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There's a very famous geometric pattern inscribed on the Temple of Osiris at Abydos, called the "Seed of Life," which looks like this:

Nobody really knows the formal significance of the shape, but I noticed something interesting about it while I was pondering the mysteries this evening.

Pythagoras was famously the first Greek to formally be initiated into the mysteries of the Osiris cult (though, of course, there must have been prior transmission since the Demeter and Dionysus mysteries are related). A generation later, Empedocles was initiated into the Pythagorean brotherhood, but later expelled for revealing the mysteries in writing. I conjecture that Empedocles' poem was derived from the Osiris cult, since it concerns the same phenomenon (the descent and reascent of the soul) and features the four gods:

τέσσαρα γὰρ πάντων ῥιζώματα πρῶτον ἄκουε·
Ζεὺς ἀργὴς Ἥρη τε φερέσβιος ἠδ' Ἀιδωνεύς,
Νῆστις θ' ἣ δακρύοις τέγγει κρούνωμα βρότειον.

First, hear of the four roots of all things:
shining Zeus and life-giving Hera and Aidoneus
and Nestis, who wets the springs of mortals with her tears.

It seems pretty reasonable to equate Osiris with Zeus, Hera with Isis, Set with Hades, and Nephthys with Nestis. Now Empedocles talks about how the roots begin united in Love, but peel off one at a time as Strife begins to intervene: first fire, then air, then water, then earth; this is the same as the first part of the Isis myth, where Osiris (fire) is killed, sealed in Set's box (air), dumped in the Nile (water), and encapsulated in a heather stalk (earth). We have a geometric symbol for the same thing: Pythagoras's tetractys, showing the progression of unity (1) into completion (10). It fits very nicely onto the Seed of Life:

Now, the second part of the myth has Osiris chopped into fourteen pieces, but his penis gets eaten by a fish and is never found, so Isis has to make do with the thirteen remaining pieces. Guess how many intersection points the Seed of Life has?

Finally, the last part of the myth has Horus (in place of Osiris) defeating Set and becoming king. This is a myth about the re-ascension of the soul back to its source: the three battles between Horus and Set are the rise from earth to water, water to air, and air to fire. (Diogenes Laertius tells us that Empedocles's Hera is earth, which makes sense to me in a roundabout way since Hera is Isis is Demeter is earth. Notice how, after the first battle, Horus deposes Isis by taking her crown, indicating the soul rising above earth.) Empedocles talks about that, too, since as Strife gives way to Love, the elements re-collapse into themselves in reverse of the way they separated. We might suppose that Pythagoras would have symbolized the regression of the cosmos from completion (10) back into unity (1) with a reverse tetractys, which, too, fits nicely onto the Seed of Life:

So if Pythagoras and Empedocles are (as I conjecture) faithful interpreters of the Isis, Osiris, and Horus mysteries (or if they aren't but my crazed speculation is at least somewhat valid anyway), then the Seed of Life is a nice little mnemonic for the exploration and contemplation of them. Hopefully that's helpful, since I continue to have a lot of contemplation ahead of me...

I've posted about Napoleon's problem before: can you inscribe a square in a given circle using no more than six circles total (e.g. including the given one)? It's tricky, but in fact you can, and there's a bunch of ways to do it. If your given circle has a radius of 1, the resulting square has a side length of √2.

Later on, I found a way of circumscribing a square around the given circle, still using six circles. This square has a side length of 2.

Later still, I found a way of making this strange one, again using six circles. It's got a side length of √6.

Now, I've played around quite a bit and figured that those were the only three sizes to find—I hadn't tried to prove it or anything, just hadn't seen anything else come up for a while. But I was playing around some more today, trying to see if there's a way to make an octagon out eight circles (my previous best was nine), and I was very surprised that there's another six-circle square out there!

Bonkers! That square has a side length of 2√2. Finding it inspired me to write a quick computer program to look for every possible size of square that can be made in only six circles, but unfortunately there are no others. I've caught 'em all.

Since I had the program put together, I went ahead and searched to see if I could make any eight-circle octagons from any of those squares, but nope, it's not possible. (There may yet be eight-circle octagons that don't start from a square, but this strikes me as pretty unlikely.) Regardless, it's amazing to me that there was more to find over ground I've gone over so many times already! It seems to be good practice to periodically double-check things one thinks one understands.

A refinement of my butterfly using 8 circles.

Several eternities ago, when I was in Sunday School—I must have been nine or ten—the church elder instructing us mentioned that the doctrine of the Trinity was not to be found in the Bible. I raised my hand and, when called upon, asked him, "By what authority do you teach something that doesn't come from the Bible? Also, if the Trinity doesn't come from the Bible, where does it come from?"

Of course, he told me to shut up.

Since then, I've always wondered where the doctrine came from, since it always seemed bizarre to me. I still don't have the "where," but at least Plotinus is good enough to tell us "why" with his usual logical rigor.

V 6: That the Principle Transcending Being has no Intellectual Act. What Being has Intellection Primarily and What Being has It Secondarily.

1. There is an Intellect.

2. Intellection implies duality (of subject and object).

3. Unity precedes multiplicity.

4. Therefore, there is something unitary prior to Intellect, something primarily Intellective (e.g. the subjective Intellect), and something secondarily Intellective (e.g. the objective Intellect; e.g. the Soul).

5. Being prior to Intellect, the First is not intellective.

6. If we consider the First to be Good, the Second is only Good to the degree it is intellective of the First, and the Third is only Good to the degree it is intellective of the Second, and so on.

