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. 

Women Changing our World

 HS #93 2013.4.13

 

Women Changing our World

 

My late Hope College mathematics colleague, Mary DeYoung, had a quote from anthropologist Margaret Mead on her office door, “Never doubt that a small group of thoughtful, committed citizens can change the world; indeed, it’s the only thing that ever has.” 

 

Indeed, Mary was one such person herself. Dying suddenly of cancer in the summer of 2011, Mary and her husband, Steve, provided my first home when I moved to Holland in 1988 – in the appendage to their house on Pine Ave and 18^th Street which was originally constructed as a doctor’s office. Through her 25-year career of teaching and inspiring K-8 math education students at Hope College, Mary has impacted hundreds of thousands of lives. 

 

Mead’s quote came to mind recently as I listened with fascination to a BBC account of a new biography, “The Empress of the Nile” by New York Times best-selling author Lynne Olson. The book cover describes its subject, Christiane Desroches-Noblecourt, as “The Daredevil Archaeologist Who Saved Egypt’s Ancient Temples from Destruction.” 

 

Born in Paris in 1913, Desroches-Noblecourt attended a progressive all-girls school in which she became hooked on Egyptology. Her life was one adventure after another. As a curator for the Louvre during WWII, she braved interrogation by the Gestapo and twice moved, hid, and secured its contents, including the “Mona Lisa” from the advancing Nazis.  

 

But her most stunning success was the following decade when Egypt’s President Nasser decided to build a dam across the Nile River. Looking to its future rather than its history, Egypt was willing to sacrifice twelve sacred temples, including the statues of Ramses II dating back 3000 years for the economic benefits that a dam would provide. 

 

Challenging Charles de Galle and the Kennedy administration for funding, planning the details with Egypt’s President Nasser, Desroches-Noblecourt won the day by gathering contributions from 50 countries.  She was a real-life Indiana Jones in her love for antiquity and her willingness to fight any battle to save it. She saved all of the sacred monuments even though the largest had to be cut into small pieces and reassembled much as once happened to the Dutch Windmill on Windmill Island.

 

Indeed, local parallels to Christiane’s life go deeper than that. Reading her story, I was reminded of a West Michigan woman who also had a unique vision for saving history and who clung to her mission like a bulldog. 

 

I’m speaking of Saugatuck’s Felt Mansion. Built in the 1920s by the inventor of the first mechanical calculator, the Felts died within a couple years of its completion. The Catholic Church owed it for a while – it was a convent for nuns secluded from the outside world. There was a turn-style in the front entrance so messages could be transported without human contact.  The grand rooms were gutted to make space for many small living quarters. 

 

In the 1970’s a state legislator had the idea of securing this beautiful land for a state park. There was no money designated for parks, but ample available for prisons. So the state bought it, built and used the prison long enough to satisfy legalities, then tore the prison down and kept the land as a state park.  (Bravo for creative imagination!)

 

The land was retained for the Saugatuck State Park, but there was no use for the mansion. So Laketown Township bought it from the state for $1 in 1996 with the plan to let it fall to ruin so that eventually the only choice would be to tear it down. 

 

But then in 2001 Patty Meyer and her husband took a walk through the park. Seeing the potential, she badgered the township for permission to restore it. After multiple refusals, the township board decided the easiest way to discourage this determined woman was to let her have it for six months so she would realize what a useless effort it was. 

 

Remarkably and gladly, their plan backfired.  Patty and her husband filled six haul-away garbage trailers with dead bats, birds, rats, foxes, raccoons, opossums and other trash. They eventually found the grand mantle for its fireplace in the barn of a local farmer, and after years of work the mansion is now restored to its former glory and available for rent and general viewing. 

 

But Patty is not satisfied with restoring historic buildings. Taking this essay full circle, her new mission is raising childhood literacy throughout the state. Mary would approve. Kudos to these women changing our world. 

Inflection Points

  

HS #92 2023.3.9

 

Inflection Points

 

You are about to experience an inflection point. In eleven days, to be exact. You won’t feel it, this inflection point is too subtle, but it will influence your life, as most inflection points do. 

 

If you drive through downtown Grand Rapids, you will encounter one there. You often create one when coming to stop lights. Two years ago during the thick of COVID the press told us of inflection points even though most of them likely didn’t know exactly what they were talking about. 

 

It’s time to learn. 

 

Imagine walking up a smooth hillside or dune from the bottom of a valley to the top.  As you begin the trek from the bottom, the slope becomes increasingly steep. Your legs get sore because each step is a bit steeper than the last. But as you approach the top, even though you’re still climbing, the RATE of climbing decreases. You can feel the gratitude in your muscles. You have passed the inflection point. 

