Unlocking the Observatory: Ideal Triangles in the Great Mirror

May 15, 2022 By polyphanes in occult Tags: analysis, geometry, great mirror, list, telescope of zoroaster, triangle, unlocking the observatory, zoroaster's telescope 2 Comments

Where were we? We’re in the middle of discussing the obscure Telescope of Zoroaster (ZT), a manual of divination and spirituality originally published in French in 1796 (FZT) at the close of the French Revolution, which was later translated into German in 1797 (GZT) and then again in an abridged form as part of Johann Scheible’s 1846 Das Kloster (vol. 3, part II, chapter VII) (KZT), with Scheible’s work then translated into English in 2013 as released by Ouroboros Press (OZT). Although OZT is how most people nowadays tend to encounter this system, I put out my own English translation of FZT out a bit ago as part of my research, and while that translation was just part of the work I’ve been up to, there’s so much more to review, consider, and discover when it comes to this fascinating form of divination. Last time, we talked about the various figures and mirrors used for divination, especially the Great Mirror itself. If you need a refresher on what we talked about last time, go read the last post!

※ For those following along with their own copy of ZT (get yours here!), the relevant chapters from ZT are the “Fourth Step” and “Third Supplement”.

In the last post, we talked about the “spreads” of tiles that get used in the divinatory method of ZT, especially the Great Mirror. While all the triangular, quadrangular, and hexangular figures get used as mirrors at some point in ZT, it’s really the Great Mirror that takes center stage as being the primary divinatory “spread” of them all. We talked about how each of the 37 positions in the large hexagon, when used as the Great Mirror, have their own meanings and semantic fields, and how such meanings can be arrived at by considering the Great Mirror a sort of self-similar, self-replicating planetary arrangement. In this, one can already perform basic divination in ZT by simply asking a query, composing a Great Mirror, and inspecting each tile in its appropriate house.

Of course, that’d just be too simple, wouldn’t it? We’re not doing Tarot, after all.

In addition to inspecting each tile in the house it appears within, ZT also accounts for “triads” of tiles in the Great Mirror in what it calls “ideal triangles”. Recall from the last post the distinction of “real figures” versus “ideal figures”: a real figure is any complete figure that is composed from tiles and named according to its overall shape, while an “ideal figure” is a subset of tiles within a real figure that forms a subfigure of that larger figure. In the Great Mirror, particular kinds of “ideal triangles” are noted as being useful and important for interpreting a Great Mirror as a whole, not just looking at individual tiles where they fall but looking at groups of tiles and how they fall together. To borrow a bit of astrological terminology, by noting which tiles form a “triad” in the Great Mirror, we can interpret any given tile in two ways: an “essential interpretation” (the tile in its house) and an “accidental interpretation” (the tile in the triads it forms).

There’s one minor hiccup in this approach, however. The earlier description (from the “Second Step”) of “ideal figures” leads one to consider actual figures composed of many tiles together all at once, but later uses of “ideal triangles” (notably the “Fourth Step” and the “Third Supplement”) only refer to triangles—because equilateral triangles are “the only triangular figure to which the Great Cabala attaches any importance”—and further only to the corners of those triangles. We’ll see why in a bit.

First, let’s recall the numbering for the houses in the large hexagon, which is used for the Great Mirror:

This is based on the numbering system given in ZT’s Plate III, which describes the cosmological layout of the Great Mirror, the placement of Sisamoro and Senamira, and the number pattern of the individual houses thereof:

Do you see all those dashed lines across the plate? There are several kinds of dashed lines: circular dashed lines around each planetary house indicate that planet’s orbit, a wavy line that goes through all the houses of the Great Mirror indicating the order of the tiles from 1 to 37, and lots of straight lines that indicate particular ideal triangles in the Great Mirror. Consider the ideal triangle composed of houses 1, 9, and 11 (noted as 1–9–11): this is a “planetary triangle” formed from the houses of Sun, Mars, and Venus. Likewise, houses 1, 37, and 22 (or 1–37–22) form a slightly larger “zodiacal triangle” that incorporates the houses given to Aries and Taurus.

ZT has this to say about the ideal triangles:

…it is good to also become familiar with a more ideal yet highly essential division into triangles of different sizes. The main ones are indicated in Plate III: all these triangles indicated by dashed lines have their vertices at the center…and are distinguished and named according to their bases. In order to not throw ourselves here into details which would exceed the framework of a key, since it would be good for the Candidate to seek them until they come across them by analogy, we will not give an account here of triangles other than the planetary, zodiacal, and external triangles.

…

The observations to be made according to the triangles, either already described or arbitrarily noted in the Great Mirror, will be infinite; the care one takes in inspecting them will cost time and cause trouble, though ever less and less until none at all, as such calculations become ever more familiar. On the other hand, as such cares and costs decrease, the variety and richness increase, above all the infallibility of what such results will reveal.

The Candidate whose eye is not well-exercised in geometry would do well, when operating, to always have a compass in hand to find without error and without difficulty the third box which must complete a triangle for any two already chosen in the Great Mirror. For example, if a compass has one point in the center of box 18 and the other fixed in the center of box 31, lifting and moving the first half of the compass will only find a third center in box 12. This is the third point completing, together with boxes 18 and 31, an equilateral triangle, the only triangular figure to which the Great Cabala attaches any importance. So it is with all of the triangles which one will is able to imagine and whose formation is possible in the space of the Great Mirror.

Likewise, later on, it suggests several rules regarding how to make use of such ideal triangles, or at least which ones to pay special attention to:

Let us carefully observe whether and where there might be a triplicity of similar numbers, i.e. what quality a third number might have to form an equilateral triangle with two other numbers sharing the same or similar property.

Let us appreciate what such a triplicity might mean, whether for good or ill.

Let us clearly note the number which forms an equilateral triangle with two Intelligences, and that one profoundly contemplates what a triplicity of Intelligences or primitive numbers might mean.

Let the same attention be paid to a triplicity of numbers with zero or of doublets.

