Easter, Heighway's Dragon, and Other Fancy Stuff

Dear Friends and Other Readers of This Blog:

A hearty hello to you all, on this feast which I do not celebrate, and on which it is impossible to have a very good time this year, especially since this spring-related festival is largely celebrated outdoors (and with friends and family).

Easter
One reason I don't celebrate Easter is because of its---admittedly tenuous---connection with older religions and human sacrifice.  Of course Christians tend to say that, well, their version of human sacrifice was really the end of the practice of human sacrifice.  In any case, they go on to say, the crucifixion, when it took place, was not seen as sacrifice, exactly; it was a political execution, which was subsequently exalted into a symbolic---whatchumacallit.  But whatever they say, there are hidden references to sacrifice all over the language surrounding Easter.  OK, let's leave that alone.

While I was thinking of Easter, I was driven to puzzle over why people who join the Christian clergy do so.  In my youth, I observed that many religious leaders were intelligent, persuasive people.  Today, some of the clergy are certainly quite intelligent and perceptive people, but I seem to notice that a great many plodders are finding their way into the leadership of organized religion.  Why does anyone become a priest or a minister?  Christians would claim that an atheist such as me could never understand this mystery.  But I have some guesses.

One reason is that some of them are impelled to be shepherds to their community.  They see their friends, and people in their group, as leaderless, and needing comfort and guidance, and they train to be clergy to provide that guidance and leadership.

Another reason is that they have 'religious experiences', and feel the need to share these with people, especially people who seem to need such a description to validate their own experiences, or to give some sort of external meaning to their lives.

Another reason is sheer ambition, which often joins up with delusions of grandeur.  It is mostly those affected by these motives that set up mega-churches, together with the feeling that their religion, whatever it is based on, needs to have a mass movement behind it to give it any power.  If it was only providing guidance that was your motive, a mega-church would not have been necessary.  Your flock need not be enormous for you to shepherd it (unless you really had an eye to someday creating a meat-packing plant).

I have no thoughts about religion generally to share with you today, except that the number of people who identify as atheists keeps growing, and the number of ordinary intelligent people who become clergy is clearly dwindling, except those who do so cynically, in their quest for personal power.  In turn, I suppose, this suggests that the people who flock to mega-churches must be, on the one hand, blind to the character-failings of their religious leaders, and on the other, have a deep need which these churches satisfy.  I feel that this indicates flaws in their own psychological make-up, and when they get together as a group, this creates a dangerous aggregation of personalities.  But what can you do?

Calendric Matters
The method, or methods, by which Easter and Passover are established each year are interesting.  (Very interesting, for those of you who need italics to be persuaded about anything.  You know who you are, and are not.)  (Amaaazing, for any followers of Trump who may have wandered in here.  Welcome, but please leave us alone in the future.)

Easter.  Well, the Wikipedia article on the calculation of the date of Easter is very learned, by which I mean: more tending to confuse than to illuminate.  (That may not be intentional, but they achieve this unfortunate end by providing a lot of history and extraneous facts, within which they hide the actual process.  There are also many technical terms, some of them in ancient languages.)  But, once I peeled away the history, this is what I gathered.  (If this is wrong, I apologize; the article was not intended for non-specialists.)

Easter is the first Sunday after the first Full Moon after the Spring Equinox.

Note 1:  The Spring Equinox is not that actual Equinox, necessarily; it is the Ecclesiastical Equinox, which means either that it is the Church-Approved Equinox, or the Equinox as calculated by traditional Church Methods.  I don't understand this; now that we have competent astronomers who can tell us when the Equinox takes place, why should we leave it to church officials whose expertise does not include calculation of an Equinox?  Actually, any idiot could do it; just hop over to Stonehenge, and do it with the stones.

Note 2:  Sometimes, at least in the past, the date was delayed by a week to avoid a certain sort of collision with the Passover.  OK, I understand this.  It has always been an uneasy truce between Christianity and Judaism, and this exception may have to do with historical matters, or matters of observation.  It could be at least the fault of just the Jewish authorities.

