Happy ~3.14159… Day!

Today is Pi Approximation Day. This day celebrates the fractional representation of π, which is 22/7 = 3.14285….

The date originates from day and month nomenclature, rather than month and day nomenclature.

Let’s look closer at π:

When we measure the distance around a circle (starting and stopping at the same point) compared to the distance measured across the circle’s center, we are computing a ratio (and calculating a number). As curious as this intuitive measurement seems — a simple ratio of girth versus breadth — the resulting number is the mathematical constant π, defined as π = C/d. Here C represents the circumference of the circle, and d is the diameter of the circle. We learn that the distance around any given circle is a little more than three times the distance across it.

Pi itself is an irrational number. Irrational numbers are numbers that have no terminating digit after its decimal point, including no terminal repeating digits or terminal sequence of digits. Nevertheless, only the first 39 digits of π are needed to accurately calculate the spherical volume of our entire universe. Thankfully, however, when thinking about π or invoking its use during hand calculations, most people (including many math students) are safe using the first 5 decimal places, such that π = 3.14159.

Pi is used in the following non-exhaustive list of formulas:

  • Circumference, C, of a circle: C = 2πr
  • Area, A, of a circle: A = πr2
  • Volume, V, of a cylinder: V = πr2h
  • Volume, V, of a sphere: V = (4/3)πr3

For each of the aforementioned equations r is radius, and pertaining specifically to the volume of a cylinder, h is height.

Celebrate Pi Approximation Day!

Biomimicry 101: Turbulence and Chaos Theory

eagles-wingsFigure: The upward curvature (called winglets) found on the tips of wings in certain bird species, such as the bald eagle, increases aerodynamic efficiency by helping reduce drag.

When you ponder biomimicry, think design, intelligent design, and creation, but please don’t think evolution. And when considering biomimetic applications, let your imagination know no bounds.

Turbulence is the name for the small eddies that break off of larger ones; it is dissipative; it is unstable and dynamical. In fact, the study of turbulence is at the core of chaos and non-linear, dynamical systems – a subfield within mathematics with special appeal to mathematical physicists. For instance, the vortices (i.e., turbulence) that whirl off the wings of Boeing or Airbus passenger jets will differ greatly from those off the most sophisticated jet fighters, such as the F-18 Super Hornet. No matter what, each plane has its own “fingerprint” when it comes to air flow dynamics across their wings. And this characteristic signature is even more complex and dynamical for sweep-wing planes, such as the now retired but amazing F-14 Tomcat.

Of interest, just yesterday, July 19th, 2019, Airbus released a conceptual model of a large passenger prop plane with splayed wingtips and a fanned tail. This “hybrid” design attempts to reduce as much of the turbulent air flow as possible by mimicking the wings of soaring eagles.

Remark: Current models of many mid-sized to large passenger jets and cargo planes already have winglets to minimize the swirling eddies at the tips of their wings.

What is more, to help promote mathematical thinking and biomimicry, Andrew McIntosh, professor of thermodynamics and combustion theory, at the University of Leeds has held paper airplane design contests. These contests offer students a hands-on approach to help creativity form in the minds of future engineers.

Something completely different but related: aerospace and aviation

Today marks the 50th anniversary of the Apollo 11 moon landing!

On July 20, 1969, Neil Armstrong and Buzz Aldrin navigated the lunar module they piloted, named the Eagle, down onto the surface of the moon. Within moments of their landing, Armstrong radioed NASA Mission Control in Houston, Texas, his now infamous message, “Houston, Tranquility Base here. The Eagle has landed.” Soon afterwards, Armstrong made a more profound statement when he became the first person to set foot on the moon, saying, “That’s one small step for man, one giant leap for mankind.”

Three years later, on July 20, 1972, the Armstrong Air and Space Museum in Wapakoneta, Ohio, — the birthplace of Armstrong — opened its doors.

We invite you to return to our 2018 visit to the museum (click here).

Can you imagine a career in mathematics that might even help mimic nature – God’s great creation?

Starting out now on your very own discovery of the intricacies of God’s creation through scientific study might very well help you in discerning what path you should take in the future. Christian scholarship that extends into professional roles — such as mathematicians, engineers, and physicists who work in chaos and non-linear dynamics or aerospace and aviation to biologists and zoologists who study ornithology — is sincerely needed in education and society.