Thus, while all possible ideal triangles within the Great Mirror should be considered, the most important ones are those that involve two or more tenfold compound Numbers, two or more doublet compound Numbers, two or more Intelligences, two or more primitive Numbers, or any triangle that contains “similar numbers…sharing the same or similar property” (i.e. those sharing a common digit or which reduce to the same digit). Presumably, we can also consider any triangle that also contains the two Spirits as also being significant.

That being said, when we say “all possible ideal triangles”…I mean, how many are we talking? Given ZT’s reference to using a compass to determine any kind of equilateral triangle formed between any of the houses in the Great Mirror, and given the example thereof where the triangle 18–12–31 has one vertical side compared to the planetary and zodiacal ideal triangles that have horizontal sides, in addition to the “external ideal triangles” formed between three of the corners of the outermost belt of the Great Mirror…well, it turns out that there are a lot of possible triads of figures we might consider. We can break all possible ideal triangles within the Great Mirror according to their orientation:

Horizontal: ideal triangles having one horizontal side (e.g. 1–9–11)
Vertical: ideal triangles having one vertical side (e.g. 18–12–31)
Skewed: ideal triangles having neither a horizontal nor vertical side (e.g. 32–26–30)

Likewise, within each group, we can classify the triangles further based on their size, although this is easier for some than others. We’ll cover all the triangles that I’ve been able to account for in the Great Mirror. I’m not too bad at those “how many squares are in this image?” puzzles you occasionally see online, so I hope to have accounted for all possible triangles. I apologize for the lack of standardization in how I might have accounted for the triangles in the lists below; this is, perhaps, something better for a spreadsheet than a series of HTML ordered lists.

Ideal Small Horizontal Triangles

Every small triangle has edges spanning two houses with one horizontal edge, and is the smallest possible ideal triangle (or ideal figure) that can be formed according to ZT. These are, by far, the most numerous kind of ideal triangle that can be formed in the Great Mirror. There are four kinds of small triangles: those that touch two signs of the Zodiac, those that have one sign of the Zodiac and one planet, those that have one planet, and those that touch only orbital houses.

Upwards (27 total)

34–35–17 (Aquarius, Moon)
33–17–16 (Capricorn, Moon)
32–16–15 (Sagittarius, Mercury)
31–15–30 (Scorpio, Mercury)
35–36–17 (Aquarius/Pisces)
17–18–6 (Moon)
16–6–5 (orbits of Moon/Mercury/Sun)
15–5–14 (Mercury)
30–14–29 (Libra/Scorpio)
36–37–19 (Pisces, Saturn)
18–19–7 (Saturn)
6–7–1 (Sun)
5–1–4 (Sun)
14–4–13 (Jupiter)
29–13–28 (Libra, Jupiter)
19–20–8 (Aries, Saturn)
7–8–2 (orbits of Saturn/Sun/Mars)
1–2–3 (Sun)
4–3–12 (orbits of Sun/Mars/Venus)
13–12–27 (Virgo, Jupiter)
8–21–9 (Taurus, Mars)
2–9–10 (Mars)
3–10–11 (Venus)
12–11–26 (Leo, Venus)
9–22–23 (Gemini, Mars)
10–23–24 (Gemini/Cancer)
11–24–25 (Cacner, Venus)

Downwards (27 total)

34–33–17 (Capricorn, Moon)
33–32–16 (Sagittarius/Capricorn)
32–31–15 (Sagittarius, Mercury)
35–17–18 (Aquarius, Moon)
17–16–6 (Moon)
16–15–5 (Mercury)
15–30–14 (Scorpio, Mercury)
36–18–19 (Pisces, Saturn)
18–6–7 (orbits of Moon/Saturn/Sun)
6–5–1 (Sun)
5–14–4 (orbits of Mercury/Sun/Jupiter)
14–29–13 (Libra, Jupiter)
37–19–20 (Aries, Saturn)
19–7–8 (Saturn)
7–1–2 (Sun)
1–4–3 (Sun)
4–13–12 (Jupiter)
13–28–27 (Virgo, Jupiter)
20–8–21 (Aries/Taurus)
8–2–9 (Mars)
2–3–10 (orbits of Sun/Mars/Venus)
3–12–11 (Venus)
12–27–26 (Leo/Virgo)
21–9–22 (Taurus, Mars)
9–10–23 (Gemini, Mars)
10–11–24 (Cancer, Venus)
11–26–25 (Leo, Venus)

Ideal Hollow Horizontal Triangles

Every hollow triangle has edges spanning three houses with one horizontal edge. A unique quality of all ideal medium triangles in the Great Mirror is that they must have one corner somewhere in the orbit of the Sun (houses 1–7), with the other two being the equivalent houses in the orbit of two other planets. For instance, given one corner in the house to the right of the Sun (house 4), then one can form two possible ideal medium triangles, both having one corner in the house to the right of Jupiter (house 28), and either an upwards triangle with the corner to the right of Mercury (house 30) or a downwards triangle with the corner to the right of Venus (house 26). This same quality is what allows such triangles to be highlighted in the “Fourth Step” as specifically being “planetary triangles”, because when an ideal medium triangle takes the Sun itself (house 1) as one corner, the other two must be two planetary houses themselves.

Given this property, it is easy to anticipate how many such ideal medium triangles there are. Given that any given house can form a triangle in one of six directions (two possible directions × three possible corners = six possible triangles) and that there are seven houses within the orbit of the Sun, there are thus 6 × 7 = 42 possible ideal medium triangles in the Great Mirror, six for each planet.

