East of Eden - As Good as it Gets

  

HS #100 2023.11.9

 

East of Eden – As Good as it Gets

 

“East of Eden” is the most powerful and insightful novel I have ever read. The characters live large, perhaps because they were real characters in John Steinbeck’s life. 

 

Whence the title? According to the Genesis account, when Adam and Eve sinned by eating from the Tree of Knowledge of Good and Evil, they were removed from Eden. Then when sin continued in Cain who murdered his brother, he was further exiled to a place east of Eden. 

 

Exiled from one’s ideal home – from where, in your soul, you know you belong. Can you relate?

 

In one way or another, many, likely most of us, find ourselves living east of Eden - removed yet in sight of our ideal homes. Perhaps a lost job or career opportunity for which you were perfectly suited. Perhaps a beloved spouse who files for divorce. Perhaps a lost child – either by death or rebellion. Perhaps the abandonment of a close friend or friends who reject or forsake.  Perhaps an illness or accident which limits one’s enjoyment of life.  Perhaps the loss of a dream – of a reality hoped and prayed for, but never actualized. Indeed, as I write these, I can think of examples of each among those close to me.  Likely you can as well. 

 

There are two kinds of poverty. If your belly hurts from lack of food, that’s absolute poverty. But if you feel poor compared to your neighbors or your past self, that’s relative poverty. Living east of Eden is poverty compared to one’s past or one’s potential. 

 

How does one live east of Eden? How do we continue our lives removed from our real home? 

 

 Cain’s appeal to God, “How am I to live in exile?” got little consolation: “If anyone kills you, I’ll avenge you.” Big deal. So Cain leaves the picture. He blew it and must bear the consequences – he is written out of history, and lives whatever remains of his life in oblivion, east of Eden. 

 

But the key to “East of Eden” is the advice God gave Cain prior to his sin. God says, “Sin desires to have you, but you must rule over it.” That is the NIV translation. However, in Robert Alter’s Hebrew Bible, the translation is “but you WILL rule over it.” 

 

Which is it? Is God giving Cain a command (must) or a promise (will)? Neither. The Hebrew word is “timshel” and that word is the foundation to Steinbeck’s novel. As Steinbeck explains, “timshel” is neither a command nor a promise. Instead it is a statement of fact. “Timshel” means, “Thou mayest”. “Thou mayest rule over sin.” God is telling Cain he has the power within himself to make the right choice. He is not a slave to his upbringing, nor to his genes, nor to his whims, nor to the expectations of others. Instead, he has the awful power and responsibility to create his own destiny – to determine his own future. He has the power to choose. 

 

There is nothing gentle or “nice” about this. It’s tough and brutal.  It’s a shake and a slap. This sort of message flies in the face of modern sensibilities. We live in an age where dentists apologize for the prick of a needle, where parents and teachers are instructed against using negative words. Some employers have difficulty keeping young workers because they will leave if criticized. Criticism is a new experience for them. We are living in an age where we presume the right not to feel badly. 

 

Indeed, it’s human nature to avoid responsibility for our mistakes. Watch any pickleball player whose overhead swing misses the ball. S/he looks at the paddle with disgust. Stupid paddle!  This behavior goes right back to Adam who said to God, “The woman you gave me made me eat of the Tree.” Yep – it’s God’s fault. 

 

But even though Cain lost Eden, according to the Biblical record he still found relationship – perhaps the greatest human need. And this too relies on timshel. Indeed, the focal point of Steinbeck’s novel is when despondent Adam Trask is told the awful truth of his unfaithful wife. Tough medicine, but it shocks Adam out of his lethargy and into relationship with his sons. Through timshel we have hope and consolation that although we may not live in our Eden, our life may still be, as per the theme of the Jack Nicholson Academy winner, “As Good as it Gets.” 

Lesson from Far Side

  

HS #99 2023.10.12

 

Lesson from Far Side

 

My favorite Far Side comic (any reader who didn’t enjoy Far Side, Bloom County and Calvin & Hobbes during the Golden Days of newspaper comics has my condolences) is of two weather-beaten men lost at sea in an inflated life raft in the midst of ocean in all directions. Drifting towards them is a wooden crate  marked, “Pins, Nails, Broken Glass.” As one of them reaches for the crate, he says to his companion, “Don’t know what use we have for pins, nails, and broken glass, but let’s bring it aboard.” 

