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The Seed of Life and the Mysteries

2024-10-09T02:35:35Z

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...

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Starfish

2023-08-08T03:11:08Z

I was looking at a pentagram the other day and noticed something. It has a threefold vertical structure to it: it has one "head" on top, two "arms" in the middle, and two "legs" on the bottom. This is a lot like the structure of a human being, both physically (you have a head, two arms, and two legs—at least, I hope you do!) and metaphysically. See, there are three worlds: the transcendent Good, the intelligible, and the sensible. The Good is inherently unitary, but you have two parts in each of the intelligible and sensible worlds, at least according to Porphyry:

world	level of being	power	element
transcendent	the Good	being	spirit
intelligible	the Intellect	intuition	fire
intelligible	soul	reason	air
sensible	pneumatic vehicle	imagination	water
sensible	body	sensation	earth

(The pneumatic vehicle is, I think, what Plotinus calls the "lower soul." It's your mind, your imagination. Also please forgive my crude table: the pneumatic vehicle isn't a "level of reality" in the same way the others are, and exists at the level of nature alongside your body.)

Each limb of the pentagram, therefore, represents a power you possess. The head is your being (thanks to the Good). The arms are your thinking capacities: the intuitive (thanks to the Intellect) and the rational (thanks to your soul). The legs are your sensing capacities: the imaginative (thanks to your pneumatic vehicle) and the perceptive (thanks to your body). The head and arms are eternal (indeed, only one of the arms can be said to be "yours"—the other arm and head are common to all), but you periodically lose the legs and regrow them until you learn how to get around without them.

While you technically have all of these capacities, most of us are weighted towards the bottom. The point of practicing meditation is to climb your way back up the ladder as far as you're able.

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More Squares

2023-07-12T02:16:50Z

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.

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Eye

2023-06-25T15:35:46Z

An eye in seven circles. (Appropriate, that.)

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Metamorphosis

2023-06-21T20:34:54Z

A refinement of my butterfly using 8 circles.

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Foils

2023-04-21T10:15:30Z

violetcabra mentioned hexafoils lately, which are a neat little geometric construction with evident spiritual properties:

I was curious what other kinds of "foils" there were. Turns out there's only two: a "tetrafoil" with four petals, and a "pentafoil" with five:

Let's consider why. First, a "foil" obviously connects the vertices of a regular polygon with circular arcs through the center of its circumscribed circle. Thus there no such thing as a "monofoil" or "difoil" since you can't have a polygon with fewer than three vertices. The "trifoil" is simply a degenerate hexafoil with half of its petals removed, and so there seems no reason to count it. (This isn't just pique, either: to construct a trifoil with a compass, one has to construct the full hexfoil first!) The "tetrafoil", "pentafoil", and hexafoil are shown above. The "heptafoil" cannot be constructed with a compass and straightedge, but even if it could, it's petals would overlap each other: this is because they are both wider and closer together than the hexafoil's. This is true for any number of petals greater than six, in fact, which means that the hexafoil is as high as we can go.

The "tetrafoil" is unusual compared to the other two in that arcs connect its adjacent vertices, rather than every other vertex. (If we connected every other vertex, the arcs would be of infinite radius—that is, they'd be straight lines—and we'd have a sun cross.) In this respect, the "tetrafoil" is more like a degenerate "octafoil," where we removed the extra petals that overlap.

The hexafoil is trivial to construct with a compass alone, but the others are quite complicated: the "tetrafoil" took me fourteen circles to make, and the "pentafoil," fifteen. (They're both pretty easy if you allow a straightedge, though.) I think this is reflected in their elegance: the hexafoil strikes me as by far the most orderly and beautiful of these, though the "pentafoil" reminds me of a sand-dollar, which has its charms...

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Enneads V 6: What Being has Intellection Primarily and What Being has It Secondarily

2023-03-12T19:02:23Z

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.

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Psyche

2023-02-22T20:49:02Z

Zhuangzi dreamt he was a butterfly: he fluttered here and there, carefree and unselfconscious. Suddenly he awoke, and there he was again: Zhuangzi the human, beyond a doubt. But... was he the Zhuangzi who had dreamt he was a butterfly, or was he a butterfly now dreaming that he was Zhuangzi?