Planet-only, i.e. Sun-themed triangles (6 total)

1–19–9 (Sun, Saturn, Mars)
1–9–11 (Sun, Mars, Venus)
1–11–13 (Sun, Venus, Jupiter)
1–13–15 (Sun, Jupiter, Mercury)
1–15–17 (Sun, Mercury, Moon)
1–17–19 (Sun, Moon, Saturn)

Lower-left of the Sun, i.e. Mars-themed triangles (6 total)

2–20–22 (Sun-orbit, Saturn-orbit, Mars-orbit)
2–22–24 (Sun-orbit, Mars-orbit, Venus-orbit)
2–24–12 (Sun-orbit, Venus-orbit, Jupiter-orbit)
2–12–5 (Sun-orbit, Jupiter-orbit, Mercury-orbit)
2–5–18 (Sun-orbit, Mercury-orbit, Moon-orbit)
2–18–20 (Sun-orbit, Moon-orbit, Saturn-orbit)

Lower-right of the Sun, i.e. Venus-themed triangles (6 total)

3–8–23 (Sun-orbit, Saturn-orbit, Mars-orbit)
3–23–25 (Sun-orbit, Mars-orbit, Venus-orbit)
3–25–27 (Sun-orbit, Venus-orbit, Jupiter-orbit)
3–27–14 (Sun-orbit, Jupiter-orbit, Mercury-orbit)
3–14–6 (Sun-orbit, Mercury-orbit, Moon-orbit)
3–6–8 (Sun-orbit, Moon-orbit, Saturn-orbit)

Right of the Sun, i.e. Jupiter-themed triangles (6 total)

4–7–10 (Sun-orbit, Saturn-orbit, Mars-orbit)
4–10–26 (Sun-orbit, Mars-orbit, Venus-orbit)
4–26–28 (Sun-orbit, Venus-orbit, Jupiter-orbit)
4–28–30 (Sun-orbit, Jupiter-orbit, Mercury-orbit)
4–30–16 (Sun-orbit, Mercury-orbit, Moon-orbit)
4–16–7 (Sun-orbit, Moon-orbit, Saturn-orbit)

Upper-right of the Sun, i.e. Mercury-themed triangles (6 total)

5–18–2 (Sun-orbit, Saturn-orbit, Mars-orbit)
5–2–12 (Sun-orbit, Mars-orbit, Venus-orbit)
5–12–29 (Sun-orbit, Venus-orbit, Jupiter-orbit)
5–29–31 (Sun-orbit, Jupiter-orbit, Mercury-orbit)
5–31–33 (Sun-orbit, Mercury-orbit, Moon-orbit)
5–33–18 (Sun-orbit, Moon-orbit, Saturn-orbit)

Upper-left of the Sun, i.e. Moon-themed triangles (6 total)

6–36–8 (Sun-orbit, Saturn-orbit, Mars-orbit)
6–8–3 (Sun-orbit, Mars-orbit, Venus-orbit)
6–3–14 (Sun-orbit, Venus-orbit, Jupiter-orbit)
6–14–32 (Sun-orbit, Jupiter-orbit, Mercury-orbit)
6–32–34 (Sun-orbit, Mercury-orbit, Moon-orbit)
6–34–36 (Sun-orbit, Moon-orbit, Saturn-orbit)

Left of the Sun, i.e. Saturn-themed triangles (6 total)

7–37–21 (Sun-orbit, Saturn-orbit, Mars-orbit)
7–21–10 (Sun-orbit, Mars-orbit, Venus-orbit)
7–10–4 (Sun-orbit, Venus-orbit, Jupiter-orbit)
7–4–16 (Sun-orbit, Jupiter-orbit, Mercury-orbit)
7–16–35 (Sun-orbit, Mercury-orbit, Moon-orbit)
7–35–37 (Sun-orbit, Moon-orbit, Saturn-orbit)

However, of the above-listed triangles, there are a few duplicates, which reduces the number of distinct ideal hollow triangles. These duplicates are formed by triangles that contain opposing planets, e.g. 2–5–18 as the perspective of Mars from the Sun, but 5–18–2 as the perspective of Mercury from the Sun; each pair of opposing planets produces two duplicates, with the third corner of these ideal triangles placed on the cross-axis that forms between them between the other two planets on a given side. As a result, there are only 36 distinct ideal hollow triangles.

Ideal Full Horizontal Triangles

Every full triangle has edges spanning four houses with one horizontal edge, allowing for one house to form the center of such a triangle. The prototypical form of these triangles are the “zodiacal triangles” (as described in the “Fourth Step”), where the base of such a triangle can be formed along one whole edge of the Great Mirror from one corner to the next. Given the sixfold divisions of life in the “Fifth Step” (which we haven’t yet covered, but we will!), it may be that these triangles relate more to the various trials, tribulations, or experiences given in similar timeframes, especially as might impact particular lusters of life.

Of all the possible ideal large triangles, only two are contained completely within the Great Mirror without touching any of its edges or corners (internal), six involve two corners and an entire edge (fully external), and all the rest touch only one non-corner edge house (partially external).

Internal (2 total)

8–12–16 (upwards)
18–14–10 (downwards)

Fully external (6 total)

1–37–22 (downwards, First Division, Aries/Taurus)
1–22–25 (upwards, Second Division, Gemini/Cancer)
1–25–28 (downwards, Third Division, Leo/Virgo)
1–28–31 (upwards, Fourth Division, Libra/Scorpio)
1–31–34 (downwards, Fifth Division, Sagittarius/Capricorn)
1–34–37 (upwards, Sixth Division, Aquarius/Pisces)

Partially external (12 total)

20–3–17 (upwards, Aries/Moon)
21–11–6 (upwards, Taurus/Venus)
23–4–19 (downwards, Gemini/Saturn)
24–13–7 (downwards, Cancer/Jupiter)
26–5–9 (upwards, Leo/Mars)
27–15–2 (upwards, Virgo/Mercury)
29–11–6 (downwards, Libra/Venus)
30–3–17 (downwards, Scorpio/Moon)
32–13–7 (upwards, Sagittarius/Jupiter)
33–4–19 (upwards, Capricorn/Saturn)
35–15–2 (downwards, Aquarius/Mercury)
36–5–9 (downwards, Pisces/Mars)

Some patterns can be noted regarding the above sets of triangles:

If an ideal full triangle touches one corner, it must touch another, with one of its corners in House 1 and one of its edges containing two signs of the Zodiac.
If an ideal full triangle touches only one non-corner edge house, then that edge house must be a sign of the Zodiac. One of the other corners must be a non-solar planet, and the last corner must be a non-planetary house in the shared solar orbit of the planet opposite the first, e.g. Moon of Venus and vice versa.
Every non-solar planet takes part in two ideal full triangles, one upward and one downward:
- Mars: Leo (upward) and Pisces (downward)
- Venus: Taurus (upward) and Libra (downward)
- Jupiter: Sagittarius (upward) and Cancer (downward)
- Mercury: Virgo (upward) and Aquarius (downward)
- Moon: Aries (upward) and Scorpio (downward)
- Saturn: Capricorn (upward) and Gemini (downward)
The internal ideal full triangles only have corners that are in the shared orbits of two non-solar planets, with each corner being adjacent to two signs of the Zodiac.