Biomimicry 101: Special Insert: Mathematical Cardiology

mandelbrot-set-x2Central illustration: At the heart of the Mandelbrot set lies the cardioid, a well-defined mathematical representation (left panel). On the cardioid’s boundary exists an assortment of circles, with each of these circles further continuing its own fractal pattern (right panel; zoomed-in view for illustrative purposes).

When you ponder biomimicry, think design, intelligent design, and creation, but please don’t think evolution. And when considering biomimetic applications, let your imagination know no bounds.

Here we touch briefly on cardiac function and fractal behavior via what we have coined mathematical cardiology. Our intent is to glorify God as the universe’s greatest mathematician.

Mathematical cardiology – a unique pattern

While we have argued that the heart itself is complex and uniquely designed, others offer claims suggesting that the heart is seminally prototypical. In part, evidence in either case points to the intrinsic fractal behavior of the heart. Specifically, fractal descriptions mark the quintessential characteristic of non-linear, dynamical systems, and with respect to the heart such behavior is clearly present. The preeminent example deals with the heart’s electrophysiology. The rhythmic impulse that is ultimately carried through the heart’s conduction pathway can, at times, become quite irregular. In these cases, the rhythm (the irregularity) is called an arrhythmia. What is more, two somewhat related arrhythmias, known as atrial fibrillation and atrial flutter, are regarded as irregularly irregular. This pattern of irregular irregularity is understood to be a chaotic temporal disturbance of atrial origin. Furthermore, although dynamical systems are non-linear and chaotic by nature, at the center of any such system is simple, identifiable deterministic activity (despite a behavior seemingly acting in an erratic way) (Bassingthwaighte & van Beek, 2002). What this means for patients with atrial fibrillation or atrial flutter is that algorithms can be developed to mathematically identify these arrhythmias based on the deterministic, fractal character. This identification translates into a more proactive means of rhythm identification and possible correction. In fact, in the case of an urgent, life-threatening arrhythmia, attempts to convert to normal sinus rhythm via implantable defibrillators become possible sooner (or even anticipated), due to resultant mathematical interpretation of the rhythm’s fractal behavior (Captur, Karperien, Hughes, & Moon, 2017).

References

Bassingthwaighte, J.B., & van Beek, J.H.G.M. (2002). Lightning and the heart: Fractal behavior in cardiac function. Proceedings of the IEEE: Institute of Electrical and Electronics Engineering, 76(6), 693-699. [View in article] [View in PubMed]

Captur, G., Karperien, A.L., Hughes, A.D., Francis, D.P., & Moon, J.C. (2017). The fractal heart – embracing mathematics in the cardiology clinic. Nature Reviews: Cardiology, 14(1), 56-64. [View in article] [View in PubMed]

311th birthday of Leonhard Euler

leonhard-euler-and-methodus-inveniendi-max-minLeonhard Euler (1707-1783) and his Methodus Inveniendi (1744)

April 15th, 2018, marks the 311th birthday of the Swiss mathematician Leonhard Euler (properly pronounced “OY-lur”), and this blog is a small tribute to his memory.

Quotes about Euler:

  • When reflecting on the honesty of Euler and his personal convictions, Dale McIntyre (2006) wrote, “Such character was truly a testimony to the grace of God at work in his life.”
  • Concerning the prominence of Euler in the annals of mathematics, Josef Hofmann (1939) wrote, “Between us and the mathematicians of the late seventeenth century [people like Isaac Newton or Gottfried Leibniz] stands Leonhard Euler.”
  • And Eli Maor (1994) gave a well-thought-out summary on the lasting impact of Euler’s mathematical prowess:
If we compared the Bernoullis to the Bach family, then Leonhard Euler is unquestionably the Mozart of mathematics, a man whose immense output — not yet published in full — is estimated to fill at least seventy volumes. Euler left hardly an area of mathematics untouched, putting his mark on such diverse fields as analysis, number theory, mechanics and hydrodynamics, cartography, topology, and the theory of lunar motion. With the possible exception of Newton, Euler’s name appears more often than any other throughout classical mathematics. Moreover, we owe to Euler many of the mathematical symbols in use today, among them i, π, e, and f(x). And as if that were not enough, he was a great popularizer of science, leaving volumes of correspondence on every aspect of science, philosophy, religion, and public affairs. (p. 153)

Indeed, Euler made many contributions to mathematics. Apart from the accomplishments mentioned above, he is known for helping establish a specific field of math known as calculus of variations. This area of math is concerned with functionals (i.e., functions of functions), and mostly deals with their extrema, specifically maxima and minima. The website WolframMathWorld defines any maximum as the largest value of a mathematical set or function, and likewise defines any minimum as the smallest value (Weisstein, n.d.).