 

What’s the humor? Obviously, there would be little if any benefit from pins, nails and broken glass. On the other hand, the negative consequence of bringing those things aboard is obvious. Might even say inevitable. One doesn’t know which particular item will puncture the raft, but it’s just a matter of time before it happens.  

 

A foolish decision.  Agreed? 

 

What’s the lesson?  Put the right conditions together and tragic results are bound to happen. This was essentially the lesson in Shakespeare’s “Romeo and Juliet.” Given the bad blood between the families, it was inescapable that some event – it happened to be a mix up of communication – would lead to tragedy. 

 

Conversely, by reducing the number of potentially dangerous factors, one can lessen the possibility of adverse consequences. 

 

 We apply the principle broadly. We know, for example, that mosquitoes have three stages of life: egg, larvae, adult. So by lowering the number of mosquitoes surviving each stage (minimizing standing water and spraying adults), we most effectively diminish the number of mosquito bites. Smart. 

 

Similarly, we know that fire requires fuel, oxygen, and heat. Thus, whenever two are present, we carefully avoid the third. If all three are together, fire is likely.  So when enriched oxygen is present, heat and combustibles are limited. 

 

This thinking led to seatbelt laws. Unrestrained human bodies in cars caused tens of thousands of deaths from auto collisions. Requiring seatbelts and mandating airbags has dramatically reduced the number of deaths. 

 

Such decisions depend upon and reveal the values of a given society. Seatbelts are annoying, but we have decided that saving lives outweighs the inconvenience, so we require them. 

 

Unfortunately, because of the messed-up values of our society, this preemptive type of thinking doesn’t apply to deaths from firearms.  

 

“Guns don’t kill people. People do.” Such is the silliness one often hears. Of course it’s false. The correct answer?  People with guns kill people. Both ingredients are necessary.  

 

If a second-grade teacher discovers that her students are using sharp scissors to jab each other, she addresses the behavior problem, but in the meantime, she takes away the scissors. If any student disputed, “I want my scissors – scissors don’t stab us, children do!”  she’d have a good belly laugh. The prevalence of easily-used fire arms together with the number of stressed/disturbed people makes the continuing tragedy of numerous gun deaths as predictable as the fate of the hapless men on the raft. 

 

Indeed, what’s easier to control, a person’s psyche or a piece of metal? If we valued human life more, we’d realize it’s ludicrous not to control both of the factors causing deaths from firearms. Stalin famously said, “The death of one man is a tragedy. The death of a million is a statistic.” He was right, and we are seeing the truth of it play out before us.

 

Yes, yes, yes, we have a Second Amendment. So let’s be faithful Constitutional originalists and allow muskets for security.  Anything beyond that is more than the Founding Fathers had in mind, so other types of firearms must be justified for non-Constitutional reasons.  

 

Interestingly, “standing on rights” is antithetical to Biblical teaching. Jesus didn’t talk about rights, but instead about love as a guiding principle for living one’s life. The Apostle Paul admonished the early church in Corinth not to claim their right to eat what they want, but instead to make choices that edified others. Even the Old Testament Yahweh was concerned about His people living in community rather than claiming their own rights and privileges. 

 

How bizarre that so many are fascinated with carrying – concealed or openly – a weapon whose only purpose is to kill others. Ludicrously, our own county (Ottawa Impact) is promoting it. Other civilized countries look at us with incomprehension. Future generations of Americans will doubtless look back in wonder at our abject foolishness. But, for the present, we must live with our choices.  Insanity. Truly. 

Discerning Moral Truth

  

HS #98 2023.9.14

 

Discerning Moral Truth

 

When teaching my senior seminar at Hope College, I gave the assignment of writing a paper selecting one of two titles: “I am a Seeker of Truth” or “I am a Child of God.” I explained that these are not mutually exclusive. However, if someone on the street asked, “Who are you?” although there are myriad possible responses, a person’s answer would reveal what is most central to their identity. Similarly, they were to pick the title which best described their core. 

 

Personally, I pick the former. I live my life very imperfectly pursuing truth, beauty and love. 

If there is a Supreme Being who encompasses these in all their fullness, then worship and love is the natural response. Thus, being a seeker of truth can set one free to become a devoted Child of God as well.  

 

How does one seek truth?  Scientific research is a powerful means to converge on truth about our natural world. The process of experimentation – repeated by others who are motivated to find flaws – doesn’t prove anything true with certainty, but it does lead to increased confidence. 