Ideal Large Horizontal Triangles

Every large horizontal triangle has edges spanning five houses with one horizontal edge, allowing for three houses to form the center of such a triangle. Unlike the other ideal triangles, large triangles are too large to have a base along the edges of the Great Mirror; instead, they can only have one of their corners on the Great Mirror’s edge. Each of these triangles involves two signs of the Zodiac, both of the same modality (cardinal, fixed, or mutable) but of incompatible elements (fire/earth or water/air). In each triangle, the third house is a non-planetary house between two planets in the planetary belt.

Upwards (3 total)

21–26–16 (Taurus/Leo)
20–12–33 (Aries/Capricorn)
8–27–32 (Virgo/Sagittarius)

Downwards (3 total)

35–30–10 (Scorpio/Aquarius)
36–14–23 (Gemini/Pisces)
18–29–24 (Cancer/Libra)

Ideal One-Skip Vertical Triangles

Now that we’re done with horizontal triangles, we have to consider vertical triangles. Due to the “grain” of houses in the Great Mirror, while it’s easy to state the base of a horizontal triangle in terms of how many houses it covers, vertical triangles are somewhat trickier. To resolve this, we’ll use the notion of “skips” it takes to go from one house along the vertical base of a vertical triangle to the next house directly above or below it. Thus, from house 34, it takes one skip to go to 34 to 18, two skips to go from 34 to 18, and three skips to go from 34 to 22.

One-skip vertical triangles fall into types: those touching one sign of the Zodiac and one planet in different orbits, those touching two signs of the Zodiac in the same orbit, those touching non-planetary and non-zodiacal houses within the same orbit, or those touching one planet and two non-planetary and non-zodiacal houses in other planets’ orbits.

Rightwards (19 total)

34–18–16 (all in orbit of the Moon, touching upper left corner)
8–22–10 (all in orbit of Mars, touching lower left corner)
14–12–28 (all in orbit of Jupiter, touching right corner)
35–19–6 (Aquarius and Saturn)
19–21–2 (Taurus and Saturn)
2–23–11 (Gemini and Venus)
4–11–27 (Virgo and Venus)
15–4–29 (Libra and Mercury)
33–6–15 (Capricorn and Mercury)
36–20–7 (Pisces/Aries in orbit of Saturn)
3–24–26 (Cancer/Leo in orbit of Venus)
32–5–30 (Sagittarius/Scorpio in orbit of Mercury)
17–7–5 (Moon)
7–9–3 (Mars)
5–3–15 (Jupiter)
18–8–1 (Sun)
1–10–12 (Sun)
16–1–14 (Sun)
6–2–4 (all in orbit of the Sun, surrounding the center)

Leftwards (19 total)

31–14–16 (all in orbit of Mercury, touching upper right corner)
18–8–37 (all in orbit of Saturn, touching left corner)
12–25–10 (all in orbit of Venus, touching lower right corner)
32–5–17 (Sagittarius and the Moon)
17–7–36 (Pisces and the Moon)
7–9–20 (Aries and Mars)
3–24–9 (Cancer and Mars)
13–26–3 (Leo and Jupiter)
30–13–5 (Scorpio and Jupiter)
33–6–35 (Capricorn/Aquarius in orbit of the Moon)
2–23–21 (Taurus/Gemini in orbit of Mars)
29–27–4 (Virgo/Libra in orbit of Jupiter)
15–4–6 (Mercury)
6–2–19 (Saturn)
4–11–2 (Venus)
16–1–18 (Sun)
1–10–8 (Sun)
14–12–1 (Sun)
5–3–7 (all in orbit of the Sun, surrounding the center)

Ideal Two-Skip Vertical Triangles

Two-skip vertical triangles fall into three types: those touching a single corner, those touching two signs of the Zodiac, and those touching three planets. As an interesting result of the astrological qualities of the signs, when a two-skip vertical triangle touches two signs of the Zodiac, the signs it touches are of opposing elements (fire/water, air/earth) but same modality (cardinal, fixed, mutable); all the rightward triangles of this sort touch only earth and air signs, while all the leftward triangles touch only fire and water signs.

Rightwards (7 total)

35–21–4 (Taurus/Aquarius)
23–27–6 (Gemini/Virgo)
33–29–2 (Libra/Capricorn)
34–8–14 (touching upper left corner)
22–12–18 (touching lower left corner)
10–28–16 (touching right corner)
17–13–9 (Moon/Jupiter/Mars)

Leftwards (7 total)

24–5–20 (Aries/Cancer)
30–26–7 (Leo/Scorpio)
32–3–36 (Sagittarius/Pisces)
31–12–18 (touching upper right corner)
25–14–8 (touching lower right corner)
10–16–37 (touching left corner)
15–11–19 (Mercury/Venus/Saturn)

Ideal Three-Skip Vertical Triangles

There are only two possible three-skip vertical triangles, one of which points to the right and one of which points to the left, and both of them involve the extreme corners of the Great Mirror. For this reason, ZT explicitly calls these two triangles “external triangles” in the “Fourth Step”, and notes that these are the “most ideal” of any triangles in the Great Mirror (possibly as a result of how they are the largest possible triangles that can be formed of any size or orientation).

22–28–34 (rightwards, extremes of Mercury/Venus/Saturn)
25–31–37 (leftwards, extremes of Moon/Mars/Jupiter)

Ideal Two-Move Skewed Triangles

Just as how we had to judge the size of vertical triangles differently from horizontal triangles, so too do we have to consider skewed triangles (which have neither horizontal nor vertical edges) differently. For this, we’ll use the notion of “moves”, how many houses one must cross to go from one corner of an ideal skewed triangle to the next. Thus, between houses 36 and 16 there are two moves, between 36 and 15 there are three moves, and between 36 and 30 there are four moves. Likewise, because there’s no horizontal or vertical base to such a triangle, it’s hard to say which direction these triangles are “pointing”. As a result, instead of going with upward/downward/rightward/leftward as we did with the other triangles, we’ll just group them into what they touch or make use of in the Great Mirror.