In fact, we catch a glimpse of Euler’s prowess (mathematical, philosophical, and theological) in the opening lines to the introduction of his 1744 book called Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, a book in which Euler instituted calculus of variations. Today we have the pleasure of reading these lines in English — via the translation offered by Oldfather, Ellis, and Brown (1933) in their article Leonhard Euler’s Elastic Curves:

For since the fabric of the universe is most perfect, and is the work of a most wise Creator, nothing whatsoever takes place in the universe in which some relation of maximum and minimum does not appear. Wherefore there is absolutely no doubt that every effect in the universe can be explained as satisfactorily from final causes, by the aid of the method of maxima and minima, as it can from the effective causes themselves. (pp. 76-77)

— The Four Causes —

Here we point out that Euler both recognized and equated final cause and effective cause (also known as efficient cause), two of the four so-called causations recognized in Western philosophy. In fact, it was Aristotle who postulated the four causes: material cause, formal cause, efficient cause, and final cause (Falcon, 2015).

  1. Material cause: “that out of which”
  2. Formal cause: “the account of what it is to be”
  3. Efficient cause: “the primary source of the change or rest”
  4. Final cause: “the end, that for the sake of which the thing is done,” in other words, “the purpose”

Creationists have no problem incorporating all four causes into their worldview; however, on average most, if not all, evolutionists (naturalists and materialists) only find significance in material and efficient causes. For an overview that offers a brief but concise presentation on the topic of causation, please see the article “Four Causes” by Peter Blair (2009) in The Dartmouth Apologia: A Journal of Christian Thought.

— The Basel Problem —

Finally, Leonhard Euler’s claim to fame came in the years he solved (1734) and presented (1735) the answer to the Basel Problem, a puzzle which by that time was a 100-year-old math problem named for the birthplace of its origin, Basel, Switzerland. This problem had notoriously eluded many mathematical minds before Euler by asking for the precise summation of the reciprocals of the squares of the natural numbers, i.e., the following infinite series:

1 + 1/22 + 1/32 + 1/42 + 1/52 + 1/62 + 1/72 + …

To see the answer to this problem, and at the same time have some fun grasping a remarkable mathematical and physical perspective on it, take a look at the short video called “Why is Pi Here? And Why is It Squared? A Geometric Answer to the Basel Problem” by Grant Sanderson (2018).

In closing, we wish to echo the sentiment of the French mathematician Pierre-Simon Laplace, who once said about Euler, “Read Euler, read Euler, he is the teacher of us all.”

References

Blair, P. (2009). Four causes. The Dartmouth Apologia: A Journal of Christian Thought. Retrieved from http://www.dartmouthapologia.org/apologia/four-causes/.

Falcon, A. (2015). Aristotle on causality. The Stanford Encyclopedia of Philosophy. Retrieved from https://plato.stanford.edu/entries/aristotle-causality/.

Hofmann, J.E. (1939). On the discovery of the logarithmic series and its development in England up to Cotes. National Mathematics Magazine, 14(1), 37-45.

Maor, E. (1994). e: The story of a number. Princeton, NJ: Princeton University Press.

McIntyre, D. (2006). The God-fearing life of Leonhard Euler. Journal of the Association of Christians in the Mathematical Sciences. Retrieved from https://acmsonline.org/the-god-fearing-life-of-leonhard-euler/.

Oldfather, W.A., Ellis, C.A. & Brown, D.M. (1933). Leonhard Euler’s elastic curves. Isis, 20(1), 72-160.

Sanderson, G. (2018, March 15). Why is pi here? And why is it squared? A geometric answer to the Basel problem [Video file]. Retrieved from http://www.3blue1brown.com/videos/2018/3/15/why-is-pi-here-and-why-is-it-squared-a-geometric-answer-to-the-basel-problem.

Weisstein, E.W. (n.d.). Maximum. WolframMathWorld. Retrieved from http://mathworld.wolfram.com/Maximum.html.

Weisstein, E.W. (n.d.). Minimum. WolframMathWorld. Retrieved from http://mathworld.wolfram.com/Minimum.html.

Pi Day celebrates 3.14159…

Happy 3.14159… Day!