 

The mathematical definition of an inflection point is where the concavity changes. Think of an “S” lying on its side. The bottom of the “S” which bends up is said to be concave up. The top of the “S” which bends down is “concave down” The middle of the “S” is the inflection point. Driving north along the S-curve in Grand Rapids, the inflection point is the place where you transition from turning left to turning right. 

 

You can feel an inflection point if you’re slowing down as you approach a stop light, but then the light turns green, so you start accelerating. The transition from slowing down to speeding up is the inflection point. 

 

What’s the inflection point we will all experience shortly? Since this past December 21, the amount of daylight has been increasing. But not only increasing, it’s been increasing at an increasing rate. In early January we got an extra minute daily. As we approach the first day of spring (March 20), we are gaining 3 minutes daily. Once we enter spring, the length of days will continue increasing, but increasing at a decreasing rate. 

 

Not to worry, both equinoxes are inflection points. In September, the days will continue getting shorter until Dec 21, but they will be getting shorter more slowly. 

 

Why are inflection points important? Suppose a graph (a curve) represents the overall welfare of a person’s life. One would think that a higher graph (better welfare) would lead to more happiness. But not the case! Studies have shown, for example, that a lottery winner and an amputee are equally happy several months after their life-changing events. Surprising? 

 

Instead, happiness seems more tied to the CHANGE in a person’s welfare. A sick person on the road to recovery is generally in good spirits. For those who watch stocks, if the market goes up on a particular day (even if down for the year), it makes one happy. Similarly, a person enjoying a luxury vacation can get bummed by burnt toast. 

 

How do inflection points and concavity affect us? I submit that they are related to our hopefulness. Again, consider your stocks. If after bad economic news, the stock market goes down daily 300, then 200, then 100, you may feel badly for the decrease, but the trend makes you hopeful that it’s reaching the bottom of the “S” and in coming days it will start increasing again. 

 

Understanding that hopefulness is related to inflection points, you can perhaps appreciate why during COVID epidemiologists were concerned about them. They accepted that lots of folks had COVID, they even accepted that the number was increasing weekly. But they were carefully watching the rate of increase. Even though the number of COVID cases was increasing, the rate of increase was decreasing. So there was hope that eventually the total number would reach the top of the “S” and start declining. Which it did!

 

Our present inflation provides another example. Prices were increasing at an increasing rate. Now prices are still increasing but at a decreasing rate.  Prices had an inflection point. We’re breathing a bit easier as a result. We’re more hopeful that the future will bring economic normality. 

 

This is all good news of sorts. No matter a person’s lot in life, some days are still better than others. And a person’s happiness is tied to those changes. Moreover, even if things are getting worse, if you can see an inflection point coming, there is reason for hope. 

Afraid of Dying?

 HS #91 2023.2.9

 

Afraid of Dying?

 

Last month in her January 14 Holland Sentinel column, Janet Weisiger raised a thought-provoking question: “Are you afraid of dying?”  She detailed her own strong belief in the caring, loving God described in the Bible and in God’s promises such as John 3:16. 

 

I am not presently reading through the Bible yearly as she is, but I know it well.  While in college, I wrote my own four-notebook commentary covering the entire Bible using (and recommending!) the three-year daily study guide “Search the Scriptures” by Intervarsity Press.  

 

Yes indeed, the Bible makes claims about eternal life. There are explicit New Testament references such as Jesus telling his disciples (John 14:2) that he will prepare a place for them. But there are Old Testament hints as well - even in the most ancient book of the Bible, Job.   After Job’s time of trial allowed by God which stripped Job of all that he had, God restores to Job double the wealth, double the flocks, double the herds, BUT just the same number of children who had died.  Was that because the dead children still lived in the afterlife – so Job’s number of children had in fact doubled too? 

 

However, Weisiger and I do apparently differ in our epistemologies, perhaps depending on our respective personalities. 

 

I am skeptical about drawing conclusions of things unseen.  Being honest with myself, I don’t share Weisiger’s confidence in the things she believes about God and the afterlife. Instead, I question if there is a creator God of the universe, or whether the physical universe itself is the ultimate thing. Either is possible. It’s a toss-up.