Single Planet + Planetary Zone + Solar Orbit (12 total)

17–14–2 (Moon)
17–8–4 (Moon)
19–16–3 (Saturn)
19–10–5 (Saturn)
9–18–4 (Mars)
9–12–6 (Mars)
11–8–5 (Venus)
11–14–7 (Venus)
13–10–6 (Jupiter)
13–16–2 (Jupiter)
15–12–7 (Mercury)
15–18–3 (Mercury)

Single Planet + Corner House + Solar Orbit (12 total)

17–31–4 (Moon)
17–37–2 (Moon)
19–34–5 (Saturn)
19–22–3 (Saturn)
9–37–6 (Mars)
9–25–4 (Mars)
11–22–7 (Venus)
11–28–5 (Venus)
13–25–2 (Jupiter)
13–31–6 (Jupiter)
15–28–3 (Mercury)
15–34–7 (Mercury)

Sun + Two Zodiac Signs (12 total)

1–20–23 (Sun, Aries/Gemini)
1–21–24 (Sun, Taurus/Cancer)
1–23–26 (Sun, Gemini/Leo)
1–24–27 (Sun, Cancer/Virgo)
1–26–29 (Sun, Leo/Libra)
1–27–30 (Sun, Virgo/Scorpio)
1–29–32 (Sun, Libra/Sagittarius)
1–30–33 (Sun, Scorpio/Capricorn)
1–32–35 (Sun, Sagittarius/Aquarius)
1–33–36 (Sun, Capricorn/Pisces)
1–35–20 (Sun, Aquarius/Aries)
1–36–21 (Sun, Pisces/Taurus)

Small Single Zodiac (12 total)

35–8–5 (Aquarius)
36–16–2 (Pisces)
20–20–6 (Aries)
21–18–3 (Taurus)
23–12–7 (Gemini)
24–8–4 (Cancer)
26–14–2 (Leo)
27–10–5 (Virgo)
29–16–3 (Libra)
30–12–6 (Scorpio)
32–18–4 (Sagittarius)
33–14–7 (Capricorn)

Ideal Three-Move Skewed Triangles

Two Zodiac Signs (6 total)

20–24–5 (Aries/Cancer)
23–27–6 (Gemini/Virgo)
26–30–7 (Leo/Scorpio)
29–33–2 (Libra/Capricorn)
32–36–3 (Sagittarius/Pisces)
35–21–4 (Aquarius/Taurus)

Small Zodiac-Planet (12 total)

20–11–16 (Aries, Venus)
21–17–12 (Taurus, Moon)
23–13–18 (Gemini, Jupiter)
24–19–14 (Cancer, Saturn)
26–15–8 (Leo, Mercury)
27–9–16 (Virgo, Mars)
29–17–10 (Libra, Moon)
30–11–18 (Scorpio, Venus)
32–19–12 (Sagittarius, Saturn)
33–13–8 (Capricorn, Jupiter)
35–9–14 (Aquarius, Mars)
36–15–10 (Pisces, Mercury)

Large Single Zodiac (12 total)

35–31–3 (Aquarius)
36–22–4 (Pisces)
20–34–4 (Aries)
21–25–5 (Taurus)
23–37–5 (Gemini)
24–28–6 (Cancer)
26–22–6 (Leo)
27–31–7 (Virgo)
29–25–7 (Libra)
30–34–2 (Scorpio)
32–28–2 (Sagittarius)
33–37–3 (Capricorn)

Ideal Four-Move Skewed Triangles

Elemental Zodiac Groups (4 total)

32–26–20 (Aries/Leo/Sagittarius, i.e. fire signs)
33–27–21 (Taurus/Virgo/Capricorn, i.e. earth signs)
35–23–29 (Gemini/Libra/Aquarius, i.e. air signs)
36–24–30 (Cancer/Scorpio/Pisces, i.e. water signs)

Large Zodiac-Planet (12 total)

20–15–25 (Aries, Mercury)
21–13–34 (Taurus, Jupiter)
23–17–28 (Gemini, Moon)
24–15–37 (Cancer, Mercury)
26–19–31 (Leo, Saturn)
27–17–22 (Virgo, Moon)
29–9–34 (Libra, Mars)
30–19–25 (Scorpio, Saturn)
32–11–37 (Sagittarius, Venus)
33–9–28 (Capricorn, Mars)
35–13–22 (Aquarius, Jupiter)
36–11–31 (Pisces, Venus)

Ideal Principle Triangles

Although not explicitly called an ideal triangle as such, the placements of Sisamoro and Senamira around the Great Mirror is suggestive of one. ZT states that Sisamoro should be placed above the Great Mirror as the corner of an upwards equilateral triangle formed with houses 28 and 37, and Senamira likewise but downwards beneath the Great Mirror. Unlike some of the ideal triangles listed in the “Fourth Step” which are explicitly without dashed lines indicating them in Plate III, the triangles formed with the Principles do have those dashed lines, suggesting that these, too, form a kind of ideal triangle, albeit a nonstandard one that cannot be formed with any other houses in the Great Mirror. Technically, these seats form four ideal triangles each (one for each “layer” of the Great Mirror through its equator), but ZT suggests that it is only the largest possible triangle with the equator as its base that counts.

Given the importance of the leftmost and rightmost houses in the Great Mirror as being representative, respectively, of Saturn/Death and Jupiter/Life (akin to the bottom and top of the Wheel of Fortune in Tarot imagery), it might make sense that the Principles would find themselves in alignment with these points and no others. This may be a hint as to how the Principles, if drawn, are to be interpreted.

