Natural Education

  

HS #85 2022.8.11

 

Natural Education

 

In the rotunda at the entrance of the Field Museum of Natural History in Chicago is a quote from the scientist/inventor George Washington Carver, “Reading about nature is fine, but if a person walks in the woods and listens carefully, he can learn more than what is in books.” Interestingly, while Thomas Jefferson learned of nature via his extensive library, his fellow Founding Father Benjamin Franklin learned from nature itself through keen observation and myriad experiments. 

 

As a child growing up on the outskirts of Des Moines Iowa, much of my early development came from exploring the ten acres of wild land behind the parsonage. My friend and I dug a fort which entertained the men of the church as they noted our progress each week. How much character development comes from digging a hole! 

 

With that in mind, I bought the plot of wooded land with a stream behind my house on East 12^th Street. It stands in contrast to the manicured designed parks all around us in Holland. Nothing wrong with them, but by their very design, children are meant to do activities that adults have intended for them. 

 

On the other hand, in my backyard plot I am continually surprised by what the neighborhood children do – building forts out of logs and branches, zip lines, dams, paths. What a great place in a residential neighborhood for children to “own.” 

 

Indeed, a running stream offers abundance of fun. How many times have I taken Hope College students and other friends to the stream that lies a half mile south of Laketown Beach. It’s just the right size for several ambitious souls to dam up, watch the water collect, and then destroy. The water movement and resulting formations of sand formed in a couple hours models geological structures built by rivers over eons. 

 

My little backyard stream also attracts and sustains abundant wildlife to watch – steelhead, crayfish, opossums, woodchucks and occasional more exotic things. I’ve seen a fox pass by my backyard window. 

 

The fact that we need such natural interaction can be seen in how we design our indoor environments. Those remodeling 19^th century homes in Holland may find layers of colorful paint on the woodwork in their homes. Indeed, after working outside among the trees all day, bright colors were a nice change of pace. But now, largely removed from nature, we not only keep natural surfaces intact, but also design the plastic surfaces of cars and furniture with the appearance of natural wood. 

 

In the U.S. we fare better than Europe. Several years ago, I took the Dutch son of the architect of the Hope College Kruizenga Art Museum with me for a trip to the Boundary Water Canoe Area (BWCA) in northern Minnesota. As we passed out of Grand Rapids going north on US 131, he noted that this was the first time he had seen a “wild tree.” As with a wild animal, a wild tree is not intended and cared for by humans, but is there of its own accord. (The BWCA was a stretch for him – he stayed in his tent safe from mosquitos.) 

 

Aristotle’s principle of moderation and the Golden mean can be applied to learning from nature. The Outdoor Discovery Center here in Holland is a great example. Children are given opportunity to play on manufactured playground items, but these are placed in a natural setting. I recently enjoyed watching my friend’s five-year-old scamper up the boulders and logs after each turn down the slide. 

 

The collection of area parks - Holland State Park and our county parks - is another prime example where humans have designed areas to make them accessible and fairly safe while still offering opportunity for natural exploration. How fun it is to kayak up the Black River enjoying ducks, king fishers, and occasional osprey all within sight of Holland homes and businesses.

 

I say “fairly safe” because there is still a balance to find. Cushioned rubber mats and rounded edges on playground equipment prevent injuries, but do not teach children the natural consequences of taking risks. Understanding the risk of harm is helpful as children grow and hopefully fully engage in living lives of learning and adventure in the wild of nature.

 

Then our children can live their lives in the spirit of Carver, “I love to think of nature as an unlimited broadcasting station, through which God speaks to us every hour, if we will only tune in.” 

 

Life Lessons from Pickleball

 HS #84 2022.7.14

 

Life Lessons from Pickleball

 

In case you haven’t noticed, pickleball has overtaken the world. Invented in the early 1970s, it sat dormant for decades with little apparent growth, but in recent years has exploded with (by my count) about three dozen courts in the Holland-Zeeland area and likely more to come. 

 

Pickleball had a natural barrier to overcome – a chicken and egg problem of sorts. No opportunity for people to play unless there were courts and equipment, and no reason to manufacture equipment and build courts unless there were people to play. It was like a boulder sitting in a depression at the top of a hill. It needed a push – what chemists call “activation energy” – to get it rolling. 

 

Apparently, a critical mass of tenacious folk in the state of Washington provided the activation energy, and now, like Starbucks and Apple and our nation’s COVID that all began there, the sport has spread across the U.S. 