 

If a creator God exists, is God a personal God who loves and watches and cares for us and “knows the number of hairs on our heads” (easy in my case)? Or is God possibly a deistic God, as believed by some of our Founding Fathers, who made the world and is now letting it go of its own accord?  Either is possible. Note that the lack of present-day miracles is not a case against a personal God. Consider a farmer’s relationship to his fields. There are special times when the farmer is active – planting, weeding, harvesting, but otherwise the attentive, ever-present farmer carefully watches. Similarly, the Biblical record suggests a loving God who has been active at crucial times in the history of the world, but otherwise lets it run much on its own without obvious interference. 

 

If a personal God does exist, is God as described in the Christian Bible? There are other monotheistic religions with different claims. Jewish faith, for example, places little emphasis or belief in the afterlife. Their faith, as I understand, is much more rooted in this present existence. 

 

And even within the realms of Christian belief, there are questions about life beyond this. Once when I pressed for an answer, a pastor told me, “We will remain in God’s love.” That was as specific as the pastor claimed. 

 

Lots of uncertainty. So to answer Weisiger’s question, I am not afraid of dying – but for reasons different than hers. 

 

Accepting that this life is possibly all there is frees one from the unease of hoping for more without being sure. Some may be energized by “things hoped for; the evidence of things not seen.” (Heb 11:1)   But for others, including myself, uncertainty generates disquietude. It feels better to learn to be content with what one has – this present life. This was the wisdom of Epicurus. 

 

But how to be content?  Briefly, I don’t feel robbed that I was not alive to experience life in the 1800s. Similarly, I accept that I will not live to experience the 2100s. I accept that I am experiencing a brief bit of the universe right NOW. So I want to make the most of it by living fully in the present while hopefully impacting the future world for good.  Then I will die satisfied.  

 

Interestingly, mathematics offers another consolation of sorts.  Using the logarithmic scale, human size is about halfway between the smallest thing (Plank’s quantum distance) and the largest (the universe itself). That is, we are as many times larger than the smallest thing as the entire universe is larger than us.   But in contrast, a typical human lifespan (70-80 years) is a full 85% of the way along the logarithmic scale from the smallest quantum time unit to the age of the universe.  Not bad! Take advantage!

35 Years Living in Community

 HS #90 2023.1.12

 

35 Years Living in Community

 

As I write this, I’m watching the original Star Wars movie on TV.  I didn’t realize when viewing it as a sophomore in college that the theme of the trilogy is friendship. When Luke chooses C-3P0, his fearless droid friend R2-D2 rocks back and forth in anxiety fearing he will lose his companion. At C-3P0’s suggestion, Luke takes R2-D2 also, so the two friends stay together. 

 

Friendship binds the humans as well. Hans Solo foregoes his own plans for the sake of his new friends, and Luke’s dedication to friends gives him the spirit and power to defeat the emperor. 

 

And I just saw an ad for “The Wizard of Oz.” Friendship is its theme too. Dorothy realizes that love of faithful friends, family and home are the only things worth living for. 

 

Yes, we are meant to live in community with friends. 

 

Indeed, why am I watching Star Wars on TV?  I own the DVD; I can watch it anytime. But it feels good knowing others are sharing the experience. I endure commercials for the sake of (unseen) community. I can even appreciate being stuck in traffic realizing that the entire lineup of vehicles is a spontaneously formed community of sorts – each of us dependent upon the actions of the others. 

 

It was this desire for community that brought me to Holland 35 years ago this week. 

 

After completing my Ph.D. in mathematics at Iowa State University in 1987, I received employment offers from Hope College and two state universities. Having done my higher education at public universities, I was tempted to make one my career home. But I was fascinated by the prospect of living an integrated life in a community where my colleagues were also my neighbors.

 

Having grown up as a PK, we moved several times during my childhood. Thus, perhaps I value more than most the opportunity to have lived in Holland for 35 years – most of it in the same brick ranch on East 12^th Street. There is something rich about riding my bicycle around town noting the homes of friends and neighbors – some departed, of walking through the cemetery on 16^th Street among increasing numbers of engraved names of longtime friends and colleagues. Indeed, my life is enriched as I live surrounded by ghosts who speak to me providing memories from the past. 

 

How neat to live several blocks from where my father and grandfather went to college and seminary. I served as a Resident Director in Archie/Cosmo Hall for five years – those were the fraternities of which they were members. 

 

How rich to have spent over 30 years celebrating holidays with my cousins at the home of my (now 94-year-old) aunt, seeing my cousin’s young boys grow to have children of their own. 

And to have a former colleague tell me recently that he prays for me as he passes my home on his daily run. 

 

There is richness in long enduring relationships. 

 

On the other hand, I tell students that the best way to learn is by keeping one foot on firm ground while venturing into unknown water with the other. That is, we should explore new territory while staying connected and secure with the old and familiar. 