Whew. Assuming that I counted them right, didn’t miss any, and didn’t repeat any, all the above would yield a total of 264 possible ideal triangles (maybe 266 if we also allow for the Principles to form ideal triangles as well and if they are drawn in a Great Mirror). And, uh…yeah, this is a lot. Even just accounting for what the triangles are and a handful of patterns among them, this is a lot to note and remember—but that’s just the point, we don’t need to remember or memorize any of this stuff. Again, just like with learning the significations of the Numbers and the semantic fields of the houses, all we really need to do is account for the fundamental patterns that play themselves out in the Great Mirror. On top of that, while surely investigating all possible ideal triangles would be a noble thing to do, ZT gives us a handful of things to look out for which would highlight and whittle down the ideal triangles to what would be most important—note how many of those pieces of advice stated two Intelligences/primitive Numbers/doublets/nilleds, as opposed to just one. If you have just one tile like that, then it could form any number of triangles, but with two, it’s (almost) trivial to see what the third might be (to result in one or maybe two triangles depending on the spacing of those two given points).

Earlier I mentioned, borrowing astrological terminology, how we might consider these triad-based relationships of tiles that fall as giving a sort of “accidental” significance to any given tile (i.e. any tile relative to other tiles), as opposed to the “essential” significance given by the house a given tile falls into (i.e. any tile on its own where it is). I think that’s a useful way to consider this approach of using ideal triangles, but it raises the question of whether the houses themselves come into play when determining the meaning of a given ideal triangle, and personally, I’m inclined to think that there is. ZT doesn’t say as much, but then, ZT doesn’t say a whole lot, either. I think it would make sense for such a relationship to account for all of this together, which would indeed require a good amount of intuition as well as investigation on the part of the diviner. While the number of ideal triangles and the possible triads of tiles isn’t really infinite (though it is indeed likely a vast number), ZT is absolutely being honest with us when it says:

…the care one takes in inspecting them will cost time and cause trouble, though ever less and less until none at all, as such calculations become ever more familiar. On the other hand, as such cares and costs decrease, the variety and richness increase, above all the infallibility of what such results will reveal.

This is something to definitely take care with when trying out the divinatory method of ZT, and given the cosmological structure of the Great Mirror, I think there’s a good amount of stuff to contemplate and consider. If you think I’ve missed any triangles or overcounted any, or if there are any patterns or qualities you think are important that I didn’t get around in saying for particular kinds of ideal triangles, dear reader, please do say so in the comments!

Unlocking the Observatory: Figures, Mirrors, and the Great Mirror

May 11, 2022 By polyphanes in occult Tags: athanasius kircher, diamond, fractal geometry, geometry, great mirror, hexagon, hexangle, house, interpretation, lozenge, meaning, mirror, planetary hexagram, planets, quadrangle, rhombus, signification, telescope of zoroaster, tree of life, triangle, unlocking the observatory, zodiac, zoroaster's telescope 2 Comments

Where were we? We’re in the middle of discussing the obscure Telescope of Zoroaster (ZT), a manual of divination and spirituality originally published in French in 1796 (FZT) at the close of the French Revolution, which was later translated into German in 1797 (GZT) and then again in an abridged form as part of Johann Scheible’s 1846 Das Kloster (vol. 3, part II, chapter VII) (KZT), with Scheible’s work then translated into English in 2013 as released by Ouroboros Press (OZT). Although OZT is how most people nowadays tend to encounter this system, I put out my own English translation of FZT out a bit ago as part of my research, and while that translation was just part of the work I’ve been up to, there’s so much more to review, consider, and discover when it comes to this fascinating form of divination. Last time, we talked about the 112 (or 113) tiles used for divination, what each needs to have on it, and what each means in divination. If you need a refresher on what we talked about last time, go read the last post!

※ For those following along with their own copy of ZT (get yours here!), the relevant chapters from ZT are the “Second Step”, “Fourth Step”, “Seventh Step”, and “Third Supplement”.

Alright! As of the last post, we now have the toolset required for divination; in Tarot terms, we’ve taken a good look at all the cards (so to speak) and know what they are, what they represent, and the like. What comes next is how to make use of such tools, and just as Tarot cards get drawn and arranged into spreads, so too are the tiles of ZT drawn and arranged into…well, there’s a bit of terminology we have to go through and sort out first, I suppose, because ZT was trying to innovate its own terms in a time when such terms were still in the process of taking shape and becoming standardized.

Figure: A geometric shape composed of tiles, named after the shape that it forms.
Mirror: A whole figure that is used for divination.
Tablature: “The reasoned and just enunciation of what a Great Mirror gives to read”, i.e. the interpretation and reasoning of a divinatory session (especially, but not necessarily, making use of the “Great Mirror”—more on that term later).

In order to form a figure from tiles, one composes a figure by arranging successive tiles in an outward spiral, starting from one tile then proceeding counterclockwise, with the second tile always to the lower left of the first then proceeding outwards from there. Tiles within figures are always densely-packed, meaning that there is no space between them and tiles are pushed together against their own edges and corners. In this way, given the hexagonal geometry of the tiles, figures can be formed in shapes that are overall triangular, quadrangular, or hexangular.

There are four kinds of triangular figures that ZT allows:

The 3-tile triangular figure, also called the “small triangle”.
The 6-tile triangular figure, also called the “simple triangle” or “hollow triangle” (because its center is a meeting of three tiles at a vertex instead of a whole tile itself).
The 10-tile triangular figure, also called the “full triangle” (because its center is a whole tile).
The 15-tile triangular figure, also called the “large triangle” or “double triangle” (because its center is another whole triangular figure).

There are three kinds of quadrangular figures that ZT allows (which it calls “lozenges” or “diamonds”):

The 4-tile quadrangular figure, also called the “small diamond” or the “hollow diamond”.
The 9-tile quadrangular figure, also called the “medium diamond” or the “full diamond”.
The 16-tile quadrangular figure, also called the “large diamond” or the “double diamond”.

There are three kinds of hexangular figures that ZT allows (which it also just calls “hexagons”):

The 7-tile hexangular figure, also called the “small hexagon” or the “orbital hexagon” (because the outer six tiles form an orbit around the center tile).
The 19-tile hexangular figure, also called the “medium hexagon” or the “double hexagon”.
The 37-tile hexangular figure, also called the “large hexagon”, the “triple hexagon”, or “the totality” (because it includes all other possible figures that are permissible according to ZT).

