers no clues to her current condition; the bone marrow is healthy and functioning. I have put Regan on bed rest and proscribed Chromagen. Poor Regan had difficulties in school last semester and has had to go to summer school. She will unfortunately not be able to attend. I have contacted Bates High School to explain her absence.

Log of Riget Hospital’s Emergency Services

July 10- July 11

6/10 11:05, 34y/o M w/ contaminated animal bite to LUQ. Dr.Stauf sutured wound. Rx: amoxicillin

6/11 3:15, 20 y/o F w/ contaminated animal bite to R ant. forearm. Dr. Brundel sutured wound. Rx: amoxicillin

Regan Torrance Obituary

July 22

Regan Torrance, 17, passed away in her sleep on Wednesday, July 19. A well-known and regarded student of Bates High School, she will be remembered by its faculty and her classmates for all the years to come. Her loss will be felt most deeply by her parents, Gomez and Morticia. She was fond of music and literature and had aspired to pursue a career as a music journalist by applying to Miskatonic University’s school of journalism. A vigil for her will be held tonight at her family’s tomb.

WE watch’d her breathing thro’ the night,

Her breathing soft and low,

As in her breast the wave of life

Kept heaving to and fro.

So silently we seem’d to speak,

So slowly moved about,

As we had lent her half our powers

To eke her living out.

Our very hopes belied our fears,

Our fears our hopes belied–

We thought her dying when she slept,

And sleeping when she died.

For when the morn came dim and sad,

And chill with early showers,

Her quiet eyelids closed–she had

Another morn than ours.

-Thomas Hood, The Death-Bed

“The Dunwich Problem, its Global Implications, and the Need for Immediate Action”

paper submitted to The Journal of Miskatonic Studies by Philip Ward M.D., D.Ph.D, D. Lit, Hon. B.D.S., H.S.T. Ph.D.,MA M.D.E., HVS, Esquire, Etc. ,Etc.

August 7

It has been known even to the layman for some time that there have been unnatural events in the New England hamlet of Dunwich, however the academic community, and through its patronage, the political community have proven all of their negative stereotypes by consistently failing to respond to what even cursory examination proves to be an imminent and grave pattern. I speak to you of the lycanthrope! Of the hemophage! Of their threat to humanity individually and plurally! Yes, Dunwich has become residence to the damned. Its nights haunted by the werewolf and the vampyr! How slovenly does this revelation prove the intellects of its academics? Why, one requires no more education than I to divine the fearful symmetry that the facts plainly harbor. Barba tenus sapientes.

The presence of these devils can clearly be seen in the face of the commonest of common people encountered along this damned town’s streets and byways. It is the human psyche that is the best barometer for the spiritual storm to come, and it takes no keen observer to see that it is low. But exactly how close are the clouds, and how violent will the thunder be? To answer these questions we will turn to the queen of the sciences, the ars magna.

It is well known that our two antagonists are plague bearers governed by their hungers. As they feed they multiply, and it is this multiplication that poses for us the greatest threat. Let us consider the case of the vampyr in a simplified case: We begin with 1 vampyr, the Primarch, and it chooses a human to convert. After a feeding period, the human dies and is resurrected as another unclean. Now there are 2 vampyrs, and they must both feed. There is another feeding period, and afterwards there are 4. Naturally, all four vampyrs are subject to the thirst, and they produce four more. Now there are 8. Allow me to make the pattern explicit: 1,2,4,8,16,32,64… After each feeding period, the vampyr population doubles. This is a pattern of geometric growth. Dat Deus Incrementum. It will be used as the basis in defining the vampyr function, v(n), which will give us the number of vampyrs after n feeding periods, with the information we have now, v(n)=2^n. Notice that when n=0, v(0)=2^0=1 (any number raised to the zero power is one), because prior to any feeding periods, the number of vampyrs is one, the Primarch.

We have assumed that our vampyrs are not gluttonous and partake of only one mortal per feeding period, however let us consider other alternatives. Consider a more robust breed of nosferatu that enjoy the ichor of two humans during each feeding period. Again, we begin with 1 vampyr, but after the end of one feeding period we now have 3 vampyrs. Each of these three vampyrs will spawn two more, yielding six new vampyrs for a total of 9. Again, these nine will add eighteen, and there will be 27 total vampyrs. When each vampyr is adding two more to its coven each period, the vampyr population progresses like this: 1,3,9,27,81, 243… After each time period, the population triples. In this situation, the progression can be modeled thus: v(n)=3^n. If our vampyrs are insatiable suckers, and they must drink from three mortals per period, the progression would be: 1, 4, 16, 64, 256… and the function would be v(n)=4^n. Notice in general the base of the vampyr function is one more than the number of mortals each vampyr feeds from. We can incorporate this into our model. Let m be the number of mortals fed upon during each period, then v(n)=(m+1)^n. Notice that if the vampyr isn’t feeding, that is if m=0, then there is no progression in the vampyr population for v(n)=(0+1)^n=1^n=1.

