Saturday, October 29, 2011

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]
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...

Wednesday, October 19, 2011

Problem Solving, the Wheel, and the Box - or when to divide by parallel fifths

I often wonder what it is they mean when they talk about "problem solving skills" - as if these were some sort of magical trickery prohibited to all students below a certain age. It's especially funny, because when this comes up, as it does all the time whenever someone is trying to justify spending even more money on schools or colleges, they usually couple it with the old line about how children ought not learn the rudiments of arithmetic such as Long Division, since they ought to be learning something called "technology" and - let us chant it together - and problem-solving skills. (No, to my surprise when they say "technology" they do not mean learning automata theory, which is fun as well as useful - in fact it is the basis of ALL video games - nor Boolean Algebra, which is loads simpler than the regular kind.)

What a shame. It's like saying one ought not practice one's instrument, because music is a matter of "feeling" the intent of the composer. Harold Hill would be proud: "Now think, men! Think the Minuet in G..."

It's true one ought to grasp the intent of the composer - but that door can only be opened if one has the right key... (probably with a single sharp, ahem!)

Now, I could give four or five examples of problems which are quite simple to solve once one knows the method called "Long Division". They are not directly solvable on a computer - in fact solving them with a computer requires that knowledge. It's very funny - by depriving students of the ability to do Long Division, these teachers actually work against themselves. Ah, but as long as they can double-click, the world will be saved. Yes, yes.

So this is a whine-post, Dr. Thursday?

No... it's just a curious introductory tune, to try to get you to think about the box. People love that epigram about "thinking outside the box" - but you have probably never noticed the box before... and you would be very startled if someone told you to think outside the cylinder! Just what DO you mean by a problem-solving skill, and why do you exclude the very elegant and simple algorithm for long division from that category? Or is it merely that you've not noticed the box?

Another way of putting it is this: At Goodyear or Firestone or Dunlap or Pirelli, no one is permitted to say "let's not re-invent the wheel!" (Not when their business is selling wheels, and their adornments.)

Or, to vary the analogy, there are times when even the great masters of music used parallel fifths or octaves... knowing that one has to have a thorough grounding in the basics if one wants to re-invent the wheel, or think about the box-in-itself.

Yeah, yeah... Gosh, Doc, you are going to get really deep-and-mystical today, we can tell.

I am. You, see, I would like you to do some "sitting and thinking" as good old H.M. liked to say. (That's "Sir Henry Merrivale", the bald-headed detective in the novels of Carter Dickson, the pen name of John Dickson Carr.)

Let's just take the alphabet... not as we take it in English, but as a computer scientist takes it: as a collection of characters or symbols. Just for fun, I would like you to think - not OUTSIDE the box, but about the box.

Now, you could play the game like the molecular biologists who have found that their four-letter DNA alphabet of A, C, G and T isn't enough to spell out the words they are reading from their DNA sequence analysis. They were forced to invent tricks so that they could have veritable "chords" of letters - yes, just like in music - they let M stand for "either A or C", and S stand for "either C or G"... there's two and three-letter symbols, and N for any of the four. (Speaking as a musician, which I must do with some trepidation, it makes me wonder if these people are string musicians... but that is a pun and I will be banned from both the music and computing spheres, alas!)

Yes... these are the famous wild-cards of DNA sequence analysis, which form a Boolean Algebra... here's the Hasse diagram for it:

But let us not do something which is so practical as to enter into matters like cancer research, which if you don't know Long Division, will be a hopeless frustration... It's not like there's something called the ribosome - I mean, they like their problems to be practical, don't they? So let us just ask ourselves another question, and try a slightly simpler model.

Whenever we think of a word - I mean we computer people - we think of a series of characters from our given alphabet. This series is in most ways just like a train - specifically a freight train (also called a "goods train" in England.) It has a starting member: the engine, followed by a series of other members, which sometimes might actually be other engines, and at the end a caboose. (We note that there could be a caboose elsewhere, and in fact there could be a caboose at the start, though I suspect that is quite rare if not actually forbidden.) My intention is not to guess at such matters, but to call your attention to the BOX - no not the boxcar! Hee hee.

I mean the very curious fact that each car of the train has exactly two couplers - one at the front, and one at the back. The train can be organized in all sorts of ways regarding the particular length and order of the cars - but one car is not placed next to another, or on top of another - or any other arrangement, be it classically measured in purely real numbers, or quantized according to Planck or any other unit. It is strictly assembled, coupler to coupler, one following another after the first, until we reach the end. There are no chords in the composition of a train, except in the case of a collision, which is (alas) no longer a train, but a wreck.