ZT gives a lot of precedence and eminence to the large hexagon, because it forms the basis of many of the divinatory processes and cosmological models used in its “Great Cabala”. Although that is one of the reasons the large hexagon is called “the totality”, the other is more in the sense of a limitation. One might wonder why we might not make larger triangles or diamonds by adding in more tiles and continuing the spiral; ZT disallows this by saying that only the figures that can be contained within the large hexagon are permissible for use in divination. Thus, one cannot make a triangular figure out of 21 tiles or a quadrangular figure out of 25 tiles because they wouldn’t be able to “fit” inside the large hexagon.

This leads to a distinction that ZT makes between what it calls “real figures” versus “ideal figures”:

Any isolated figure is called “real”; it therefore forms a picture, a mirror. Any included or contained figure is called “ideal”.

In other words, a whole figure that is composed from tiles and seen as a whole is considered “real”, while any subset of tiles within such a figure that could also be composed as a separate figure on its own is called “ideal”. Let’s say that we draw three tiles and form a small triangle; this would be a real figure. If we draw another 16 tiles and, with all the tiles put together, make a medium hexagon, then this is another real figure. However, if we look at the bottom “pie slice” of that medium hexagon (tiles 1, 2, 3, 9, 10, and 11), and note how those tiles form a sort of sub-figure in the shape of a hollow triangle, then this sub-figure is an ideal figure, because it is not a figure on its own but is part of a larger figure that it is found within. In that light, a real large hexagon contains all other possible figures as ideal figures within it; thus, although one might consider the large hexagon to be the goal of being built-up from smaller figures, ZT takes the opposite approach and says that the large hexagon is what “came first” in a sense, from which the smaller figures could be broken out. Although this seems like an odd distinction to make, it forms the basis of a powerful interpretive technique later on, so it’s good to start paying attention to the possible ideal figures that might occur within a larger real figure.

When it comes to the structure of the large hexagon, it helps to consider it in terms of its general structure as having one center and three “belts” or “zones:

Center: house 1 (also called the “focus”)
Inner belt: houses 2 through 7
Middle belt: houses 8 through 19
Outer belt: houses 20 through 37 (also called the “frontier”)

With all that out of the way, we’re finally able to talk about ZT’s main approach to divination: the Great Mirror. This is a large hexagon formed in the usual way, but each tile-position (what I’ll call “house”) in the Great Mirror has particular cosmological signification. As a result, ZT also talks about the large hexagon as using the “astronomical aspect” or “sidereal aspect” (in contrast to the “temporal aspect” or “chronic aspect” which is another use of the large hexagon we’ll get to later). In many ways, the Great Mirror is the ZT equivalent of the Celtic Cross spread in Tarot or the Grand Tableau in Lenormand.

The above diagram is a reproduction of ZT’s own Plate III, which includes a bit more information than what’s shown above but which we’ll get to in a bit:

The Great Mirror is generated the same way as with any large mirror: counting in an outwards counterclockwise spiral starting from the center and proceeding to the lower left. The Great Mirror is broken down into four regions based on its overall structure:

The center, which is the single house 1 in the middle of the Great Mirror.
The solar belt, which consists of houses 2 through 7 (i.e. the Great Mirror use of the inner belt), immediately around the center. This belt is also called the “central belt”.
The planetary belt, which consists of houses 8 through 19 (i.e. the Great Mirror use of the middle belt), immediately around the solar belt. This belt is also called the “sidereal belt”.
The zodiacal belt, which consists of houses 20 through 37 (i.e. the Great Mirror use of the outer belt), immediately around the planetary belt.

Of special significance in the Great Mirror are houses 1, 9, 11, 13, 15, 17, and 19, because these are the houses given (respectively) to the Sun, Mars, Venus, Jupiter, Mercury, the Moon, and Saturn. All the other houses are said to be “in the orbit” of one or two planets; thus, when we look at Mars (the “inner corner” of the Great Mirror on the lower left), then we can say that house 9 is Mars itself; houses 21, 22, and 23 are houses exclusively in the orbit of Mars; house 10 is in the shared orbit of Mars and Venus, house 2 is in the shared orbit of Mars and the Sun, and house 8 is in the shared orbit of Mars and Saturn. Note how, while all the non-solar planets have three houses that are in their own orbit exclusively, every house in the Sun’s orbit is shared with another planet. Thus, if we consider a planet together with that planet’s orbit, then what we’re doing is effectively considering an “ideal small hexagon” within the larger Great Mirror. This is why the other term for a “small hexagon” is “orbital hexagon”, because it represents a single planet in one of the focal points of the Great Mirror and the six houses that surround it.

And yes, we’re being bumped back down to seven planets here, not to nine planetary intelligences. As opposed to a 9-fold system of numerology, we’re using a 7-fold system of geometry here, which necessitates that we talk about seven places of interest and not nine. To that end, it would be inaccurate to say “house of Adamasto” or “orbit of Seleno” (rather “house of Mars” or “orbit of Moon” respectively), because the Great Mirror focuses on planets and not planetary intelligences, even if the intelligences have their seats in their own planets. Thus, both Genhelia and Psykelia share the same orbit of the Sun, just as Seleno and Psykomena share the same orbit of the Moon. (It gets a little more complicated later, but that’s later, not now.)

So, as you might have predicted, each house in the Great Mirror has its own set of meanings, its own contextual or semantic field, just as the twelve houses do in an astrological horoscope or each of the card positions in a Tarot spread. When a tile is placed in one of these houses, the meaning of that tile is to be interpreted in the scope of the house it’s found in. Thus, there are 37 such houses, each with their own meaning—but again, there’s a system behind this. Recall how when we were talking about the Intelligences and Numbers how, although each Number has its own signification, the significations didn’t have to be memorized but rather “generated” based on their smaller digits? A similar approach is used for the meanings of the individual houses of the Great Mirror.