 

One reason for its success is that it is cost effective. The area needed for a pickleball court is ¼ that of a tennis court, and because most pickleball is team play (2 against 2), pickleball offers 8 times the bang for the buck in real estate.  Also, since its devotees include middle aged and retired folk who serve on town councils, it has a voice at the table. 

 

Pickleball is not just for worn out tennis and racquetball players, put to pasture in their later years. As a former enthusiastic of both of those sports, I initially viewed pickleball with a bit of disdain. No more. Granted, not as much running as tennis or pounding as racquetball.  A different skill set is needed. Fast reflexes are essential and spin can be helpful – similar to table tennis. Unlike tennis, serving is not important; instead volleying and coordinated team play are key.  

 

I became a convert two years ago – a new religion of sorts. All religions have informal rules not stated in the catechismal creeds, such as when it’s appropriate to clap or shout “AMEN!” I’ve discovered and list below some unstated rules of pickleball - which also provide good advice for living: 

 

·      As with all recreation, it’s more fun to play if you should be working. We were put on earth to be productive, whether it be employment or community volunteering or raising families. But, as evidenced by human history, including the ancient Olympic games and Native American lacrosse, we all need a break. 

 

·      Any player who conscientiously learns from her mistakes will steadily improve. After each lost point ask, “What should I have done differently?”

 

·      Watch for opportunities to compliment.   Complimenting your partner causes them to play better; criticizing often leads to self-consciousness and worse play. 

 

·      What initially seems to be your partner’s mistake is, with more reflection, often realized to be your own. 

 

·      The mark of an experienced player is not only the person who makes a good shot, but also the person who hides his surprise when he successfully executes one.

 

·      Be deliberate, but not timid. Hit with conviction. Boom!

 

·      Don’t hit balls that are going out of bounds – somethings in life take care of themselves. 

 

·      Move your feet – you can’t hit a good shot from a poor position.  An Olympic table tennis player once told me the order of importance: i) feet, ii) head, iii) hands. 

 

·      Everyone – even good players – has an Achilles Heel. In pickleball, it’s the heel. Aim for it. 

 

·      Run for everything unless your partner is under 20. 

 

·      Playing half-heartedly is worse for developing skills than not playing at all. 

 

·      Line Calls: Always give your opponents the benefit of the doubt, and give them the benefit of the doubt that they are giving you the benefit of the doubt.  

 

·      Profanity: Mark Twain said “In desperate times profanity provides comfort denied even by prayer.” My favorite is “BLAST!” It has lots of spitting consonants and can be used in all occasions without undue embarrassment. 

 

·      Patience is a virtue. This actually IS written in the official literature. The description of a competent (3.5) player includes having developed the patience to wait for a good shot.

 

·      Better for both partners to swing than for neither. Take ownership and responsibility. 

 

·      When returning an errant ball over a fence, it always takes two attempts. We often misjudge. 

 

·      A good rally is fun for everyone – win-win situations occur in life. 

 

·      Trash talk:  I seldom hear it, so have to limit it myself.  Oh well, no sport is perfect. 

High School Commencement Address

  

HS #83 2022.6.9

 

High School Commencement Address

 

Faculty and staff, relatives and friends, 2022 graduates: “Tell me, what is it you plan to do with your one wild and precious life?” This line of poetry from Mary Oliver is powerful and profound. Say it aloud – commit it to memory. Take it with you as a guide in life. 

 

Let’s ponder it a bit. “Tell me” forces us to clarify our thoughts. It’s not enough to have just a vague answer to so important a question. 

 

“What is it you plan to do?” That is, YOU are in charge. You have the freedom AND the awful responsibility. The Statue of Liberty needs a counterpart, “The Statue of Responsibility.” As John Steinbeck explains in East of Eden, the Hebrew word “Timshel” – “Thou mayest” captures both ideals. We have the agency and must bear the consequences. 

 

“With this one” – there are no second chances. As the TV soap opera is titled, “One Life to Live.” 

 

“Wild and precious life” – “Precious” since every moment is valuable – each moment provides opportunity for us to experience and interact with the universe. What about “wild”? In Chronicles of Narnia, CS Lewis describes Aslan the Lion as wild. “He’s not a tame lion.” Wild means unpredictable and not controlled. Even with best laid plans, things won’t go as you anticipate or want. Sometimes better, sometimes worse. The floating feather which begins and ends “Forrest Gump” catches the inherent randomness of our lives. No one saw COVID coming. Read Endurance. Only the singular grit and focus of Earnest Schackleton saved his Antarctic crew from unexpected twists of nature.   