Now, if you make the stunning leap from the railroad to the print shop and take up a handful of type instead of begging access to a switch engine and riding around a yard (oh what fun that could be!) you will find something much the same. Or, if you have no old-fashioned print shops, maybe you have an old "Scrabble" game around the house. You can have all sorts of fun sticking those letters together and making curious things... but even when you play with a crossword puzzle or any of its relatives, you will find yourself recurring to the analogy of the train and joining the letters end to end.

SO the question is: what sort of a thing could it be if the letters were - uh - sort of like the chemical elements? And some had one hand, or two, or three, or four, or more, or none - or could bind with double or triple bonds...

Yeah, Doc (you yawn) you probably need some more sleep. Or maybe a cup of coffee. Or another beer or two.

Thanks, most generous - don't mind if I do. But maybe it's a matter of thinking about the box, and knowing a little about the mystery of Long Division - which is a key to unlock these and other mysteries.

Ah do you know why? You DON'T? What a shame. Glad you have YOUR problem-solving skills... but I have some useful tools - yes, I even know how to do Long Division - and I've got some work to do.

More on this another time.

P.S. I am well aware that certain ideogrammatic languages permit a certain sort of overlaying of simpler symbols to form more complex ones. I am not referring to these; I want to consider the idea of a "molecular alphabet" which is not just the usual linear two-coupler railroad car sort of thing. These things have some curious properties, you know, just as the wild-card alphabet - or - gosh - or even regular numbers. Division, you know, is not commutative - but that's another sort of forbidden problem-solving skill, and I apologize if it's not in your toolkit. It should be... it comes up in other realms besides mathematics.

Sunday, October 16, 2011

I had an interesting idea to tell you - and by the time the "posting window" finally opened it had evaporated. All I can tell you is that it had something to do with the "Five Regular Solids" - that is, the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron - which I have recently been preparing models of, for a curious young man whom I see every so often on his way to school. I don't know if you've ever tried to make them - it may be worth your expending some energy the next time you are looking for an interesting little crafts project. I am thinking of making another set to use for a future Christmas project... hm. Very interesting... as you know, I pay attention to words and numbers and their inter-relation, be it of the classical (psephy) or modern (ASCII) approach... But I have no time to explore that just now.

Rather, I was pondering the idea of translations and their variants, especially as this touches critical matters like the Five Regular Solids, or the Holy Sacrifice of the Mass. No, this is NOT a moan or a critical post - it is an observation about two translations I have noted of Biblical passages. Each has had a profound impact on me as I think and ponder my work or my writing - and I feel you ought to know about both. And no, I am not going to analyse them, or explain their other versions. I prefer you to consider them as they are.

The first is just a vrey brief phrase from the Divine Office:
Dedicate yourselves to thankfulness.
[cf Colossians 3:15]
The other is from the "Jerusalem" edition of the Bible:
...the elements fight for the virtuous.
[cf. Wisdom 16:17]
There was some chance alignment here - and somehow it was spurred on by the fact that just now the sun is shining on me as I sat at my computer - and then I recalled this sparkling gemstone from a wonderful book:
Considered as a collector of rare and precious things, the amateur astronomer has a great advantage over amateurs in all other fields, who must content themselves with second and third rate specimens. ... [he] has access at all times to the original objects of his study; the masterworks of the heavens belong to him as much as to the great observatories of the world.
[burnham, Burnham's Celestial Handbook, 5, emphasis added]
And then I recalled St. Francis and his Canticle of the Sun, and Chesterton's audacious literary analysis of the Gospels:
Even in the matter of mere literary style, if we suppose ourselves thus sufficiently detached to look at it in that light, there is a curious quality to which no critic seems to have done justice. It had among other things a singular air of piling tower upon tower by the use of the a fortiori; making a pagoda of degrees like the seven heavens.
[GKC The Everlasting Man CW2:332]
In fact, Burnham's thesis applies a fortiori to the sun. We humans have access at all times (that is, during the daylight hours!) to this most original object of study - that is, our local star, "Ole Sol" the Sun. Not for nothing does St. Francis seem to exalt it as a sort of Christian god... but of course you will note that he does not praise the sun, but God Who created the sun. And if we want to begin to grasp even in some faint fashion the amazing wonders of that creation, and the gift which is ours in our local star, we need to go and enjoy it, or at least read about it.

