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.