The Shape of the Universe

How does one describe the shape of an ever-expanding universe? It is not easy to imagine what it might look like. Neither was it easy for some to imagine a spherical earth. For many centuries, the best model for the shape of the earth was a flat disk. This model held sway for at least five centuries in the Common  Era (CE); and, even then, a spherical model was only adopted in certain Greek philosophical circles. Imagine for a moment how difficult it would have been to conceptualize that if you traveled west long enough you would arrive back at your point of departure. Returning to the shape of the universe, apply such imagination to its shape. What if someone told you that if you kept going in one direction in the expanse of the universe, you would one day come back to your point of origin? Would that be similarly hard to comprehend? That is precisely what would happen in the current best model for the shape of the universe. The model is represented by a toroid. It is sometimes described as a donut shape but a donut shape is only the simplest of the toroidal shapes. The shapes pictured in this blog also define toroids.

The best calculations by mathematical physicists tell us that the universe is close to completely flat and could fit on the surface of such a toroid. This is a shape that is definitely hard to conceptualize for very long. I think it must have been brief glimpses into the mysteries of the universe such as this that led Albert Camus to say, “Beauty is unbearable, drives us to despair, offering us for a minute the glimpse of an eternity that we should like to stretch out over the whole of time.” We can only hold such images in our minds for the briefest of moments before they escape us or drive us mad. Those who lived in a "flat disk world-view" must have felt the same. How could one hold onto this concept for more than a few seconds? It was outside of the experience of all people until a few brought back reports of strange journeys and new understandings of geography and geometry. Now, although we take it for granted that it can be done, few have circumnavigated the globe to prove the theory. I can only dream of what the experience might be like and only astronauts have a true, immediate, experience of circumnavigation. But, in my mind, momentarily, I can capture the mystery.

Works cited:
Nature; Published online 23 May 2008 | Nature | doi:10.1038/news.2008.854; "Doughnut-Shaped Universe bites back: Astronomers say Universe is small and finite."

Slide Rules and Gopher Protocols

(Click on the thumb-nail image for a larger picture.)

In my junior high education we used logarithm tables to do specific types of math and in grade 10 science I purchased a slide rule to do my calculations. In 1975, calculators were not very sophisticated, they were expensive, and my physics teacher believed we all should learn how to use a slide rule. So when I read a paragraph in a 2014 Science News article, I felt like someone had put my picture in the dictionary beside the definition of old. For there I read:

You may find this hard to believe, but there are people still alive today who once did their mathematical calculations by sliding sticks back and forth. No keypads, no batteries, no LEDs. Just sticks.
Yes, it sounds like a device from the Stone Age, but as late as the 1970s scientists and engineers commonly used such a stick-sliding device, known as a slide rule, to perform multiplication and division and other tasks like extracting square roots. Working versions of these instruments still are on display in museums today.1

As I sat reading these words in my office I reached over to the top drawer of my desk and pulled out a familiar cloth case containing my 1975 slide rule. A thing of beauty, a logarithmic scale on two sticks. I slid the sticks and cursor and did a quick calculation to convince myself that my brain was not ready to be on display in a museum. "I've still got it," I thought, and then marveled at the rapid advancement of science, math, and communication witnessed in those 39 years. Programmable calculators were one of the next significant developments and in 1977 I had to shell out $120 for my first Texas Instruments (TI) calculator for a calculus course I was taking. It had less computing power than the GPS watch now on my wrist or the phone now in my pocket but was capable of solving formulas with the push of just a few buttons. Of course that was only after I had correctly entered all of the formula operands into the temporary memory of the TI (sometimes no small feat). The home computer revolution followed quickly after this and as the computers got more powerful their footprint got smaller.  Calculations that had previously been done on devices that took up a whole room were now worked out on top of our desks.

Computing took another huge leap forward when ARPANET allowed communication between computers around the world. As the World Wide Web (or Internet) grew, so did our ability to share and store information. My first forays into the "bulletin boards" of the ARPANET were to find information that research labs had stored about the chromosomal locations of common genetic disorders in humans. This information allowed me to work out experiments in which I was able to determine the probability that a patient in our clinic would develop a certain genetic disorder present in her family. I used a search program known as the Gopher protocol which was fast and efficient . . . just like my slide rule. Advancements in knowledge and search engines mean that we now have a much larger array of information available to us on a massive range of topics. Much of this began 400 years ago with the invention of the logarithm.


1 https://www.sciencenews.org/blog/context/logarithms-celebrate-their-400th-birthday

Group Dynamics

Tucked in a 64 year old book is a mathematical formula that gives clues about grouping procedures. Bossard1 points out that the number of people in a group may increase by simple mathematical progression, but the increase of relationships comes through geometric progression.

Two variables are defined. Let Y equal the number of persons in the group, and X the number of personal relationships between the members. Then using the formula X = (Y2 - Y)/2 (that is, X equals Y squared minus Y all divided by 2), as Bossard1 has, we find that the larger the group, the more disproportionate the increase in personal relationships.