To even the amateur cryptozoologist, it is evident that the werewolf population follows a similar pattern. Are they of equal threat then? To answer this question we must be able to compare the vampyr function, v(n), to the werewolf function, w(n). However, there is a problem: Both functions rely on the breeding periods of their respective devils, and these periods are not equal. Also, these periods are non-human temporal units and it would be far more useful if they were expressed in something more tractable to the layperson. So this will be our goal: To render these periods in mortal days, so as to achieve a better count of the sands in our glass- technically speaking, we are normalizing the time periods. For werewolves, this task is simple, for it is well known that these creatures are lunatics, married to the moon, and therefore the solar month serves as their period, that is the werewolf’s feeding period is once per 29 days.

For vampyrs, this task is slightly more complicated, for they are as a rule more idiosyncratic. If any one member of their sinister species can serve them as a model it is that treacherous Transylvanian, Count Dracula. Luckily for us, the records of his activity are well preserved and the mean feeding habit of a vampyr can be extracted from it. We will assume that he began his awful conquest of Lucy Westenra in the early morning of August 8^{th} and continued this cruel task until her death on September 20^{th}, this gives the vampyr feeding period as 43 days.

We can now create more precise models. New members are added to a werewolf pack once every lunar month, that is 1/29. A vampyr coven expands once every 43 days, giving 1/43. Given these new facts, we can modify our functions like this: v(d)=(m+1)^(d/43), w(d)=(m+1)^(d/29), where d is the number of days since the monster arrived and m is the number of mortals they feed on each period.

We are now ready to assess which one is the greater threat. Will vampyrs or werewolves consume us first? We will use 6.8 billion (in scientific notation, 6.8*10^9) as the current world population. When will vampyrs wipe us out? When there population equals or exceeds ours. When will that be?

v(d)=6.8*10^9

(m+1)^(d/43)=6.8*10^9

log(m+1)^(d/43)=log(6.8*10^9)

(d/43)log(m+1)=log(6.8)+log(10^9)

d/43=(log(6.8)+9)/log(m+1))

d=43(log(6.8)+9)/log(m+1)

For our purposes, we will assume temperate terrors, so each monster has one victim per period.

d=43(log(6.8)+9)/log(2)

d~=1405

Therefore, vampyrs end us all in approximately 1,405 days, or about 3 years and 10 months.

On the other hand, for werewolves we have:

w(d)=6.8*10^9

2^(d/29)=6.8*10^9

d~=948

Werewolves get us in 2 years 7 months. Of the terrors, werewolves are decidedly more terrible.

Our campaign is clear. First kill the lupine, and then stake the undead. Only this way will we ensure our survival. We have little time, but at least we possess it. If we had been plagued by another fearsome fauna, our time would already have been up. For the lethality of the walking dead has been well demonstrated:

The implications are clear. Pax Dei. Pax Americana. Pax Mundus.

Memorandum from “The Journal of Miskatonic Studies” to Philip Ward.

August 13

TO: Philip Ward

FROM: Paul Sheldon, PhD. Chief Editor of The Journal of Miskatonic Studies

DATE: August 13, 2006

RE: “The Dunwich Problem, its Global Implications, and the Need for Immediate Action”

Your article, “The Dunwich Problem, its Global Implications, and the Need for Immediate Action” fails to meet our publication standards, and will not be included in the next volume of this journal. Furthermore, we will decline to review any additional articles bearing your authorship until you have provided us a full and verifiable curriculum vitae. It is the opinion of this journal and its editors that several if not all of your degrees are fictitious. Your consistent misspelling of ‘vampire’ has not improved your reputation in these offices. Please take these criticisms into consideration before again submitting a paper to us.

Sincerely,

Paul Sheldon

Service report from Ripley’s Large Vehicle Towing Inc.

August 14

DATE: August 14

LOCATION: Elm St.

VEHICLE TYPE: Chemical Truck

REGISTERED: Umbrella Corporation

INSURANCE: Milton, Chadwick & Waters

MISC: vehicle ran off road into Crystal Lake. Majority of shipment lost. Carrying: T-(illegible)

Ah, you are back. You’re braver than you look, that’s good. You’ll need all the nerves you can muster here. You see, we’ve created a little problem: Our mathematical, metaphysical musings have created quite a stir. The deceased our crossing Heaven off of their plans for the hereafter, and Hell is full to bursting, and when there is no more room in Hell, well, you know what happens:

Yes pilgrim, I’m sorry to say it, but we are on the brink here. Today is what Max Brooks calls Z-Day, the beginning of the Zombie Apocalypse! That cold chill you are feeling is the natural reaction to this news. These cannibalistic cadavers are gruesome guests indeed! And look- there comes one now! What a sinister, shambling, spook! Watch as the lurching living dead approaches – it is really quite pathetic with its ponderous pace. Does it really pose such a grave threat? Well let’s find out.

