Factorials: A Christmas Postscript

I had just posted my previous blog about Kevin's fascination with factorials when Kevin's wife asked about the origins of the Christmas Carol, "The Twelve Days of Christmas", on Facebook. That was quite serendipitous...

Verse 12
On the twelfth day of Christmas,
My true love gave to me:
Twelve Drummers drumming,

Eleven pipers piping,

Ten lords-a-leaping,
Nine ladies dancing, Eight maids-a-milking,
Seven swans-a-swimming,
Six geese-a-laying,
FIVE GOLDEN RINGS!
Four calling birds,
Three french hens,
Two turtle doves
And a partridge in a pear tree.

Whether the carol was coded with Gnostic or Roman Catholic allusions didn't concern me that much. Instead the coincidence reminded me of the magic of Pascal's Triangle. (See http://dimacs.rutgers.edu/~judyann/LP/lessons/12.days.pascal.html.)

Pascal's Triangle

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 330 165 55 11 1

1 12 66 220 495 792 924 792 495 220 66 12 1

.

.

Pascal's Triangle is used for determining Probabilities, Combinations, and even Fibonacci Numbers. With the aid of Pascal's Triangle one can also: 

• find the number of new gifts given each day of "The 12 Days of Christmas.".
• the number gifts given on each consecutive day.
• the total number of gifts given by a particular day.

Embedded within the triangle of arithmetic data is useful information. Uncovering this detail evokes a sense of wonder, the same wonder scientists, poets, and exegetes experience when they "see" something more in the ordinary. Wonder is a moment of disclosure or revelation. We might call this process of discovery "mathematical deduction", "data processing", "the scientific method", "hermeneutics", or "exegesis" depending on the discipline we come from but the sense of satisfaction is the same. Contemplating Pascal's Triangle yields unexpected moments of surprise much as the thought that there's an oak within an acorn might.

With Twelve Drummers Drumming in my ears and Pascal's Triangle in my mind's eye, my Christmas thoughts turned to two fascinating Advent theological insights:

 The Logos

"And the Word (logos) was made flesh, and dwelt among us and we beheld His glory." (John 1:14) The Neo-Platonists and especially Philo used the word logos in a dual sense. For Plato and Philo the logos was the collective term for the Ideal World. The Divine Word contains within itself the archetypal forms of all things from which everything else is manifest, "made flesh", much as information leaps out at one from Pascal's Triangle. Philo spoke of the logos in a twofold sense (http://www.duke.edu/web/classics/grbs/FTexts/44/Kamesar.pdf):

i.  Logos endiathetos. This was the Transcendent Word, the Uncreated Template or Archetype.

ii. Logos prophorikos, the Manifest Word or created type.

"Some theologians distinguished between the logos endiathetos, or the Word latent in the Godhead from all eternity, and the logos prophorikos, uttered and becoming effective at the creation." (http://mb-soft.com/believe/text/logos.htm) In a symbolic way Pascal's Triangle (Logos endiathetos) embodies patterns that come to be manifest (Logos prophorikos), incarnated, or projected, and applied in the created world.

(Compare "Archetypes" and "Projection" in Jungian Psychology. For a helpful article on Philo and the distinction he made see http://www2.nd.edu/Departments/Maritain/etext/hhp49.htm. Also Wikipedia http://en.wikipedia.org/wiki/Logos. And Krishna in the Bhagavad Gita 9.)

Recapitulation

"Recapitulation" is the doctrine originally expressed by Irenaeus that all things are "summed up" in Jesus Christ. John Paul II explained: "God's saving plan, "the mystery of his will" (cf. Eph 1: 9) for every creature, is described in the Letter to the Ephesians with a distinctive term: to "recapitulate" all things in heaven and on earth in Christ (Eph 1: 10)...Irenaeus extols the one Lord, Jesus Christ, who in the Incarnation sums up in himself the entire history of salvation, humanity and all creation: "He, as the eternal King, recapitulates all things in himself" (Adversus Haereses, III, 21, 9)."  Theologically Christ is the Alpha and Omega of all Cosmic space and time. Another theologian, Julie James, wrote: "Irenaeus describes Humanity and God as coming together in Christ, an eventual restoration of the separated Human into the original divine form. Since the dawn of mans' entrance into the world, God has had a dialogue with humanity through the prophets, through his Divine Word, through rules and guidance, drawing mankind ultimately to the final goal of divinity, the Omega point." Irenaeus anticipates the Christogenesis of Teilhard de Chardin. Christ is the summation of all there is.....by analogy, the Divine Logarithm, the Beginning and the End. He encapsulates and recapitulates all. He is the "capstone".

oOo

Long ago, Magi from the East looked up into the night sky and were amazed by an astonishing configuration of the planets and stars. Suddenly it was pregnant with meaning just as Pascal's Triangle came to be for me. They discerned patterns hitherto never recognised before. Suddenly the ordinary took on a radical new import, wonder, and significance for them. The Divine was swaddled in a manger as revelations are swaddled in Pascal's Triangle. Immanuel! It could be a "partridge in a pear tree". It is the smile on an infant's face...

Still the night, holy the night!

Son of God, O how bright

Love is smiling from Thy face!

Strikes for us now the hour of grace!

©Colin G Garvie Home Page: http://www.garvies.co.za/