 

Similarly, some relationships and communities are eventually replaced by new ones. Since coming to Holland, I have taught at four different colleges, sung in six different choirs, and changed churches. New hobbies also bring new communities. Racquetball and ultimate frisbee partners have given way to pickleball and kayaking friends. Since 2015, writing this monthly column has formed new connections.  Few of these changes were planned in advance, and that very fact makes the future both hopeful and exciting. Retirement will bring losses, but also new opportunities and relationships. Even as life changes, new communities emerge and buoy us up. 

 

What about you? Teddy Roosevelt’s daughter said her father would be the bride at every wedding, the corpse at every funeral. Yes, it’s hard to attend a wedding or funeral without thinking of oneself. M.C. Escher’s self-portrait shows his reflection front and center in a crystal globe held in his outstretched hand. We are all the central image in our own lives. 

 

So even as you, my Holland neighbors, read my life details, I hope and assume that you have been remembering similar stories and evolving communities of your own. As life changes, we are blessed to live in community – with each other. 

Calvin and Hope

 HS #89 2022.12.8

 

Calvin and Hope

 

Calvin VanderWerf, President of Hope College 1963-1970, had a couple of noteworthy ideas. One was his humorous definition of an atheist as someone who doesn’t care who wins a Hope-Calvin basketball game. A second was separating Hope College from the control of its parent denomination, the Reformed Church of America. 

 

Both of those ideas are front and center these days. Calvin University recently announced that football is in their future. Given the rivalry of other Hope-Calvin sports teams, one can anticipate full stadiums and cheering crowds on both sides. West Michigan will be the richer for it. 

 

Indeed, the rich rivalry between Calvin and Hope goes beyond sports. Traveling around the country giving talks at various colleges and universities, I have grown to appreciate the uniquely balanced competition between these West Michigan institutions – iron sharpening iron. Years ago, when touring the Art Institute of Chicago, I spied a group of Calvin-clad students. Tapping one on the shoulder, I asked, “Calvin College – that’s that nice school in Holland Michigan?” “No – that’s Hope.” “Oh – you’re the one that won MIAA basketball this year?” “No – that was Hope.” “You’re the one with all the undergraduate science research?” “No – that’s Hope.” I then admitted I was a Hope professor. 

 

But the joke could easily have gone the other direction. Especially in the humanities, Calvin has produced noteworthy scholars including Alvin Plantinga and Nicholas Wolterstorff, not to mention the recent thoughtful challenge, “Jesus and John Wayne” by Kristin Kobes Du Mez. Both institutions are making their respective impacts. 

 

Their “respective impacts” – that brings us to VanderWerf’s second idea. As is happening in many Christian denominations, the RCA (Reformed Church in America) and the CRC (Christian Reformed Church in North America) are wrestling with the LGBTQ+ issue. However, since Calvin University is under the ownership and direction of the CRC, the CRC’s decisions in this matter have a greater impact on their institution. 

 

As is seen on the CRC website, the denomination holds official positions on a number of matters. Some of these, such as belief in the infallibility of inspired scripture and that certain historic creeds faithfully represent scriptural doctrines, form the bedrock of the faith.  These are raised to confessional status, while most, including observance of the 4^thCommandment (Lord’s Day), baptism, divorce, and creation and science are not. 

 

This summer the 2022 Synod (CRC governing body) affirmed that the sin of “unchastity” in the Heidelberg Catechism includes homosexual sex (even within a monogamous marriage) and raised this issue to confessional status. Interestingly, while this biblical position can perhaps be defended (as can the opposing position), it is not clear from scripture why it was raised to confessional status. Which issues would achieve confessional status if scripture were one’s sole guide (Sola scriptura)? Perhaps the Ten Commandments - including swearing and Lord’s Day observance? Perhaps justice and mercy? 

 

But while I don’t understand the action taken by the CRC, I still respect it. Those making the decision undoubtedly did so from an inner conviction that they were doing the right thing. Unfortunately, future high school students may increasingly have inner convictions that the decision was the wrong thing. Given the increasing challenge attracting enough students given declining high school demographics, this presents an additional hurdle. 

 

But while the CRC’s authority over Calvin University presents challenges, it also provides Calvin  with a steady anchor and clarifies its identity.  This, by contrast, reveals its rival’s challenge. 