We all know that the most dangerous part of a zombie is its maggot-ridden maw. A love bite from that and you will soon be shambling yourself. Yes, the bite is the zombies’ main form of propagation. Inevitably, someone bitten by the undead becomes undead themselves. This pattern will serve as the basis of our model.

For purposes of simplicity, let us assume that the current world population of about 6.8 billion is evenly distributed across one massive, flat continent, and that each person is within one hour’s walking distance of at least one other person. In this world, a zombie can reach at least one victim an hour. Under these conditions, we will try to define a function that predicts the population of zombies at any time after the progenitor zombie has been encountered.

We will define three stages of the zombie disease: 1.) Exposure: This is when a person is bitten. 2.) Metastasis: The time during which a person is infected, but not yet a zombie. For our purposes we will assume this to be one hour. 3.) Zombie: Without exception, exposure leads to a person becoming a zombie. So zombiism progresses like this: Exposure – Metastasis – Zombie.

Now, consider the first five hours of the outbreak:

Hour One: There is 1 zombie (the progenitor zombie) and it bites one person (1 person is exposed).

Hour Two: There is 1 zombie, 1 metastatic person (the previous exposed), and 1 exposed (the progenitor has bitten another person).

Hour Three: There are 2 zombies (the progenitor zombie and the previously metastatic person), 1 metastatic person, and 2 exposed (both zombies bite people).

Hour Four: 3 Zombies, 2 metastatics, 3 exposed.

Hour Five: 5 Zombies, 3 metastatics, 5 exposed.

Hour Six: 8 Zombies, 5 metastatics, 8 exposed.

Look at the progression of the zombie population: 1,1,2,3,5,8. Do you notice a pattern? Each new term is the sum of the proceeding two: 1+1=2, 1+2=3, 2+3=5, 3+5=8… A sequence that is determined by its preceding terms is said to have a recurrence relationship, and we can use this to help us define our Zombie Function. Let the function Z(t) indicate the zombie population t hours after the appearance of the progenitor zombie, using the recurrence relationship we just observed, we define Z(t) like this: Z(t)=Z(t-1)+Z(t-2), given Z(1)=1, Z(2)=1.

Now that we know the Zombie Function, how does that help us? Well, we can use it to see if the human race has any chance of survival. That is, when and will the entire human population become zombies? To help us answer that question, here the Zombie Function values for the first fifty hours:

…So short answer: Yes, and soon. In less than 49 hours, that is, in just over two days, only the dead walk the Earth. Today being the 29^{th}, that will make for a gruesomely, ghoulishly appropriate Halloween.

I can understand those of you who may no longer wish to read this, and would prefer to spend your remaining time huddled futilely with your friends and families, but for those that choose to stay, there is more to consider. You have probably realized that the numbers produced by the Zombie Function are indeed the Fibonacci numbers. This famous sequence (1,1,2,3,5,8,13…) naturally occurs in all manner of phenomenon from biology to finance for reasons similar to the reasons it describes zombie outbreaks. It was first studied by Indian mathematicians, most notably, Pingala, Virahanka, Gopāla, and Hemachandra. Drawing upon their work, Leonardo of Pisa, better known as Fibonacci, discussed the sequence in his 1202 book, Liber Abaci. We owe a great deal to this book, for it was this book that popularized the use of the efficient numeral system we use today (1,2,3,4,5,6,…) and our method for writing fractions, both of which Fibonacci borrowed from the Indian mathematicians. In Liber Abaci he discusses the Fibonacci sequence in an application dealing with rabbit populations. For as you know, rabbits breed like zombies, and it makes you wonder which we should truly fear more.

There was a flash of lightning and in its glare I noticed that Mr. Monroe was carrying a little bundle- a bundle with tiny glistening eyes.

-Bunnicula, James and Deborah Howe

Tune in tomorrow night, as the monster math continues

Through me you pass into the city of woe: Through me you pass into eternal pain: Through me among the people lost for aye.

Justice the founder of my fabric mov’d: To rear me was the task of power divine, Supremest wisdom, and primeval love.

Before me things create were none, save things Eternal, and eternal I endure. All hope abandon ye who enter here.