(Zhuangzi II, as adapted by yours truly)

When the soul sleeps, the body feels;
when the body sleeps, the soul reveals
in dreams, the coming woes or weals.

(Pindar, as quoted by Plutarch, and as adapted by yours truly)

The construction from scratch is left as an exercise for the reader, but as a hint, the top two circles are centered at <±3,0>; the bottom two circles are centered at <±4,3>; and all four of them intersect at <0,0>.

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The Roots of Sacred Geometry

2022-12-09T23:24:33Z

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...

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Cracking the Chestnut

2022-12-06T11:34:31Z

I found something neat! Let me share it with you.

Consider this lovely construction:

It's similar to the various solutions I've posted to Napoleon's problem—it's a square made out of six circles—but it's not quite the same since it's a circumscribed square rather than an inscribed square. I think it's beautiful because the construction exhibits so much symmetry.

(Did any of you ever play the game Riven way back when? If not, I recommend it—it's one of a very few I consider edifying—but in any case, this construction is terribly reminiscent of the art of that game.)

Anyway, if that were all, it'd be cute but not terribly exciting. But what's neat about it is that, by reflecting a few points around,...

...poof! you have a regular octagon! This construction takes only nine circles, compared to my previous best, which took ten. The trick, of course, is that I wasn't specifically trying to place this octagon anywhere in particular, and it happens to be more natural to construct √2 (as I did here) than it is to construct 1/√2 (as I did previously).

Still, those reflections aren't very efficient, and it feels like I ought to be able to construct a regular octagon using eight circles somehow...

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Yin and Yang

2022-12-05T16:22:27Z

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.

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Ankh

2022-12-04T20:45:27Z

Starting to play with a few constructions of my own. Here is an ankh, using eight circles:

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Inscribed Regular Polygons

2022-11-29T20:04:04Z

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... )

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Moss's Egg

2022-11-25T21:11:55Z

The highlighted curve is called Moss's Egg, though I'm not certain who Moss is. (The link cites Dixon's Mathographics—I ordered a copy, perhaps it'll tell me.) I was surprised to find a way to construct it using only 8 circles! This is because circles naturally want to generate √3's, so usually it'd take extra effort to generate the √2 proportions the egg demands, but it turned out to only require going a single circle out of my way.

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Construction of a Regular Heptagon

2022-11-23T21:58:30Z

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

circle AB,
circle BA intersecting circle AB at points C and D,
circle CD intersecting circle AB at point E≠D and circle BA at point F≠D,
circle EB intersecting circle BA at point G,
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.

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Kepler's Triangle

2022-11-23T17:12:00Z

Here is another construction for you all. (It must seem like I must do nothing but these, but in my defense, I've been sick as a dog for a long time and I seem to be fit for nothing but mathematics when I'm sick.)

Given points A and B, draw

circle AB
circle BA intersecting circle AB at points C and D,
circle CD intersecting circle AB at point E≠D and circle BA at point F≠D,
circle AF,
circle BE intersecting circle AF at points G and H,
circle CH,
circle HE intersecting circle CH at point I;

then triangle CDI is Kepler's triangle. (Yes, that Kepler, the pre-eminent astronomer.) This triangle is the unique right triangle with edges forming a geometric progression: 1:x:x². (Curiously, the Great Pyramid of Giza when it was built—it has now weathered considerably—had proportions matching Kepler's triangle to three decimals.)