 

Unmoored from the RCA, Hope College has been free to evolve depending on its leadership and board. While this has resulted in a dynamic chapel program, Hope would do well, especially in hiring/tenure, to clarify its identity and settle into a stable niche, distinct from Calvin’s, but also distinct from secular colleges. Presently, Hope asks candidates for an expression of faith and relationship with God. But with no foundational doctrinal statement, who then decides if it is adequate? And how? (For example, Nietzsche observed, “Only a person of deep faith can enjoy the luxury of skepticism.”) Hope might instead consider using SOLELY the mission statement developed by VanderWerf’s successor, Gordon Van Wylen: “The mission of Hope College is to offer academic programs in liberal arts …  in the context of the historic Christian faith.” Instead of attempting to evaluate the adequacy of a personal faith expression, simply ask: “How have you, do you, and will you support this mission?” Simple and straightforward. 

 

This would provide two distinct and distinguished institutions of higher education in West Michigan. 

 

 

The Pull and The Fall

 HS #88 2022.11.10

 

The Pull and The Fall

 

Ideas for these monthly columns are often inspired by a confluence of events. Such happened last month. The Fall (the Biblical depiction of the origin of human evil and pain) was on my mind while writing a column on Genesis. (See timothypennings.blogspot.com)  I penned,  “As with Icarus, humans sought to fly too high, to know more than was intended and instructed, causing a cascade of calamity.” 

 

The second event was watching the grueling last half hour of “The Pull” of Hope College. I’ve regularly attended this event since coming to Hope in 1988. Until recently, it was held on the two – usually muddy – sides of the Black River. The new location, on East 11^th Street by the DeVos Field House, loses some of the ambiance, but compensates by allowing spectators to easily watch both teams. 

 

Since the Even Year team runs by my house on East 12^th Street for three weeks of multi-hour practices, they are my default favorite. The male chorus of chants has been enriched by higher-pitched female voices as modern sensibilities affect even this century-old tradition. 

 

Over the years I have learned various stories from pullers. My favorite is that during the first week of intense training – including hundreds of sit ups and push-ups – the pullers are promised a day of recreation at the beach the coming Saturday. Arriving ready for sun and surf, they are then directed to continue training by carrying each other up the dunes. 

 

True or not, I believe the claim that for most of these students, the three weeks of Pull practice is, bar none, the most intense and single-focused physical endeavor they have ever experienced. Their raw hands, strained voices, and cramped legs are all testament to giving this team effort their all. 

 

Thus, I choked up at the conclusion when, standing on the Even Side, I heard the junior year coach announce to the exhausted team, “You have done all we have asked and expected. You have carried on the great tradition and have made us coaches, the Even Year alumni, and Hope College proud. You have done your very best, and you - - are victorious!” It was a speech worthy of Vince Lombardi and similar to Marshall’s commendation of Eisenhower. 

 

After enjoying their celebration for a few minutes, I shifted to the Odd Year side. I expected to find low spirits, but was unprepared for the tears and genuine heartache. I’ll not forget the look of shock, despondency and loss as they consoled each other with long embraces. During the final minutes of the event, I overheard a student spectator remark, “What’s it like to spend three hours pulling on a rope – and then lose?” Indeed!

 

These students didn’t know it, but they just experienced the Fall. Both sides. I didn’t understand the richness of the Fall until reading C.S. Lewis’s “The Space Trilogy.” He explores, as only Lewis can through fiction, that, yes, the Fall brought misery and heartache, but, living through and conquering those challenges adds richness and purpose to our lives. As per ABC’s Wide World of Sports, there is no thrill of victory unless there is also agony of defeat. 

 

And these students are getting just a taste of what’s to come – the thrill of marriage, the agony of conflict and divorce, the thrill of child birth, the agony of health issues, the thrill of new jobs, the agony of layoffs. Is the Fall then a curse or a disguised blessing? It’s both. 

 

Beautiful patterns on diamond willow walking sticks are the result of infection in the wood. A broken bone heals stronger. Mutations, while usually harmful to the individual, are beneficial to the species since the species can then survive changing environments. (For more examples, read “Antifragile”). U2’s Bono in his new memoir, “Surrender” explains it was the loss of his mother that led to his life of creative brilliance.  Someone has noted that in the centuries during which turbulent war-torn Italy gave the world unsurpassed music and art, placid, staid Switzerland produced  - - the cuckoo clock. 

 

There’s a story about an Irishman who, exiting a pub and seeing two men slugging it out, taps one on the shoulder and asks, “Is this a private fight or can anyone join?” Some enjoy the fight. 

 

Indeed, the richness of striving, of struggling, of laying oneself bare with no guarantee of victory – this is the legacy of the Fall. A tough life – I’d have no other.