 

That was poetry – now a couple short stories remembered from my high school senior year. I didn’t understand them then, but they keep recurring to me. 

 

The first tells of a person visiting a friend’s home and noticing a goldfish bowl with nothing but a single motionless goldfish hovering in the water and a penny lying at the bottom. The owner explains that the penny keeps the goldfish alive indefinitely without feeding. Then the person discovers how: The penny leaches copper into the water which all but kills the fish. The goldfish is alive – but just barely. It continues living because it doesn’t have energy to move – it just exists. A long life of meaningless existence. 

 

The second complimentary story tells of a flea which has spent its entire adult life on a leaf of a shrub by a forest path. The flea just waits, waits, waits for a warm-blooded animal – perhaps a deer – to amble by. Then it LEAPS! If successful, it lands on the back of the deer, then buries into the fur, drinks blood, develops and lays eggs, and dies – its life-mission fulfilled.  If it misses the target, it falls to the ground withers and dies. No energy for a second chance. 

 

“A Noiseless Patient Spider” by Walt Whitman captures a similar theme. Find it and read it. We are like a spider that sends out filament, filament, filament to catch in the wind and attach ourselves to the world. Notice the recurring theme of this essay? The combination of intention and randomness. The spider has no idea which web will catch – or if any will. But it DOES know that if it is to successfully live, it must send them out – it must keep trying. Failure is guaranteed if it quietly exists like the goldfish. 

 

Poetry, short stories – what’s left? Today’s art form is video games. Question: If you could plot out your life trajectory, what would you choose? Most might choose a gradually inclining line in which the good things of life are steadily increasing. Picture that – like rolling a ball up a gentle slope until it gets to the top. Sounds good, doesn’t it? Tell me, would you play a video game like that? Of course not – how boring! If such a monotonous video game isn’t appealing, why do you want such a life? Instead, a game worth playing is one with challenging pitfalls and risk-laden opportunities to seize the moment. 

 

Tell me, what is it YOU plan to do with your one wild and precious life? As I write this, NPR radio is telling of Calvin University’s Prison Initiative.  Twenty inmates each year – most of them serving life sentences – are graduating from Calvin’s prison campus. Their leap, their filaments, their choices brought them a lifetime in prison. But they are now flinging out new filaments and making the most of their lives. Will you?

Ramadan

 HS #82 2022.5.12

 

Ramadan

 

I have a new friend in Holland – an engineering graduate from Texas whose family hails from Pakistan and who follows the Islamic faith. He recently observed the month of Ramadan. More on that later. 

 

Those who visit the cathedrals of Europe, and local ones such as the Cathedral of St. Andrews in Grand Rapids, immediately realize that such structures were built for the glory of God. Soaring arches, stained glass windows, murals and iconography all point one’s thoughts to the Almighty. Other things are obviously less important. Straight-back wooden pews are hard and often uncomfortable. 

 

In contrast, modern Christian churches are largely designed for the comfort of us. Some to the extreme – well equipped with the amenities of 4-star hotels. Leather lounge chairs, polished tiled floors, coffee shops, cushioned seats, recreation centers. Belonging to a church is not unlike being a member of a family country club. It all makes practicing the Christian faith comfortable and enjoyable. Today’s reality compares with 20-year-old humor of a Doonesbury cartoon which depicted a couple deciding which church to join. Speaking to a pastor, the husband explains that he doesn’t like that church’s emphasis on sin, but observes, “However -  you DO have racquetball courts.” 

 

To be clear, I am not (necessarily) criticizing. Indeed, as an avid pickle ball player I appreciate the generosity of local churches in making their recreational facilities available to the community. Instead, I’m just observing a clear trend in the practice of the Christian faith. 

 

As buildings and programs have become more user-friendly, the more austere practices of the Christian faith are disappearing. As a 9^th and 10^th grader in an RCA church, I joined all my peers for two years in spending an hour each Saturday morning learning the Heidelberg Catechism including homework and memorization.  Other more fun activities were put aside. In contrast, churches now are more likely to offer (and pay for) weeklong adventure mission trips where young people and other members can visit and offer help in exotic faraway places – even (I’m told) as remote as Nepal/Tibet. 

 

Indeed, Christian Smith describes the modern religion of young people as Moralistic Therapeutic Deism. Religion is meant to make our lives easier without imposing any costs. Convenience and comfort is the modern creed. 

 

This is not to say that no Christians sacrifice. Indeed, the impressive church facilities in Holland have been financed with the tithes and donations of members. Some give to causes that help those beyond the walls of their church. Many donate their time generously. 