Note 3:  Carl Friedrich Gauss provided a rule to calculate Easter that substituted a formula to obviate the need for having to observe the full moon immediately after the Equinox.  (This is known as The Paschal Full Moon.  The term Paschal means many things, among them Easter-related.)  All of these rules and formulas were in aid of determining the Paschal Full Moon (Ecclesiastical variety) without exposing oneself to the elements (which would be unscientific).

Passover.  The Hebrew (Jewish) calendar is a Lunar-solar calendar; that is to say, it calculates by lunar months for months, but by the solar calendar for years.  This means that (just as our modern calendar has to add an extra day in February for Leap Years) the Hebrew Calendar has to add not only extra (leap-) days to certain months, but extra (leap-) months to certain years.  There are more than 12 lunar months in a year, so every once in a while, they have to add a whole extra month.  My brain is somewhat overburdened with all this information, so I do not know whether the Hebrew Year is exactly as long as a modern astronomical year (and the extra days are filled in with a padding month), or whether they just let their year be shorter (unjustifiably considering that my readers are all non-Jewish; if some of you are Jewish, Welcome!  My pride knows no bounds!!) and every once in a while, use a leap month to align the years.  If they line their years with ours every year, well, they will have an extra short month every year.

Anyhow, the Passover is a week-long observation.  It begins on the 15th of the month of Nisan, as far as I can tell, and lasts for a week, though extra-religious Jews will observe for eight days, though I don't know whether they put the extra day at the beginning of the period or at the end.  I am also not clear whether there is a day called The Passover, and in which case, which day it is; the first day, which I presume is The Sabbath, or the last, which is either the day before the Sabbath, or the Sabbath.  (Though I am interested for strictly cultural but non-religious reasons, these matters are of interest in the relationships between the observations of the two religions.  Numerous Christians, I have learned, partake in the observation of Passover with their Jewish friends.)

Heighway's Dragon

And now for something completely different.  When I was a mathematics teacher (the word professor has unfortunate connotations, and in any case at our school you were not considered a professor unless you were a full professor, which was a rank to which I did not aspire) I became interested in fractals, a topic that gained a great deal of interest among the public for a decade or two, and has now been thrown on the garbage pile of mere curiosities.

However, I happened to mention to my wife and her sister that I was vaguely interested in creating a certain fractal, called Heighway's Dragon---which is considered of even less importance than most fractals---out of embroidery, that is, using, er, I forget the term needlepoint.  To my horror, nothing would do for my wife but that I was hustled to a craft store, at which I had to buy a huge embroidery frame (I don't know what the proper word is; they're a sort of pair of concentric hoops that hold your base fabric like a sort of vise), and special embroidery base fabric, on which you sew your design, like a tapestry.

One of the chief properties of Heighway's Dragon is that it is a space-filling curve.  Many space-filling curves exist; they are basically designs that consist of an entire sequence of curves, where the first is a simple line, and each successive curve is more complex and more dense, until they appear to be a solid block.

This is one of the most famous, called the Hilbert Space-Filling Curve.  The actual Hilbert S-F-C can be considered to be the entire sequence of images, or its limit, which is, contrary to what a non-mathematician would think, not just a solid block.  It certainly would appear to be a solid block, but of course things are often not what they appear to be.

Heighway's Dragon (or the Heighway-Harter Dragon) is illustrated by an old post on YouTube by me:

Heighway Dragon

Here's another representation of this dragon.

As you can see, from the YouTube video, each successive dragon is obtained by (A) dividing each straight-line 'cell' in the middle, and bending it into a 90-degree angle.  This doubles the number of 'cells' into 2, 4, 8, and so on.

The animated GIF shows that the dragon can also be obtained by joining two dragons, head-to-head, at 90 degrees.  You get the same kinds of dragons in both ways, except for them being rotated.

Heighway's Dragon has nothing to do with Easter (nor has Easter anything to do with the Dragon).  They're both things I happen to be interested in today.

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