-Dante Alighieri, Inferno

Good evening lycans and gobble-uns. Welcome to Spherical Cow’s continuing Halloween festivities! But don’t allow yourself to get too comfortable, this is merely the assembly point. Account for all your limbs and baggage, and dress light because we are going on an expedition, and tonite I’ll play your Virgil. Ah, now by candlelight, I see the apprehension in your face- the unspoken realization… Yes, heed the poet’s warning well, for our destination is the Valley of Jehoshaphat, Gehinnom, Jehennam, Gehenna, Tartarus, Hades-Hell!

La Porte de l'Enfer, Auguste Rodin 1837

I shall audience no arguments against our descent for the reservation of our reprobation is nontransferable, and where we’re going, people are dying to get in.

“Where is that,” you ask. To which I respond:

“Come now, you must have heard of it. People have been screaming about it. Why, it is all the rage. It’s hot! I tell you- hot! Here, peruse the brochure.” I hand you The Good Book (King James) opened to Revelations 21:8 :

the fearful, and unbelieving, and the abominable, and murderers, and whoremongers, and sorcerers, and idolaters, and all liars, shall have their part in the lake which burneth with fire and brimstone: which is the second death.

Yes, our vacation spot – The Second Death. While the climate may not be everyone’s cup of tea, you can’t beat the view. But how bad is heat? Well, our best clue is given by the fact this is a lake of brimstone. Brimstone is an archaic word for sulfur, which became conflated with the concept of Hell through its odious odor. The historic Valley of Hinnom is the model for Hell in Judaism, Christianity, and Islam. This site essentially served as Jerusalem’s landfill. It had previously been a ritual site for the worshipers of Moloch, an ancient Semitic god, and the ancient Jews felt that its transformation from shrine to junkyard was poetic justice. Like the tire yard of Springfield, constant fires consumed the detritus in the valley. The rot, smoke and heat created an intolerable place – a place of fire and brimstone. Thus Hinnom became legendary. It became Gehinnom then Gehenna, and latter when the Hellenes were converted to Christianity it was forced into Hades, which became Hell.

But how bad is the heat? Well it is a lake of sulfur, and therefore the temperature of Hell must be great enough to melt the element but less than or equal to its boiling point (otherwise it would be a vapor). The melting point of sulfur is 239.38°F, and its boiling point is 832.3°F. From this we can conclude that an average day in Hell has lows of about 240°F and highs around 830°F. Hellishly hot indeed!

Oh! Leaving so soon? We have only just arrived! Is the heat getting to you? That would explain why you are behaving so rashly. Before you go running back to your Beatrice, you may want to consider this:

Moreover the light of the moon shall be as the light of the sun, and the light of the sun shall be sevenfold, as the light of seven days

-Isaiah 30:26

Isaiah paints a bright picture of Paradise. Perhaps too bright. Did you bring enough sun-block? Let’s check: I propose we keep a tally of Sunshine. He tells us the Moon in Heaven is as bright as the Sun on Earth, so currently the amount of Earth Suns shining on Heaven is one. Next we learn that Heaven is irradiated each day by sevenfold the light of seven days, that is 7×7=49 Suns. Adding this to the light from the Moon we find that salvation has the equivalent of a blinding 50 suns! To get the requisite 15SPF per sun, you would need to apply 750 layers of Banana Boat.

However, it is not just your sun block budget that would be hurting. A useful thermodynamics law will allow us a clearer view of the sky firmament. The Stefan-Boltzmann law for radiation allows us to compare the absolute temperatures of Heaven(H) and Earth(E) with the amount of radiation being absorbed from the suns. Without getting too technical, here is the formula simplified. The 50 is representative of the radiation from the 50 suns.

(H/E)^4=50

The UN’s Intergovernmental Panel on Climate Change gives the Earth’s temperature as 255K. Setting E=255, we obtain:

(H/255)^4=50

Solving for H reveals that the temperature of Heaven is approximately 678K or 760°F! As you can see, sometimes Heaven is hotter than Hell! And even when it isn’t, it’s still hot as hell. Perhaps it would be better to avoid Heaven and Hell altogether and aim strictly for an extended time in Purgatory.

So dear reader, I bid you farewell once again. But the Witching season is not over yet, and the haunting here continues. If you haven’t read part one of my Tricks and Treats, do so now, and be sure to join me again soon as Spherical Cow’s Halloween continues.

You better mind yer parunts, an’ yer teachurs fond an’ dear, An’ churish them ‘at loves you, an’ dry the orphant’s tear, An’ he’p the pore an’ needy ones ‘at clusters all about, Er the Gobble-uns ‘ll git you Ef you Don’t Watch Out! – James Whitcomb Riley, Little Orphant Annie

This post was inspired by the classic article, Heaven is Hotter than Hell, published anonymously in Applied Optics (1972). I have made a number of additions and changes to enhance and update the original concept. You can read the original here.