 

But this all stands in stark contrast to the personal sacrifice of Muslims in observing Ramadan. Ramadan is an annual observance during which Muslims fast from sunup until sundown. Not just for a day or even a week, but for an entire month. Here in Holland when Ramadan, which follows the Islamic lunar calendar, occurs during the summer months, this means going daily from 6:00am until 9:30pm without food and water. 

 

What is the contrast with the sacrifices Christians make? Two important ones. 

First of all, Ramadan is practiced worldwide by over 90% of able-bodied Muslims past puberty.  That’s difficult enough when living in a predominantly Islamic country where laws and customs accommodate it. It is much tougher when observed in America where Muslims must continue with their jobs and other activities all while fasting. 

 

But more important is this essential difference. The sacrifices that Christians make, as explained above, are either done with an eye towards their own comfort and enjoyment, or are activities that even non-religious folk may undertake motivated by humanitarianism – such as giving to and helping the poor and needy.  

 

 In contrast, Ramadan is a singularly religious practice. Unlike Christian activities motivated by  self-serving or humanitarian reasons, Ramadan is observed solely because it is commanded by Allah. There is no other reason to make this extended annual sacrifice of time and comfort than to honor and obey Allah. 

 

That’s pure and unadulterated worship.   Impressive.

 

That’s not to say that Ramadan is without side benefits. This demanding practice promotes community and shared sacrifice similar to military traditions which develop esprit de corps. It also helps one develop self-discipline – discipline needed in observing other practices of the Islamic faith and virtuous living. Indeed, it was this discipline and community that transformed the life of Malcolm X from one of crime and self-centeredness to a life dedicated to changing the world. 

 

Impressive indeed. No matter our own beliefs, would that we all should live so well. 

Changing Colors

 HS #81 2022.4.14

 

Changing Colors

 

I heard recently that a common mistake middle aged folk make in planning for the future is assuming their interests won’t change. For the young, it’s expected that changes occur. One of the richest verses from scripture is Luke 2:52: “Jesus grew in wisdom and stature and in favor with God and man.” Getting annual Christmas cards each year, it’s clear (especially from the pictures) that attention and pride resides with the growth and development of children and grandchildren. 

 

But once we become adults and pass into and through middle age, it’s easy to suppose that our interests and passions will remain constant, even if our bodies give out a bit. Indeed, as adults our desire is to remain the same. Middle age and beyond might be defined as that period of life where we are jealous of our former selves. 

 

Poets certainly have expressed this thought. I’ve had this poem by Yeats on my refrigerator for years: “I thought no more was needed // Youth to prolong // Than dumbbell and foil // To keep the body young. // Oh, who could have foretold // That the heart grows old?” Yeats realized that our noble ambitions wane as we age. 

 

Counterexamples exist. The great Hungarian mathematician Paul Erdos, the most prolific of the 20^th century with 1500 articles, continued publishing until his death at age 83.  Similar to the Kevin Bacon Number among actors, mathematicians have an Erdos Number depending on whether they coauthored an article with Erdos (#1), coauthored with someone who coauthored with Erdos (#2) and so on. Driven by his love of mathematics, Erdos described fellow mathematicians who stopped researching as “they died.” When someone truly died, he said simply, “They left.” 

 

My 93-year-old aunt (Wanda Hendrickson) who learned how to roller blade at 75 and still takes classes at Calvin University is a personal counterexample and inspiration. Our regular visits provide an intellectual challenge. 

 

Robert Frost also contrasted between the glory of youth versus the sameness of middle age in his poem, “Nature’s first green is gold. Her hardest hue to hold. Her early leaf’s a flower, But only so an hour. Then leaf subsides to leaf. So Eden sank to grief. So dawn goes down to day. Nothing gold can stay.” 

 

Indeed, each Tulip Time reminds us that the “early leaf” – the petals of tulips – are appreciated just because we know they won’t last. In early May my front yard is gorgeous as the dogwood tree blossoms in pink and white. Then leaf subsides to leaf – to the necessary chlorophyll-green leaves which drink up the sun’s energy. 

 

 

Garrison Keiller (Prairie Home Companion) once said he’d trade his entire decade of 40’s to be 19 again. As a mathematician, I think he was on to something. Since each new year of life adds just a fraction to the memory of our past years, an alternative way to determine our age is by adding: 1 + ½ + 1/3 + 1/4 + . . . That is, early years of life are the most colorful and rich, we learn more than in our later years. High school seniors missed much more during the COVID down shutdown than did us 60-somethings. 

