Saturday, October 29, 2011

Some Things about Five

Five is a very curious number, and not just because it is half of ten, or the number of digits on one human hand. It gets bad press in some circles since it seems (by certain views) to be unnatural. For example, by the Laws of Symmetry, when we are packing balls of various sizes, which is what God does when a crystal is formed, there are only 32 possible arrangements for them:
The restrictions placed on space lattices impose addtional restrictions on the motifs that populate the lattice points. The motifs themselves must outline units that fill all space. For example, 5-fold arrangements cannot exist in crystals for they will not pack together without leaving spaces between. It has been shown that there are a limited number of ways of rrangeing objects, such as atoms of a motif, about a point (in our example, a lattice point). Only 32 such ways exist; these are known as the 32 point groups.
[Hurlbut, Dana's Manual of Mineralogy, 5-7, emphasis added]
Ah, well. Of course we know our hands are not symmetrical - that's why we have to buy two gloves, and these are different, not only from each other, but within themselves - there is a "thumb" end and a "pinkie" end. You may say that our hands do not crystallize, and nod knowingly - that is true. The same goes for starfish, or for flowers with five points, or with other living things that have five-ish-ness about themselves.

Now, when it comes to symmetry in things that are NOT crystals, anything goes, and we have some wonderful things like the dodecahedron with its pentagon-shaped faces, or the icosahedron which seems to be a dodecahedon wearing a pentagonal pyramid disguise. (Hee hee) But without going into three-space, we have the pentagon, and its interior diagonals, sometimes called a pentagram. Here's both at once:

This sort of thing was used as a heraldic device - that is, something on a coat-of-arms, linke this:This symbol is known as the Five-Knot, and no less than J. R. R. Tolkien wrote about it in his fascinating hyper-annotated Sir Gawain and the Green Knight - in those days it was considered a holy emblem, suggesting the Five Wounds of our Lord, or other five-shaped ideas from the Bible. It's hilarious to think that some people now consider this a demonic symbol, but then others have gotten confused about Christ, or so it is reported - remember, "Your Master casts out demons by the prince of demons" [Mt 9:34] and so on. Here's what Tolkien had to say:
The pentacle was an ancient symbol of perfection which was used by the Pythagoreans, the Neo-Platonists, and the Gnostics. ... Throughout the Middle Ages it was a mystic symbol, and was popularly thought to have power to repel spirits. It was also called the endless knot, "because its interlacing lines are joined so as to be continuous, and if followed out they bring the tracer back always to the same point..." "In five ways, and five times in each way ... the five wits, the five fingers, the five wounds [of Christ], the five joys [of Mary] and five virtues."
[Sir Gawain and the Green Knight ed. by J.R.R. Tolkien and E.V. Gordon, 91-2]
Usually the Five Joys of Mary were the Annunciation, the Nativity, the Resurrection, the Ascension, and the Assumption. And, just for completeness, here's some of the actual stuff from the Gawain story:
[The pentacle] was the sign Solomon set ere-while, as betokening truth, for it is a figure with five points and each line overlaps the other, and nowhere hath it beginning or end, so that in English it is called "the endless knot." And therefore was it well suiting to this knight and to his arms, since Gawain was faithful in five and five-fold, for pure was he as gold, void of all villainy and endowed with all virtues. Therefore he bare the pentacle on shield and surcoat as truest of heroes and gentlest of knights.

For first he was faultless in his five senses; and his five fingers never failed him; and all his trust upon earth was in the five wounds that Christ bare on the cross, as the Creed tells. And wherever this knight found himself in stress of battle he deemed well that he drew his strength from the five joys which the Queen of Heaven had of her Child. And for this cause did he bear an image of Our Lady on the one half of his shield, that whenever he looked upon it he might not lack for aid. And the fifth five that the hero used were frankness and fellowship above all, purity and courtesy that never failed him, and compassion that surpasses all, and in these five virtues was the hero wrapped and clothed. And all these, five fold, were linked one in the other, so that they had no end, and were fixed on five points that never failed, neither at any side were they joined or sundered, nor could ye find beginning or end. And therefore on his shield was the knot shapen, red-gold upon red, which is the pure pentacle.
[Sir Gawain and the Green Knight tr. Jessie L. Weston, 13-4]
There are all sorts of other things to be said about five, as I hinted about in my sadly truncated series of comments in Advent of 2008 about water: the typical molecule may be linked by hydrogen bonds to four others, thus water is really groups of five, like this:Someday I hope to post some more. I had wanted to post something as a study of what I call "The Five Verbs of the Gloria: Laudamus, Benedicimus, Adoramus, Glorificamus, Gratias-agimus... these are the five we-verbs which we chant in union with the Angelic Choir, and about which at least a fat book ought to be written. Oh, my - yet another project, yes, but at least with this posting I've managed to say something about five. Sure, even without knowing a lot about math, one could make a case about every number - that is what one expects as a Chestertonian, or even as a believer in God - it is predicted in the famous Apocalypse/Revelation that "all things in creation" [see Apo/Rev 5:13] will glorify God, which does not exclude mathematics... And five is linked to so many things in strange ways - this is helpful, not as a superstition, but as a reminder and a remembrance:
Per sua sancta vulnera gloriosa custodiat et conservet nos Christus Dominus. Amen.

By His holy glorious wounds may the Lord Christ guard and protect us. Amen.
That is from the ritual of the Great Vigil - and if you are wondering where the "five" is, all you have to know is that the priest says this prayer while he inserts five grains of incense into the Paschal Candle...


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