Point 6, above, is of course why Plotinus considers matter to be evil: something can only be good in participating with that which is above it. Don't look down!

Plotinus gives us a very elegant analogy in §4: the One is light, the Intellect is the sun (something giving off light), and Soul is the moon (something reflecting light).

Incidentally, most modern commentators describe Plotinus as advocating a trinitarian view; I don't believe this is so! While he only proves the top three here, both he and Porphyry frequently refer to the four highest beings: the One, the Intellect, Soul, and Nature. (Unless I am much mistaken, Plotinus likens these to the Hesiodic Ouranos, Kronos, Aphrodite Ourania, and Eros, respectively.) What's more, it's hard not to see the Pythagorean one-two-three-four and the Empedoclean fire-air-water-earth in these. (One might think this sequence can continue indefinitely, but Plotinus explicitly says it does not: beyond this, the creative power is spent and too weak to continue further; all that remains is to play with every possible combination of principles within these.)

While playing around with these notions, I found a neat geometric correspondence. If one wishes to produce a tetractys with circles alone, it takes six circles, to wit:

Note how we are given two points to begin the construction with. We might as well assign these to the dyadic Intellect, right?—since where else would we assign them? But this is kinda like how Plotinus proves the whole structure also beginning with the Intellect.

You guys tired of geometry yet? No? Great! Neither am I! I found another neat thing I'd like to show you.

Back in 2002, Kurt Hofstetter showed how fundamental the golden ratio is by demonstrating a very simple construction using five circles:

I discovered today that he considerably undersells that construction. Not only does it contain the golden ratio, but it also contains the other two fundamental ratios of sacred geometry, √2 and √3. Consider this pentacle-like diagram:

In this diagram, notice the the dark circle in the bottom-center. If it has a radius of length 1, then the blue lines are of length √3, the green lines are of length √2, and the red line is of length ϕ. Thus it is almost trivial to construct any of the classical regular polygons—the triangle from √3, the square from √2, the pentagon from ϕ, or their multiples—within the dark circle. Indeed, the triangle is already present; the square and hexagon take a single additional circle each, while the pentagon requires two additional circles:

I haven't had much time today, but I'd like to explore this construction further...

Here is a construction of the most common modern taijitu (Yin and Yang) figure, using twelve circles. It is not very elegant—I have the feeling it could be constructed using fewer—but it's the best I've managed so far.

The interesting lesson here, if there is any, is that it's much simpler to double (or triple, etc.) a length than it is to half (or third, etc.) it.

A very common traditional construction was, given a circle and a point on it, inscribe a regular polygon within it with a vertex at that point. I've completed my collection of non-exotic inscribed polygons using (I believe) as few circles as possible, and figured I'd share them in case anyone is interested.

(Regular decagons (10 sides) and pentadecagons (15 sides) are also commonly encountered, due to their appearance in Euclid, but I've omitted them for being of less interest with compass-only constructions.) ( Read more... )

The construction of a regular heptagon was unknown to the Greeks, and was only finally managed by Renaissance geometers.

Given points A and B, draw

1. circle AB,
2. circle BA intersecting circle AB at points C and D,
3. circle CD intersecting circle AB at point E≠D and circle BA at point F≠D,
4. circle EB intersecting circle BA at point G,
5. circle FG intersecting circle AB at point H;

then BH is the side of a regular heptagon, and may be copied around the edge of circle AB to form the other sides.

...just kidding! The regular heptagon is impossible to draw with a straightedge and compasses (or indeed with compasses alone), and this fact was known at least as early as Kepler. In fact, the regular heptagon wasn't even unknown to the ancients: Archimedes managed to construct one with only slightly more sophisticated tools. The one I've constructed here is just an approximation, though a very good one, and is related to a construction by Albrect Dürer.

It always seemed odd to me that one can trisect a line but not an angle. It's one of those things that makes sense arithmetically—trisecting an angle requires a cubic equation, and circles are only quadratic—but I don't have a good visual intuition for why that's so.

I set out to play around with trisecting a line in an attempt to get a better feel for the problem, but it didn't help: the numbers fell out very easily and I feel like I have no better an understanding than when I started. Oh well.

Given points A and B, draw

1. circle AB
2. circle BA intersecting circle AB at points C and D,
3. circle CD intersecting circle AB at point E≠D and circle BA at point F≠D,
4. circle EB,
5. circle FC intersecting circle EB at points G and H,
6. circle GB,
7. circle HB intersecting GB at I≠B;

then I trisects AB. This one's straightforward to analyze using trigonometry. Let's first observe that EABF are all collinear, so we'll just worry about distances along the line EF. If we define AB=1, then EA=BF=1 as well, making EF=EA+AB+BF=3. FC=CD=√3. Suppose GH intersects EF at X: using the formula for the intersection of two circles, EX=(3²-√3²+2²)/(2·3)=5/3, therefore AX=EX-EA=2/3 and XB=AB-AX=1/3. I is the reflection of B about GH, therefore IX=XB=1/3 and AI=AX-IX=1/3.

I've been having quite a lot of fun playing with geometric constructions lately, and I have another one for you all today: a solution to Napoleon's Problem (that is, inscribing a square in a given circle using only a pair of compasses).

It's a famously tricky problem, and to be honest, I came up with this solution by accident while exploring something else. Nonetheless, I suspect this solution—involving six circles—is the simplest one possible, but I haven't managed to prove it yet.