 

But how does one discern moral truth – what one “ought” to do? Especially for the Child of God, how does one combine scripture, culture, and one’s own experience? A recent Holland Sentinel column by Bill Sutton (July 30) on the LGBTQ+ issue suggested that the Bible preempts all else. I think it’s more complicated than that. Why? Whether using the Bible or any other source, EVERYTHING is interpreted and understood by our faulty minds. And everything, even the Bible, comes to us through imperfect humans. Thus, all must be interpreted in light of the others. 

 

It wasn't long ago (1970's) that same-sex attraction was considered by psychologists to be a malady - something from which to be cured. Then, gradually, two things were realized: i) sexual attraction doesn’t change, ii) gays and lesbians enjoy healthy happy wholesome lives (except for rejection by some) by living in accordance with their same-sex attraction. Since a guiding principle of the medical world is “Do no harm” the medical community changed its stance on the issue. Was this wrong, or a step towards finding moral truth?

 

Also, sometimes culture seems to be ahead of the church in recognizing Biblical truth. 

Women’s suffrage, abolition of slavery, the 1990 Americans with Disability Act were all issues which are in step with biblical precepts, but where secular society took the lead. I can imagine a child of a slave holder in the 18^th century understanding the Bible as endorsing slavery. Then, the child becomes friends with children of slaves and sees life through their eyes. Suddenly, it is OTHER verses of the Bible which impinge upon the youngster. The Bible hasn’t changed, but the eyes reading the Bible have. They have been sensitized by experiencing a different culture. Is this wrong or a step towards discerning moral truth? 

 

In explaining why there is pain in the world, there is a “baby test.” Any explanation must be able to be given to a couple who has just lost an infant child. I thought of that recently when I met two beaming and wholesome gay 30-something men vacationing in the UP. Having met at “Teach for America” they have been a couple for seven years and were obviously enjoying their vacation and their lives together. What would anyone against gay marriage say to them? 

 

Should they separate?  No longer be friends? Should they be friends, but not live together - robbing them of the joy of each other's constant companionship? Should they live together, but unlike every hetero couple, resist the urge to embrace each other fully in love? Pats on back? Quick hugs? Long hugs? Walks hand in hand? Sleep next to each other? 

 

What if one gets aroused? Then quickly move to another room or bed? Take a cold shower? Repent of their sinfulness and return to separate beds with guilt and sense of their fallen nature? 

 

Years ago, after Hope College professor David Myers gave a talk defending same-sex relationships, a burly construction-clad middle-aged man stood up and remarked that no matter Myers’s argument, he still believed it was wrong. He was followed immediately by a similar looking man who said, “Last year I would have said the same, but this past Christmas my son told me he is gay – and I love my son.” Should such be discounted when finding moral truth? 

 

All worth pondering. 

Kayaking the Au Train

  

HS #97 2023.8.10

 

Kayaking the Au Train

 

As you read this, I am likely kayaking the Au Train River in the UP of Michigan. Since Henry David Thoreau wrote an entire book detailing his life on Walden Pond, I will allow myself a newspaper column detailing how to enjoy the good life in Au Train, which sits at the widest North-South point of Lake Superior.  

 

I do so with a bit of trepidation. In eight years of writing columns for the Holland Sentinel on myriad topics including religion and politics, nothing brought the chastisement as did last year’s column on the Michigan UP.  I lovingly described it as a place of fresh air, clean water, and rusty pickup trucks. A few (possible) BMW owners took offense. Oh well, here goes: 

 

First, the kayaking. Genesis 5:24 says that Enoch walked with God. Interesting word, “walked.” It connotes making continual progress, but at a non-tiring rate. Kayaking is even better since one can let one’s mind wander while drifting and/or paddling with no need to give attention to curbs and traffic.  A broad stream or river allows one to be close to nature – hidden water falls, turtles, muskrats, bald eagles, blue herons, and occasional otters – while immersed in conversation or lost in one’s own thoughts. 

 

Do you know the song, “The Happy Wanderer”?  “I love to go a wandering – along the mountain track. And as I go, I love to sing – my knapsack on my back.” I changed the lyrics to “I love to go a kayaking, along the forest stream. And as I go, I love to sing, and think and plan and dream.” Filling the silent woods with robust vocalizing is perhaps the reason I don’t see deer or moose. 