One little item - which ties into that mystical quote from Wisdom - is the curious fact that one of the chemical elements was disccovered on the sun before it was discovered on earth. Yes - the element we know as helium, which is just the Greek for "the sun-element", proclaimed its presence by its spectrum - certain lines of color (or of darkness) which could not be traced to any known element on earth! True, helium was later found to be present on earth; it can be obtained from certain natural gas deposits. But it is astounding to remember that the presence of helium was announced not in any test tube or chemical reaction apparatus, but from beams of light: it was (as Chesterton once pointed out in the case of the writer Alice Meynell:
She could always find things to think about; even on a sick bed in a darkened room, where the shadow of a bird on the blind was more than the bird itself, she said, because it was a message from the sun.
[GKC Autobiography CW16:269]

And with that observation, perhaps I have given you a little hint about something else... but for today, I will let you do your own sitting and thinking. Perhaps then you will also dedicate yourselves to thankfulness...

Tuesday, October 11, 2011

Just a quick note to call your attention to the fact that this year our Advent will consist of 28 days - that is, four full weeks:

So you ought to plan accordingly. Note also that 28 is a perfect number:

28 = (1*28) = (2*14) = (4*7)
and
28 = 1 + 2+14 + 4+7

as well as triangular:

28 = 1+2+3+4+5+6+7

which means there's all sorts of possibilities for fun, be it psephy or otherwise.

A note: "psephy" was a sort of game, sometimes used for attempting to foretell the future or gain insight into mystical writings. In several ancient languages (such as Greek), the symbols of the "alphabet" were also used as what we now call "digits". Thus, any given word could be converted into a numerical value, which (by some views) was thereby related to other such words. Its most famous instance is found in Apocalypse/Revelation 13:18 in the "number of the Beast" 666, which has been given a whole long list of meanings. Very few people seem to recall the number of Christ, which is in some ways more interesting - it is "alpha omega" (see the same book, 1:18) which in the old Greek notation comes out to what we now write as 801.

Wednesday, October 05, 2011

Something simple

Well, maybe not so simple. But it was something I needed, and of course, since I could not lay my hand on a convenient reference, I decided to go through the exercise of finding out for myself.

What?

Nothing more exotic than the "central angle" of a tetrahedron.

So... first, let us take a regular tetrahedron...

with its corners (vertices) labelled P,Q,R,S.

The length of all six edges is s. Each of its faces is an equilateral triangle with side s and angle 60°, or p/3 radians.

Here you see the face PQR with its altitudes shown in red:

Note that the altitudes bisect the angles and the sides, and are perpendicular.

Since this is equilateral, and the altitude RA is the side opposite to the 60° angle with the hypotenuse of length s, the length of RA is given by:

|RA| = s*sqrt(3)/2

We also have a 30-60-90 triangle PMA, where the side opposite to 60° is s/2.

Each of the segments PM, QM, RM are the radii of the circumscribing circle of the triangle. Their lengths are the same, and is:

|PM| = s*sqrt(3)/3

The lengths of the segments from the center M to each edge is:

|AM| = s*sqrt(3)/6

As a check we see that

|RM| + |MA| = s*sqrt(3)/3 + s*sqrt(3)/6

or, since RM and MA are colinear,

|RA| = s*sqrt(3)/2

Now, let us go back to our tetrahedron and put in those medians in red, and some other lines from the center C to each vertex in green:

We know that C is in a vertical line between M and S. Consider the triangle AMS, where A is the midpoint between P and Q. We know that it is a right triangle with its right angle at M; we also know (from above) the length of AS is s*sqrt(3)/2. We also know the length of AM, which is s*sqrt(3)/6, and so by the Pythagorean theorem, we can determine the altitude a of the tetrahedron, which is the length of SM:

|SM|2 + |AM|2 = |AS|2

Substituting, we get:

a2 + (s*sqrt(3)/6)2 = (s*sqrt(3)/2)2

This reduces to:

a = s*sqrt(6)/3

Next, we consider the triangle PCM. This has a right angle at M, and we know the length of PM to be s*sqrt(3)/3. We note that the length of PC is equal to that of SC - all these are the radii of the circumscribing sphere for the tetrahedron. We also note that the altitude SCM of the tetrahedron is colinear, so |SM| = |SC| + |CM|.

To simplfy things, let us call
r = |SC|

and
w = |CM|
, thus
r + w = a = s*sqrt(6)/3

Solving for w, we have:
w = a – r

Then, again by the Pythagorean theorem, we have:
r2 = w2 + |PM|2

Applying the value of |PM| and the above equation for w, we get:

r2 = (a – r)2 + (s*sqrt(3)/3)2

Now, it's just algebra to solve for r...

r2 = a22*r*a + r2 + (s2/3)

2*r*a = a2 + (s2/3)

but we also know that a = s*sqrt(6)/3, so we can reduce this to:

r = s*sqrt(6)/4

Recall that r is the radius of the circumscribing circle.