When we measure the distance around a circle (starting and stopping at the same point) compared to the distance measured across the circle’s center, we are computing a ratio (and calculating a number). As intuitive and curious as this ratio seems — a simple measurement of girth versus breadth — the resulting number is a mathematical constant known as pi, which we notate by the Greek letter, π, and define as π = C/d. Here C is the circumference of the circle, and d is the diameter of the circle. We learn that the distance around any given circle is a little more than three times the distance across it.

Pi itself is an irrational number. Irrational numbers are numbers that have no terminating digit after its decimal point, including no terminal repeating digits or terminal sequence of digits. With these concepts in mind, we invite you to view the first million digits of pi at the following link: One Million Digits of Pi. Interestingly, only pi’s first 39 digits are needed to calculate the spherical volume of our entire universe (Pi Day Website).

Yet, even though pi cannot be expressed as a rational number (and for that matter a common fraction), it is approximated by the fraction 22/7. Most people, however, choose to simply memorize a certain string of digits from pi, such as π = 3.14159.

Pi is used in the following non-exhaustive list of formulas:

  • Circumference, C, of a circle: C = 2πr
  • Area, A, of a circle: A = πr2
  • Volume, V, of a cylinder: V = πr2h
  • Volume, V, of a sphere: V = (4/3)πr3

For each of the aforementioned equations r is radius, and pertaining specifically to the volume of a cylinder, h is height.

Today is the 30th anniversary of Pi Day!

— Pi Day Contest (sponsored by Ashland Creation Colloquium) —

Celebrate Pi Day by taking NASA’s out-of-this-world mathematical challenge (click this internal link to download the novelty poster for the Planet Pi challenge).

External link to NASA’s website for the Planet Pi challenge novelty poster

Entry Deadline: Please submit your answers by the 22nd of July, 2018 (also known as Pi Approximation Day) to the provided address.

writing.studiesoncreation.org@gmail.com

Contest Rules: Entries with correct answers will be placed into a raffle, and three winners will be drawn. Each winner is awarded his or her choice of a pie pan and server, or a straightedge and compass! Serve up a piece of your favorite pie and contemplate the volume of the universe; use the compass and straightedge to harness your mathematical prowess by attempting to square the circle (see WolframMathWorld: Circle Squaring). This contest is open to all ages.

Disclaimer: By participating, contest participants agree to release all liability against studiesoncreation.org and Ashland Creation Colloquium, e.g., waive any injurious liability that may occur through use of prizes. This contest is open to all ages. We value safety first; winners are encouraged to follow intended use guidelines associated with prizes, which may involve proper supervision from parents, teachers, or guardians.

Fun Fact: Today is also the birthday of Albert Einstein, who if still alive, would be 139 years old.

Upcoming Celebrations: Leonhard Euler’s 311th birthday, April 15, 2018, and NASA’s 60th anniversary, October 1, 2018.

2017 marked the 100th anniversary of the Radon transform

1917-2017-100-years-of-the-Radon-transform

God certainly inhabits the honest work of believers and unbelievers alike and uses it to ends far beyond the original vision of those workers! And how often this has happened in mathematics!

— Dale McIntyre, PhD

The year 2017 celebrated the 100th anniversary of an important and unique yet somewhat obscure mathematical operator that came to be known as the Radon transform — aptly named for its discoverer, the Austrian mathematician Johann Radon. In 1917, Radon introduced this transform early in his career in a paper titled “On the Determination of Functions from Their Integral Values along Certain Manifolds.”

On a technical note, transform denotes special meaning in math. Intuitively, however, we may describe a mathematical transformation as an operation that changes the mathematical nature of some shape, entity, or thing. With this concept in mind, the Radon transform is defined as an integral transform, which means it makes use of integration to achieve this purpose.

Now, when we think of integration (and an integral), our thoughts lean us towards calculus (with its two main divisions, differentiation and integration, divisions which constitute differential calculus and integral calculus). And as the case is in most areas of math, in calculus each function or equation we work with can itself be represented as a graph. And when we consider the most basic approach to integration, we teach students to calculate the area under a given curve. This basic premise gives us a number, which in this case represents area under the curve. Integration is not, however, capable of producing the actual equation whose graph is the given curve. On the other hand, the Radon transform offers more detail. Notably, when integration is carried out within the principal constraints and context of the Radon transform, the process finds the equation.

The Radon transform uniquely offers us the ability to integrate, and as a result, get back the equation and not simply a number.