 

Mark Twain caught the idea with his usual wit: “Youth is so precious, it’s a shame it’s wasted on the young.” 

 

However, nature has one more lesson for us. Not only is the “early leaf” colorful, so also is the autumn leaf, albeit for a different reason. The brilliant fall pigments of red, orange and yellow are contained in the leaf all along, but they are dominated by the green chlorophyll. With less sunlight in the fall, the unneeded chlorophyll subsides, and the other colors – just as rich but not as assertive – have opportunity to express themselves. 

 

Similarly, as the passion of youth and focus of middle age subside due to changes in body chemistry and the accumulation of life experiences, perhaps that allows some subtleties in our personality – there all along – to be revealed. All of which shows that, indeed, even as we age, we continue to change – new colors, long muted, being revealed in our personalities. 

 

Is death the end of this process? Are other colors waiting to be expressed? Even the most creative among us cannot imagine the other colors in the electromagnetic spectrum left of red (infrared) and right of violet (ultraviolet). So also, any color changes to come are beyond 

ours to know and even imagine.

 

Travels with Elvis

 HS #80 2022.3.10

 

Travels with Elvis

 

I listen to audio books from Herrick District Library when driving. Biographies are my favorite. Ben Franklin, Thomas Jefferson, Theodore Roosevelt, and Leonardo da Vinci inspire and help keep one’s life in perspective. Presently I’m reading John Steinbeck’s “Travels with Charlie.” Whoever has not read his “East of Eden” should put it on their bucket list. Nothing better. 

 

Listening to Steinbeck’s adventures touring the country with his pet dog spurs me to tell adventures I had with my Welsh corgi, Elvis. When climbing the Laketown Beach steps between a friend and me, my friend observed, “He thinks he’s human, doesn’t he?” Indeed, it never occurred to Elvis that he wasn’t. 

 

It was at Laketown Beach that Elvis answered the question, “Do Dogs Know Calculus?” the title of my research paper co-authored with him. Essentially, Elvis, starting on shore, retrieved a stick thrown into Lake Michigan by first running and then swimming to fetch it as quickly as possible. The paper provided fame for both of us and gained his picture on the cover of numerous journals and several front pages of the Holland Sentinel.  I met one of his biographers (Keith Devlin, NPR’s Science Guy) at a national math convention.  Introducing myself after his talk, he exclaimed, “Tim Pennings! – I didn’t recognize you with your clothes on!” He then explained to the curious crowd (and me!) that he had sought a picture of Elvis and me for his book, but only found ones of us at the beach – me sans shirt. 

 

I readily admit that Elvis was the star of our dog and pony show – or as I’d describe, “Dog and Jack-ass Show.” A former student asked how it felt to be famous just because “you have a dog who is better looking than you even when he is running away?”  No argument. 

 

I sent two copies of the Elvis-covered math journal to fellow-corgi-lover Queen Elizabeth – one signed by me and the other requesting her signature. I received the journal from Buckingham Palace a month later with a note from her “Lady in Waiting” assuring me the Queen found our story interesting, but that she gave her signature only to those she personally knew.  Not having that rule for myself, I’m assuming she still appreciates having my signature in her library.  I kept the returned unsigned journal – after all, how many have been spurned by Her Majesty? 

 

That article led to another, “Do Dogs Know Bifurcations?” describing how Elvis, when starting in the water, would first swim to shore, then run along the shore, then swim to the stick, so that the resulting trek was the quickest. Not only did that paper earn me my highest professional honor (MAA George Polya Award), it also earned Elvis an honorary doctor’s degree from Hope College, awarded by Provost Jim Boelkins. The certificate was inscribed with the proper Latin phrase, which I shortened when I made Elvis business cards. I was later told that the abbreviated phrase had a different meaning – it conveyed “doctor” and “dog”, and because of the feminine sense of the Latin word was perhaps best translated, “Dog Gynecologist.” I’ve given away thousands. 

 

Elvis and I gave close to 300 talks all around the country, and I’ve lost count of the number of newspaper articles about him. Colleges would often invite the local press, so as we’d traveled through the state, I’d see his picture on the front page of newspapers at rest stops.  

 

He contributed to recreation trips too. As the advisor for Hope College Outdoor Adventure Club, Elvis and I led groups of students to Pictured Rocks for Fall Break. In lieu of an alarm clock, I’d unzip the tents, gently usher Elvis in among the sleeping students, then quietly command, “Speak. Speak.” Always worked. 