 

My favorite kayak is the Pungo 120. It is long enough to maintain a straight line when desired, and short enough to turn easily. The three-way adjustable seat and easy-access storage are the best I’ve found and the open cockpit allows for bent knees. The burnt orange color repels deer flies that are attracted to the blue one, but the blue color is more easily hidden in the weeds. 

 

Camping with a kayak or canoe is also a great experience. Canoes carry an SUV load of supplies, while still providing almost effortless movement. There are challenges, especially if you have never been without a biffy. Campers devise their own method – some sit on a log, others squat. I’ve called my own idea the “crap-strap.” You likely get the idea from the name - you sit suspended in it as it hooks on the branch of a tree. I’m told someone else had the same idea and name – and took the effort to manufacture them. 

 

Another challenge – finding wood and breaking the larger logs for firewood. With a canoe, it’s easy. Float around the shore where usually there are dead sun-dried limbs suspended above the water. Break them off into the canoe – you’re all set. 

 

How to break the thick ones? I call it “prop and drop.” Prop the branch with one end on a stone or log  and the other end on the ground. Drop a heavy rock on it, then pick up the pieces. 

 

Of course, one can also go exploring by car. I spent two weeks in England this past June, but nothing in London’s museums surpassed the novelty of Lakenenland Sculpture Park 17 miles west of Au Train on HYW 28. The 40 acre park (easily driven or walked) contains 80 sculptures by local artist Tom Lakenen.  The medium is mainly scrap metal and each has a thoughtful and humorous political message. 

 

The most intriguing item is the welcome sign at the entrance which warmly invites everyone to enjoy the park free of charge 24/7 – EXCEPT the “local members of the planning and zoning boards” who apparently obstructed his plans. There is a legal NO TRESSPASSING sign there forbidding them from entering. 

 

Want to bring some of Au Train home with you? Go berry picking. How to find wild blueberries? Look for cars parked on the side of the road for no apparent reason. Search around tree stumps in particular. Stop after 90 minutes (everything becomes work after 90 minutes – ask any teacher) or when you find yourself bargaining with the mosquitoes. 

 

Wild raspberries are even better – great for freezer jam. Wear long pants and best to use a container that is graduated, large open top, and hangs on the belt. A hospital urinal is perfect. 

The Present Age

  

HS #96 2023.7.13

 

The Present Age

 

I always advise college students to take courses in Introduction to Psychology and Modern Philosophy. In these two courses one meets thinkers from the past who still influence our lives: Skinner, Freud, Kant, Marx, Nietzsche, Hagel, and Kierkegaard among others. 

 

Case in point: This past spring, playing pickleball on a Saturday morning at Central Park Church, I found myself thinking of Kierkegaard’s, “The Present Age” which I read in college. 

 

Though written in the mid 19^th century, it describes aspects of our present age. Kierkegaard tells a story of a man who has fallen through thin ice on a river. As townsfolk gather on the nearby shore, individuals venture out on the ice toward the drowning man. As each gets closer to the man on the dangerous thin ice, the crowd cheers in admiration for their bravery. Receiving the accolades, they return to shore, giving others the opportunity to do the same. 

 

Does the drowning man get saved? Who knows – it’s not worth mentioning.  The important thing is that people have shown their courage and have been recognized for it. 

 

Absurd!  What’s the point? 

 

Perhaps it’s making a similar point as the story of the parent of an infant who enters a burning house to retrieve a favorite picture of their beloved infant, but leaves the infant. 

 

I think the point of both is the folly of valuing symbolism over substance. I don’t know how that inversion showed itself in Kierkegaard’s day, but there are clear present-day examples. 

 

President George H. W. Bush certainly put symbolism over substance when he (among others) advocated for a Constitutional amendment prohibiting the burning of the U.S. flag. Our Pledge to the Flag concludes “with LIBERTY and justice for all.” Perhaps the most important of these liberties is free speech – including the right to proclaim one’s disagreement, disgust, even loathing towards one’s country. How can one express such sentiment more forcefully than by burning a U.S flag? So those in favor of this amendment chose to value the symbol – our flag – more highly than the substance – our liberty. They were willing to trade the infant for a picture of it. 

 

I also think of the symbolism-over-substance fallacy whenever I hear of Dance Marathons in which participants spend an entire night dancing for a good cause. Townsfolk and businesses donate money so that participants will spend a fun night dancing. 

 

Granted, money IS being raised, which is a good thing. But just like the townsfolk applauding those who ran onto thin ice, they are rewarding a symbolic act rather than a substantive act.  What added value does the symbolic act contribute? Are participants adding real value by dancing, or just gaining self-satisfaction by dancing towards a good cause? 