Now, consider the central triangle SCP, its two legs are radii and equal to r. The third leg is an edge of the tetrahedron which is given as s. Now, a line from C to the edge PS will bisect PS at a right angle. The central angle in SCP (at C) is twice the angle between that bisector and the radius PC, which means the desired angle is twice that which has a sine given by (s/2)/r. That is, the central angle q is given:

sin(q/2) = (s/2)/(s*sqrt(6)/4)

or
sin(q/2) = sqrt(6)/3)

As expected, the s term vanishes, since the central angle is independent of the size of the tetrahedron.

To complete the exercise, the angle q is approximately 109.471 degrees.

Wow. That was fun. Now, it's time to go and use that fact.

Maybe another day I'll tell you what it's for.

Oh yes. I expect that someone, somewhere is wondering whether this post has anything to do with yesterday's. As strange as it must sound, it does.

Tuesday, October 04, 2011

Searching in Reverse: the Rosary and Science

No, this is not a posting about my doctoral dissertation, which really did accomplish what the TV version of "The Cat In The Hat" called Calculatus Eliminatus, or finding where something is by finding out where it isn't. (Yes, I really did that, and it was useful for biologists, and not just yet another goofy academic exercise - but more on that topic some other time.)

Rather, it's about how I found out someone is hunting for something by seeing that someone used one of the so-called "search engines" to come here, looking for something called "Scientists and the Rosary".

Possibly they were hunting for the famous vignette told about Louis Pasteur, quoted in Key To Happiness Oct 1986; see here. I have not yet encountered the original source of this, so for the moment it is just another story - but it sounds like something that did happen.

Or perhaps they were curious about the great Andre Ampère, who was also known to say the rosary (a vignette about him was told in Key To Happiness for Sept/Oct 1990), as was Allesandro Volta - his dedication to the rosary is documented in Kneller's important work, Christianity and the Leaders of Modern Science, available from Real View Books.

A very interesting point ought to be made here: the names of these two great Catholic scientists appear on uncountably many things all over the world. A certain dead musician (I use the term very loosely) once claimed to be more famous than the One Whose birth has fixed the naming of the very year - but that dead musician's name is not more ubiquitous than Volts and Amps. But let us proceed.

Now... it is remotely possible that the ones who were searching for "Rosary and Scientist" were not so much seeking such vignettes, be they legends or accurate descriptions. Perhaps they were seeking some sort of commentary... that is, What Does a Scientist Think Of When He Says the Rosary? or, perhaps, What Sort Of New Things Does a Scientist See When he Brings His Sort of Training To Assist In His Praying, Specifically the Rosary?

Oh, I wish I had the time to write these texts... though I expect they would be large. And perhaps you will wish I had saved this post for Friday... or perhaps had planned out an actual series of lectures - er, I mean postings - for this blogg about such topics!

Well... I can't promise anything. Like most of us, I get very busy, and there is hardly time to turn a quick AMBER concordance of GKC's use of the word "dragon" into a post (like I did last week). But since I think of myself as a scientist, and I do say the rosary, I ought to at least make an attempt. I know I did something back in 2005 called "Light From the Rosary" - but that only goes so far, and six years have passed, with a steady accumulation of new details and data and insights. There was also an Advent series on the analogy between the 20 mysteries and the 20 amino acids, which was done in 2006; see my index for the specific entries, it also gives the links to "Light From the Rosary".

It's a rich field for study - far richer since the addition of the Five Luminous Mysteries in 2002. I think I may have mentioned in one of those collections that the Luminous Mysteries might just as well be called "The Aquatic Mysteries" or "The Mysteries of Water" because of the critical role water plays in each one. But more importantly, there is a lot to see just in the very idea of the rosary as "the Hand-held Gospel" - it is a truly scientific thing, a way of developing one's mental focus to consider a topic or series of topics, slowly and solemnly, but without the dangers of trying too hard, or getting lost in the details or all the other risks encountered in less well-developed methods. Please note: I do not suggest that there be "mysteries of Science" though if the atheists were serious about their faith they would have such things, just as Comte proposed things like "Darwin Day":
A feeling touching the nature of things does not only make men feel that there are certain proper things to say; it makes them feel that there are certain proper things to do. The more agreeable of these consist of dancing, building temples, and shouting very loud; the less agreeable, of wearing green carnations and burning other philosophers alive. But everywhere the religious dance came before the religious hymn, and man was a ritualist before he could speak. If Comtism had spread the world would have been converted, not by the Comtist philosophy, but by the Comtist calendar. By discouraging what they conceive to be the weakness of their master, the English Positivists have broken the strength of their religion. A man who has faith must be prepared not only to be a martyr, but to be a fool. It is absurd to say that a man is ready to toil and die for his convictions when he is not even ready to wear a wreath round his head for them. I myself, to take a corpus vile, am very certain that I would not read the works of Comte through for any consideration whatever. But I can easily imagine myself with the greatest enthusiasm lighting a bonfire on Darwin Day.
[GKC Heretics CW1:87]