I found this by doodling around and ending up with this bizarre construction:

Lines EABF and IDJ sure look parallel, don't they? But that's weird, I was just doodling randomly and these circles have pretty arbitrary radii, so it wouldn't make sense for the intersection points to line up so nicely. So I just had to prove to myself that they were, in fact, parallel. I'll spare you my original trigonometric proof, which involved walking through five triangles using the law of cosines,* and give you a much simpler sketch using our circle-circle intersection formula. Due to Kurt Hofstetter, CH has a radius of √3ϕ. One can use the Pythagorean theorem to derive that circle HE has a radius of √6. The intersection points of circles CH and HE are therefore located at distance ((√3ϕ)²-(√6)²+(√3ϕ)²)/(2√3ϕ)=√3 from C to H. But CD=√3. Therefore CDI is a right angle, and lines AB and ID are, indeed, parallel. But the magic trick is in applying the Pythagorean theorem to triangle CDI to find DI=√3√ϕ—but √ϕ is a pretty funny number, isn't it? It means that CD:DI:CI=1:√ϕ:ϕ—which is indeed 1:x:x²—and therefore triangle CDI is Kepler's triangle. Crazy!

* Fun fact: back in high school, I ignored all the various theorems that my geometry teacher wanted me to use and instead drew tons and tons of triangles, proving whatever construction was requested using trigonometry. This was nice and easy to do—no thinking required—but the proofs routinely ran to multiple pages. About halfway through the school year my teacher stopped bothering to check my proofs. I know this since I started giving her faulty proofs in order to test her. ;)

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Trisection of a Line

2022-11-21T21:54:55Z

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

circle AB
circle BA intersecting circle AB at points C and D,
circle CD intersecting circle AB at point E≠D and circle BA at point F≠D,
circle EB,
circle FC intersecting circle EB at points G and H,
circle GB,
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.

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A Complicated Construction of Something Simple

2022-11-20T13:14:04Z

The 3-4-5 triangle is my desert-island geometric fact: if you have a triangle with a sides of length 3, 4, and 5, it's a right triangle. This is great because it's super easy to mark a rope into 3+4+5=12 equal lengths, and this means it's super easy to make yourself a right triangle. I've used this before to lay out an orchard, making sure all the rows were nice and parallel, and it worked beautifully.

Because it's so easy to make a 3-4-5 triangle directly, it seems pretty silly to go to much greater lengths to make one using a pair of compasses, but let's not let mere uselessness stop us! After all, there's the Taoist saying: "When purpose has been used to achieve purposelessness, the thing has been grasped." ;)

I'm feeling playful, so in honor of the sovereign Sun whose day it is, and his dutiful son Pythagoras, let's pick up our compasses and hop to it:

Given points A and B, draw

circle AB,
circle BA intersecting circle AB at points C and D,
circle CD intersecting circle AB at point E≠D,
circle EB intersecting circle BA at points F and G,
circle FA,
circle GA intersecting circle FA at point H≠A,
circle HA intersecting circle EB at point I;

then triangle HIE is a 3-4-5 triangle. I'm not going to write up a full proof right now, but a quick sketch runs like this: let's define AB=2. It can be shown that EAB is collinear, therefore EB=EI=EA+AB=2×AB=4. Suppose line FG intersects line AB at X: it can be shown that AX=3/2. H is the reflection of A about line FG and FG is perpendicular to AB, therefore EABH is collinear and AH=HI=AX+XH=2×AX=3. Finally, EH=EA+AH=AB+AH=5.

I believe this to be the simplest possible construction (that is, using the fewest circles) of such a triangle using only compasses. (There are much easier ways to draw a right angle, though!)

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Napoleon's Problem

2022-11-12T19:40:15Z

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.

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Construction of a Regular Pentagon, Using Only a Compass

2022-11-07T20:03:15Z

As a computer programmer, I'm enamored with the notion of economy: that is, doing as much as one can with as little as possible. It is perhaps because of this that I have always found geometry appealing: one endeavors to begin with the simplest tools possible—a collapsing compass and a straightedge—and, using these, build oneself ever-more-powerful tools, allowing one to easily draw figures of tremendous sophistication.

So it was with some amusement that I recently learned that one does not even need a straightedge: everything that can be constructed with a compass and straightedge can be constructed with a compass alone. (We're assuming that one can draw straight lines after the construction is complete to make the construction clear, but one cannot use a straightedge to find the endpoints of any of those lines!)

I haven't studied the proof of that in any depth, yet, but since we have a few fans of geometry here, I thought I might post one of my favorite compass-only constructions, which is due to Kurt Hofstetter. ( Read more... )

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