 

Other examples abound: runs, walks, dunk tanks – all activities where participants are rewarded for their symbolic sacrifice.

 

And that is why I was so impressed and pleased by what I saw at Central Park Church this past spring.  For two Saturdays as we adults played pickleball in the recreation room, the young people of the church were vigorously engaged in making pig-in-blankets and apple pies. It was clear that they came to work. In an organized and coordinated effort that would impress any local industry, we watched as they loaded tray after tray of their assembled goodies. Though obviously having fun together, they were on a mission.  These items were to be sold to raise funds for a youth trip to Nashville so they could spend a week volunteering and helping there. Impressive! It tears me up to think of it. 

 

In a recent (June 20) Holland Sentinel column, Aging 101, Jane Weisiger detailed the maladies of aging. It brought to mind Ecclesiastes 3:20, “Remember thy Creator in the days of thy youth before the days of evil (no pleasure) come.” Indeed, one key to aging well may be taking the opportunity to look back with satisfaction on these kind of efforts accomplished during the vigor of one’s youth. 

 

 Where/how did these young people get this passion to serve others even before graduating from high school? Nothing merely symbolic about their actions. They were working hard to produce a product, selling them at a fair price to earn money, and then using the money to provide an opportunity to help others. All substance. I don’t remember thinking that way as a 16-year-old. Kierkegaard would be impressed!

Three Body Problem

 HS #95 2023.6.8

 

Three Body Problem

 

Have you ever noticed the allure of the number “three”? In sports, for example, hockey has the hat trick (three goals), three strikes in baseball, three medals in the Olympics, three hits in volleyball, three seconds in the lane in basketball. What other examples can you think of? 

 

Interestingly, “three” sometimes gives rise to stability and sometimes to instability. Wind turbines have three blades. Those are needed to keep it balanced. If just two blades, then it could change direction too quickly when the blades are vertical.  A stool or camera TRIpod needs three legs for maximum stability. More than three legs leads to potential rocking. 

 

Relationships, on the other hand, seem most stable between just two people. When dining at friends’ homes, my father liked to quip, “I brought my bad wife with me and left my good wife at home. Isn’t that big – a’ – me?” But he also conceded that bigamy was forbade in the Bible. His proof text: “No man can serve two masters.” 

 

Non-PC joking aside, folks in the job market sometimes reference the “Two body problem” - the challenge of a married couple both finding employment in a given location. 

But this phrase actually comes from a deeper scientific challenge called the "Three Body Problem." It turns out that the relationship between three things often leads to surprising difficulties and counterintuitive surprises. 

 

The three-body problem originated in physics. Understanding and predicting the motion of two heavenly bodies under the influence of mutual gravity is rather simple. It’s a college-level physics problem. But, introduce a third body and calculating the motion of the three bodies becomes impossible, except with the aid of computer approximation and simulation. This led the 19^th century physicist/mathematician Henri Poincare’ (one of my heroes) to develop chaos theory, an apt term since the motion of the bodies was often chaotic and unpredictable. 

 

The intriguing mutual relationship of three things is seen even in the children’s game Rocks-paper-scissors. Isn’t it counterintuitive to you that three things can each beat something that beats something that beats itself? 

 

I use a physical example of the game as a party trick using three knives and three cups.  Set the cups on a table in the shape of a triangle so that the distance between each pair of cups is a bit more than a knife’s length. The challenge then is to place the three knives – extending horizontally from the top of the cups – so as to form a stable platform in the center of the triangle. You can do it – just interweave the ends of the knives. 

 

A more profound example of this three-way relationship exists in Christian doctrine – in fact it’s exposed in the Apostle’s Creed. Any mathematician (or lawyer or philosopher) will tell you that you need to define a term before you can use it. Yet the Apostles Creed seems to violate this principle. Reading through it, one comes to the phrase (concerning Jesus) “conceived of the Holy Spirit” and only afterwards does one recite the phrase “I believe in the Holy Spirit.” 

 

Isn’t that faulty writing? Shouldn’t the existence of the Holy Spirit be acknowledged before the action of the Holy Spirit is referenced? Seems so. However, the Spirit was sent by the Father to glorify the Son, so Jesus needs to be acknowledged before the mission of the Holy Spirit can be understood.   So just like Rocks-paper-scissors, one is forced to cut into the three-way relationship of the Trinity at some point.