Comte had a complete new religion, or rather, a new Church; for it was modelled throughout on the Catholic Church. It had a liturgy. It had a calendar. I believe it had vestments. I am sure it had saints' days dedicated to Darwin or Newton. I do not know in what the ceremonial consisted, or what were the vestments worn. Perhaps they all wore tails on Darwin Day. Perhaps they celebrated Sir Isaac Newton by dancing round an apple-tree and pelting each other with apples. I do not know exactly what was done in Comte's cathedral, indeed, I do not know whether anybody ever went to Comte's cathedral, even Comte. But certainly Comte founded, whether or no Harrison followed, the strict system of a regular religion externally very like the Roman religion, except that it was to worship Humanity instead of God.
[GKC ILN Jan 27 1923 CW33:30-31]
Again, I am not suggesting anything of that sort. I do not mean to apply "the scientific method" (about which I am also trying to write over on the blogg of The Duhem Society) to the rosary, or anything like that. I am simply thinking of several interesting ideas I have noticed as I say the rosary, considering this very simple (but powerful and profound) technique for exploring the Gospels - that is, the Life of Christ - yes, as a scientist.

We forget that a Scientist (writ large, as Father Jaki so often wrote) is not rigidly a "way of thinking"... it is something much simpler. It is a WAY OF SEEING. It means somehow factoring one's own self out of the viewpoint, and at the same time, seeing as much as one can see: especially those things that others don't see. And then, of course, telling others about it. A scientist is not a philosopher, whose gravest risk is turning into a navel-gazer. Nor is he a literary man, whose writing might be pulled up and down and all around, like a buoy on the tide, just to satisfy the current market. Being a scientist means being humble... as I said, having the will to remove one's own self, one's own bias, in order to get the view of Reality. Now, it is a wonderful paradox that the best sort of philosophy, the best sort of literature, the best sort of every Art (writ large) must also do that very same thing - but then that is why all those disciplines can rightly be called sciences as well, since they all seek Truth.

In this case, then, to apply one's scientific powers to the Rosary (or the Gospels, as the two are interconvertible)... excuse me... I think I mean to say "to the recitation of the rosary " - which is of course interconvertible with "to the reading of the Gospels". The idea of having a rosary at all is very simple, and is even more clear in this age of hand-held devices. We might not be able to carry our labs around with us. But most likely we keep a fairly good internal vision of our current work. In the same way, we might not own a copy of the Gospels to carry around with us - though again (let us hope) we keep a fairly good representation of the major events of that Real Story. The rosary is a discipined way of perusing that mental image... it is simultaneously Art and Science - the vivid imagination and the accurate memory and the skill of reasoning power all conjoined, and applied to the most fascinating thing that Is.

The funny thing, as you will see if I do get to write more, is that the more science one knows, the more one sees about the Gospels.

And vice versa - which is in keeping with those mystical words of our Lord:
Seek ye therefore first the kingdom of God, and his justice, and all these things shall be added unto you. [Mt 6:33]
Hey scientists - don't you think it's about time we start some sort of new exploration? You think it's all done? Ha. You ain't seen nothing yet. Open your eyes, time for a surprise... You don't have to wear a lab coat to pray, but you might want to get a rosary to keep in your pocket, and use while you're waiting for that experiment to complete. Other great scientists carried one... who knows what may result!

PS: I am well aware that there are Christians who will be delighted in anything which leads to a greater knowledge of the Life of Christ, and yet may have some issues about the rosary as such. It is very likely that I will address some of those concerns... but my purpose is to UNITE, not divide, even if certain sorts of divisions (in the classical sense of distinguo = "I distinguish") are required as we proceed. But if my plan goes well, it may be helpful to understand just what it is a serious Catholic does when he does such a curious thing, and actually THINKS about what he is doing, and why. May God guard and guide us, and bring all who love Christ into the unity He prayed for. {Jn 17:20-24]