To be clear, transforms in the mathematical sense were not new in 1917, but what Radon offered mathematics was. What Johann Radon discovered was a pure mathematical construct, and he applied it that way — purely within mathematics. In fact, most of the early applications were in a field of advanced math called differential equations.

But are there physical applications? On this note, a distinguishing feature of the Radon transform is its ability to take one thing or body in space and represent it elsewhere (for example, transforming some entity that exists spatially to a graphical representation, such as a picture or image). Outside of mathematics, the Radon transform famously underpins medical x-ray computed tomography (also called CAT scans, or the preferred designation today, CT scans). However, it would be a half century — following Radon’s 1917 published paper — until the transform’s application and usefulness to CT theory became recognized. And with this realization, a new era in medical imaging emerged, subsequently allowing CT scanners to leap out of the minds of engineers and become a possibility (rather than existing merely as thought experiments). According to Wininger (2012), the CT scanner consequently became one of the most significant inventions witnessed in the field of radiology and medicine since the discovery of x-rays in 1895 (p. 2).

Men became scientific because they expected Law in Nature, and they expected Law in Nature because they believed in a Legislator [Lawgiver].

— C. S. Lewis

Now when giving this mathematically-related discussion honest thought, it is actually quite interesting the way in which abstract mathematical concepts end up having significance in the real world, especially since mathematical abstractions so often lead us to muse over how and why mathematics plays any role in our lives. Or we may keenly ask what it is about science that promotes mathematics as its language. It is for these reasons, then, that we humbly consider the thoughts of Christian apologist C.S. Lewis (1947) from his book Miracles: A Preliminary Study, who said, “Men became scientific because they expected Law in Nature, and they expected Law in Nature because they believed in a Legislator [Lawgiver]” (p. 140). This quote points to the inherent order in the universe and in nature — as opposed to naturalistic or materialistic chaos.

And continuing this line of rationale we claim that a Christian worldview imparts order and meaning on mathematical abstractions and their bearing in the physical world (i.e., cause-and-effect and the laws of physics). For example, the Bible records two pertinent and related questions that God directed to Job: “Do you know the laws of heaven? Can you impose its authority on earth?” (Job 38:33). Here God is saying that He is the Creator, so He understands. Therefore, the more we learn about God, the more we get to know and learn about our universe — including the role of mathematics — and our place in it.

According to noted mathematician (and Christian apologist) John Lennox (2009), Professor Emeritus of Mathematics at the University of Oxford, the real world and mathematics are traceable to the mind of God, the Creator of the universe and our human minds (p. 61). Thus, it ought not to surprise us that mathematical theories can be spun and woven within the tapestry of our thoughts. After all, we are each made in the image of God (including our minds which reflect His image). Yet, God’s thoughts supersede our own immeasurably.

Order is the fingerprint of God upon all His works.

— Jim Berg

But perhaps Jim Berg (2008) sums it up best in Essential Virtues: Marks of the Christ-Centered Life, when he wrote, “It is the predictability of order within the creation that allows scientists to theorize, research, and experiment. It is this predictability that is the foundation of the creation’s stability” (p. 63).

In closing, the framework embedded in the transform that Johann Radon introduced, and intended purely as a mathematical operator, ended up to be instrumental for medical x-ray CT scanning. In fact, it is equally astounding and worth celebrating that the Radon transform found similar usefulness in other areas of medical imaging, as well as importance in the fields of astronomy and geophysics. Ultimately, in the Christian worldview, the well-ordered concurrence and explanation found in the universe is grounded in the ultimate rationality of God — in His stability and trustworthiness. As a result, we also discover that questions that exercise the heart the most, such as why we are here and what our purpose is, make the most sense when we have a personal relationship with the Creator, the God of the Bible.

References

Berg, J. (2008). Essential virtues: Marks of the Christ-centered life. Greenville, SC: JourneyForth Books – BJU Press.

Lewis, C.S. (1947). Miracles: A preliminary study. New York, NY: Macmillan.

Lennox, J. (2009). God’s undertaker: Has science buried God? Oxford, England: Lion Hudson.

Wininger, K.L. (2012). On the foundations of x-ray computed tomography in medicine: A fundamental review of the ‘Radon transform’ and a tribute to Johann Radon. History of Mathematics Special Interest Group of the Mathematical Association of America: Student Writing Contest Winners Archive. Retrieved from http://historyofmathematics.org/archive/.