Size of group: 2 3 4  5   8   12   15  35
Relationships: 1 3 6 10  28  66 105 595

Note how radically the number of relationships increases with the addition of one or two people. What does this do to the individual in terms of communication, understanding, and ability to participate without pressure or frustrations?

Hundreds or thousands may be spectators. Working, interacting groups seem to do best when composed of five to eight members. If the group is larger, some become performers and others spectators. At age six, spontaneous groups seldom exceed three or four children. Sizes now accepted for school classes are much too large for good cooperative work.2

If the mathematical formula is a hang-up, try drawing the relationships on a page of paper to convince yourself of the truth of this work. See the example at the end of this blog.

What implications do such formulas have for those of us who work with small groups in education or church leadership? How might we be excluding people in some of our educational contexts? We talk a lot about being communities of believers or communities of learners but this information suggests that members could easily be left on the fringe and never truly feel part of the group. The person who can entertain a large group of 1000 people may not be the best person to teach people to care for others or interact with others. Some of the goals we seek to accomplish in school or church cannot be accomplished in the size of groups we seek to use. We would be better served to restructure some of our groups so that peer learning can happen in groups of twos and threes. Neil Cole has some further insight on group dynamics in his book Cultivating a Life For God: Multiplying Disciples Through Life Transformation Groups.3 It may well be worth another look.

1 James H. S. Bossard. The Sociology of Child Development. New York: Harper and Brother, 1948. p. 146.
2 This whole section is adapted from an article by Mary Margaret Scobey, entitled "Developing and Using Classroom Groups," 1960. See http://ascd.com/ASCD/pdf/journals/ed_lead/el_196312_scobey.pdf.
3 Cole, Neil. Cultivating a Life For God: Multiplying Disciples Through Life Transformation Groups. Carol Stream: ChurchSmart Resources, 1999.

Newton

"Gravity explains the motions of the planets, but it cannot explain who set the planets in motion. God governs all things and knows all that is or can be done." - Isaac Newton*

Isaac Newton is often referred to as an English mathematician & physicist who lived from 1642 - 1727. Yet few remember that he was also a theologian. In fact, he wrote more on theology than he did on natural science. The intriguing thing about this man is that he was a brilliant scientist who believed that he could maintain faith in God while exploring the intricacies of the universe created by God. He had this to say about the uniformity of creation, the ability of eyes to capture light, and the Being who made them:

. . . Can it be by accident that all birds, beasts, and men have their right side and left side alike shaped, (except in their bowels,) and just two eyes, and no more, on either side of the face; and just two ears on either side of the head, and a nose with two holes; and either two fore- legs, or two wings, or two arms on the shoulders, and two legs on the hips, and no more? Whence arises this uniformity in all their outward shapes but from the counsel and contrivances of an Author? Whence is it that the eyes of all sorts of living creatures are transparent to the very bottom, and the only transparent members in the body, having on the outside a hard transparent skin, and within transparent humours, with a crystalline lens in the middle, and a pupil before the lens, all of them so finely shaped and fitted for vision, that no artist can mend them? Did blind chance know that there was light, and what was its refraction, and fit the eyes of all creatures, after the most curious manner, to make use of it? These, and suchlike considerations, always have, and ever will prevail with mankind, to believe that there is a Being who made all things, and has all things in his power, and who is therefore to be feared.#

I bring Sir Isaac Newton to our attention not so that we can follow his theological writings. He was not an orthodox Christian and did reject some doctrines that most contemporary theologians and Christians would view as essential (most notable would be his view on the trinity). Yet, he can teach us a way forward in our contemporary world. Many today assume that we must make a choice: scientist or theologian; believer in science or believer in God. What if, like Newton, we chose to say, "I can trust scientific technique and I can trust God."? Scientists of the 17th century had no problem with this. It is only our contemporary "scientism" and "evolutionism" as philosophical constructs that are in opposition to faith and theology. What might we learn? How much more of God might we understand if we started with the assumption that science can lead us to the Creator rather than starting with the assumption that there is no God? Newton even goes so far as to say that a natural consequence of recognizing that God created this universe ought to be that we also "fear" Him. Indeed, if we allow that there just might be a God behind all that we see would it not be reasonable to respect this awesome Being who has created a universe? In fact, He would be a reasonable authority on the best way to live in this universe in which we find ourselves.


*Tiner, J.H. (1975). Isaac Newton: Inventor, Scientist and Teacher. Milford, Michigan, U.S.: Mott Media. See also http://en.wikipedia.org/wiki/Isaac_Newton#Religious_views

#Memoirs of the Life, Writings and Discoveries of Sir Isaac Newton by Sir D.Brewster Volume 2. A portion of this is also quoted in Brouwer, Sigmund. Who Made the Moon? Nashville: Thomas Nelson, 2008.

Mathematics

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning. - Eugene Wigner