 

This challenge of three-way relationships also exists in voting. In 1972 the economist Kenneth Arrow received the Nobel prize for showing that it’s impossible to form a voting method to choose the winner among three candidates that will always satisfy three commonsense objectives. In particular, if you want a system where the final outcome i) ranks A over B if each voter ranks A over B, and ii) is not affected by including irrelevant alternatives into the voting choices, then that voting system is susceptible to the Rocks-paper-scissors dilemma. That is, A beats B, B beats C, and C beats A. (How? Suppose Beth’s order of preference is A, B, C. Tim’s order is B, C, A; and Dan’s is C, A, B. Then in a two-way vote, A beats B, B beats C, and C beats A.) 

 

Alas, my parents were all too familiar with this shortcoming since those often seemed the voting outcomes of my two siblings and me.  

The Cycloid

 HS #94 2023.5.11

 

The Cycloid

 

 

Last month several friends alerted me to a New York Times essay, “The Wondrous Connections between Mathematics and Literature.” It began with Melville’s reference of a “cycloid” in “Moby Dick.” Do you know what a cycloid is? 

 

You can draw one. Secure a ruler to a piece of paper. Practice rolling a disc (a quarter or the top of a jar will do) along the edge of the ruler. Then put a pencil against the disc and trace its path as you roll the disc along the edge one full revolution. You just made a cycloid. 

 

If that’s too much work, you can visualize a cycloid by imagining the path taken on a dark night by a lightning bug that gets caught in the tread of a bicycle tire. As the tire rolls along the sidewalk, the hapless lit-up firefly traces out cycloids. 

 

What’s so special about them? If you want to build a slide that allows one to slide from point A to point B in the minimum time, a cycloid is your answer. Intuitively, one might think that a straight line is the best shape. A cycloid path is longer, but you have the advantage of getting a faster start. 

 

There is sordid history to the discovery. 

 

The Bernoulli brothers of the 17^th century, as with many brothers, were competitive.  Johann Bernoulli discovered that the cycloid gave the optimal path, but instead of just publishing his finding, he posed it as a challenge to the entire mathematical community of his day in an attempt to best his brother, Jacob. One person submitted a solution anonymously. When Johann Bernoulli saw the exquisitely beautiful solution, he said “This is from Sir Isaac” (referring to Isaac Newton). He was correct. When asked how he knew, he said, “You can tell the lion by the size of the paw.” 

 

Called the Brachistochrone (shortest-time) Problem, it inspired an entire new area of mathematics called Calculus of Variations. I gave a talk on the problem at Hope College in 1987 when I applied for the mathematics position. So it was profitable for me as well. Making use of the alphabetical order of the names involved, the announcement read, “Inspired some of the greatest mathematicians in history including Bernoulli, Euler, Leibnitz, Newton. Pennings will talk on Tuesday.” 

 

In fact, more recently I got wondering what sort of optimal path one would make if limited to adjoining straight line segments rather than a smooth curve. Getting a couple mathematician friends to join me (including former Hope student Mark Panaggio – now a researcher at Johns Hopkins), we solved it – to be published this fall. All a testament to the ever-growing circle of knowledge inspired by past discoveries. 

 

The cycloid has another interesting property as well. A pendulum swinging a big arc takes longer to go back and forth than one swinging a small arc. That’s not surprising. But what IS surprising is that a pendulum constrained to follow the curve of a cycloid will always take the same amount of time, no matter how high or low the starting point. The physicist Huygens actually constructed an accurate clock using this property. 

 

One last notable fact about cycloids is also related to literature. Sir Arthur Conan Doyle, the author of the Sherlock Holmes stories, wrote “Silver Blaze” in which the mystery of a stolen horse was solved by Holmes noting the curious event of the dog that didn’t bark in the night.  The non-barking led Holmes to realize the dog must have been familiar with the theif. 

 

Similarly, the cycloid has a curious “negative” quality – one I have never seen noted as such in mathematics literature. Everything else involving the circle involves the number pi. The circumference is pi times the length of the diameter. The area of a circle is pi times the square of the radius (hence the area in the rim of a basket is almost 4 times the cross section of a basketball!). The area of a sphere is 4pi times the square of the radius, and the volume is 4/3 pi times the cube of the radius. Pi is ubiquitous! Except the cycloid. The length of a cycloid is exactly 8 times the radius of the circle. How cool is that! Hope the firefly appreciates it.