 

He was also fun at home. I once mistakenly fed him an Easter ham bone which plugged up his digestive track. Per the vet’s instructions, I put him on the back porch that night with the door open accessing the back yard. Feeling sorry for him, I bundled in a sleeping bag and joined him. After a chilly night’s sleep, I awoke shivering to find him warm and cozy INSIDE the house. He had pottied, gone to the front door, barked, and my neighbor let him in. 

 

What a guy. Initially Elvis was my dog who was a good friend.  Before he died Elvis was my good friend who just happened to be a dog.  Good memories. 

Three Mysterious Numbers

 HS #79 2022.2.10

 

Three Mysterious Numbers

 

 When I teach precalculus, I bring rope of various lengths to class and invite students to pair up. One holds an end stationery, and the partner – holding the other end and keeping the rope taught –  counts paces around the circumference of the resulting circle, and then across its diameter. When I did it, I counted 50 paces around, and 16 across. The ratio: 50/16 = 3.125 is an approximation of pi. The Greek letter pi represents the ratio of the length of circumference to the length of diameter. 

 

This number, so easily understood, is mysterious.  If you try to write it out as a decimal, it never stops and never starts repeating. Is there a place in the decimal expansion where there are 1000 sevens in a row? Most certainly. 

 

Pi shows up other places as well. If you continue the pattern: 1 - 1/3 + 1/5 - 1/7 + 1/9 – 1/11 + . .  and multiply the result by 4, you’ll get pi. 

 

Or try this:  Snip a piece of paper clip to the length of the distance between the lines of ruled paper. Then randomly spin and drop the clip onto the paper T times, counting the number of times (C) the clip crosses one of the lines. Then pi = 2T/C. How cool is that! 

 

Perhaps the most surprising result involving pi is also the simplest. A 25,000 mile length of string will lay around the equator of the earth.  How much longer does the string need to be if the string is raised one foot above the earth along the entire equator? I guessed 10,000 miles.  Answer: If r is the length of the radius of the earth in feet, then the equator is 2(pi)r feet. The circumference of the larger circle is 2(pi)(r+1) feet. The difference in lengths is 2(pi) feet – just over 6 feet longer! Blows my mind.

 

Here’s another mysterious number: If you have a calculator, multiply (1+1/2) by itself twice. Then multiply (1+1/3) by itself three times. Then multiply (1+1/4) by itself four times, etc. What happens to this sequence of numbers? As the sequence continues, the portion inside the parenthesis decreases to 1, but since it is multiplied by itself more times, the resulting number grows larger. So there is tension – a struggle. Some students think the resulting numbers will get closer to 1. Others think they will get arbitrarily large. Turns out that neither happens. The struggle ends in compromise.  The numbers get closer and closer to e = 2.718 . .  It’s another number that never stops. 

 

The number e is at the heart of the “normal distribution” or “bell shaped curve” which is used extensively in statistics. So, like pi, it is super important in understanding the world. 

 

One other mysterious – beautiful – number:   Consider the sequence: 1, 1, 2, 3, 5, 8, 13, 21, . . where each successive number is gained by adding the previous two. This is the famous Fibonacci Sequence. These numbers are found throughout nature – count the petals on a flower or the seeds in an arch of a sunflower, or the knobs in a row of a pineapple and you will often get a Fibonacci number. If you divide a Fibonacci number by its predecessor, the result will be about phi = 1.618 . .  Yet another number that goes on forever. This number is called the Golden Ratio. If you construct a rectangle so that the length of the longer side is phi times the length of the shorter side, you get a Golden Rectangle. Remove a square from a Golden Rectangle, and the remainder is another Golden Rectangle. When I ask 5^thgraders how many times this can be done, they shout “Infinite!” Their response gives me goosebumps; how interesting that even children imagine infinity.  (See Ecclesiastes 3:11.) Golden Rectangles are found throughout art and architecture. da Vinci used them extensively. Google it and see for yourself. 

 

On the other side of the interesting-spectrum of numbers are 0 and 1 - the most basic and ordinary numbers around. Adding 0 doesn’t change a number, and multiplying by 1 doesn’t change it. 1 has a counterpart, i, which is its imaginary cousin. How eerie then that 0, 1, i, e, and pi can all be combined in one simple equation – called Euler’s Equation, perhaps the Holy Grail of Mathematics. Yet one more surprise: “Euler” is pronounced, “oiler.” Pronouncing it correctly will impress any mathematician.