Take a look at the layout of planets on the Great Mirror: we have the Sun in the Middle, Mars to the lower left, Venus to the lower right, Jupiter to the right, Mercury to the upper right, the Moon to the upper left, and Saturn to the left. Each planet has its own orbit of six houses, but if you consider things at a grander scale, the large hexagram is a collection of seven orbital hexagrams in the same geometric arrangement as an orbital hexagram has seven houses, kinda like a fractal. If we consider a smaller “fractal” of the Great Mirror…

…then we can overlay this on each orbit of the Great Mirror as a whole to get a sort of “main planet vs. sub-planet” arrangement.

Consider house 24. This is a house in the orbit of Venus, but it’s to the lower-left of this planet, which is the “fractal direction” of Mars. In this light, we might say that house 24 is the house of “Mars of Venus”, so even though this house fundamentally has something to do with Venus, it’s about the Martian aspects of Venus’ domain. Thus, this house has the meaning of “intense or violent passions and senses”. By taking the overall planetary layout of the Great Mirror and applying it on a smaller scale to an individual planet’s orbit, we can arrive at a specific context through pairwise planetary interactions—not unlike the how we paired together the tens-digit vs. ones-digit of the compound Numbers to arrive at a specific indication through pairwise interactions of the primitive Numbers.

What about the planetary houses themselves? Well, they have the “fractal direction” of being in the center, which is the house of the Sun: thus, the “Sun of whatever-planet” house is just that planet itself; just as the font of all power in the solar system is the Sun, the font of all power within a given planet’s own orbit is that planet itself. Thus, house 9 (the house of Mars itself) is given to “military status, valor or bravery”, and house 13 (the house of Jupiter) is given to “high wisdom, science”. The system checks out pretty well in this case.

What about houses that are in two orbits at once? Consider house 7: this is a house in the orbit of the Sun, but it’s to the left of the Sun, the “fractal direction” of Saturn. Thus, house 7 is the house of “Saturn of the Sun”, so this house is about the Saturnine aspects of the Sun’s domain. Thus, this house has the meanings of “advanced age, health”. At the same time, house 2 is also in the orbit of Saturn, to the right of the planet and thus the “fractal direction” of Jupiter. The indications of “advanced age and health” can be thought of as much as the Saturnine aspects of Saturn’s domain as it is the Jovian aspects of Saturn’s domain, in this regard. Likewise, if we consider house 10, the house between the positions of Mars and Venus, this house has the meaning of “romantic adventures” (again, “romantic” in the sense of being chivalrous and novel-worthy events). From the perspective of Mars, house 10 is to Mars’ right in the “fractal direction” of Jupiter (so “Jupiter of Mars”), but at the same time, it’s also to Venus’ left in the “fractal direction” of Saturn (so “Saturn of Venus”). Sure, it might be about “romantic adventures”, but the different perspectives here can shine a different light on that same topic.

So, in that light, here’s what ZT gives as meanings for the 37 houses of the Great Mirror, along with what the planetary considerations are of each house.

Grandeur, power. (Sun of Sun)
Strength, triumph, glory. (Mars of Sun, Mercury of Mars)
Beauty, happy love. (Venus of Sun, Moon of Venus)
Genius, great reputation. (Jupiter of Sun, Saturn of Jupiter)
Treasures, gains of all kinds. (Mercury of Sun, Mars of Mercury)
Domestic prosperity, inheritance. (Moon of Sun, Venus of Moon)
Advanced age, health. (Saturn of Sun, Jupiter of Saturn)
Severe bodily injury. (Moon of Mars, Venus of Saturn)
Military status, valor or bravery. (Sun of Mars)
Romantic adventures. (Jupiter of Mars, Saturn of Venus)
Good fortune. (Sun of Venus)
Marriage, pure feelings. (Mercury of Venus, Mars of Jupiter)
High wisdom, science. (Sun of Jupiter)
Magistracies or judiciaries, equity and fairness. (Moon of Jupiter, Venus of Mercury)
Finance, trading or business. (Sun of Mercury)
Maladministration or bad management. (Saturn of Mercury, Jupiter of Moon)
Family, sedentary or domestic life. (Sun of Moon)
Melancholy, weak health. (Mars of Moon, Mercury of Saturn)
Envy, sorrows, setbacks or reversals of fortune. (Sun of Saturn)
Birth, candor, inaction. (Mars of Saturn)
Infancy, playfulness or mischief. (Saturn of Mars)
Puberty, turbulence, quarrels or squabbles. (Mars of Mars)
Adolescence, sympathy. (Venus of Mars)
Intense or violent passions and senses. (Mars of Venus)
Debauchery, infidelity. (Venus of Venus)
Tenacious passions, constancy. (Jupiter of Venus)
Celibacy, marital fidelity. (Venus of Jupiter)
Moral perfection, maturity. (Jupiter of Jupiter)
Prudence or caution, good philosophy. (Mercury of Jupiter)
Bad faith, decline. (Jupiter of Mercury)
Illicit and perilous fortunes. (Mercury of Mercury)
Travel, hectic life. (Moon of Mercury)
Inconstancy, wasted or lost time. (Mercury of Moon)
Ancestors, old age. (Moon of Moon)
Apathy, waning of fortune. (Saturn of Moon)
Infirmity, indigence. (Moon of Saturn)
Ruin, death. (Saturn of Saturn)

The list of meanings above makes sense, given the structure of the Great Mirror and these “fractal directions” that allow for different planets to overlap their meanings. According to ZT’s own admission, however, the table given above is intentionally limited and limiting:

Be extremely careful to not take the Table that follows for a fixed indication of the significations of each of the 37 boxes from which the Great Mirror is composed. This Table is only a vehicle by which the Candidate should orient themselves, especially in the approaches which have as their goal only the ordinary career of human life.* However, if the Candidate has retained well all that we have established as precepts up until this point, then they will soon regard this Table as of little use, since there is not a single piece of the Great Mirror which does not modify, either for weal or for woe, the box assigned to it—and here we say “modify”, not “distort”.

* It will be seen, for example, that this table would furnish nothing to whoever would occupy themselves with the future destinies of empires, nations, &c.

What ZT is saying is that, even though it gives this table of contextual and semantic meanings for each of the 37 houses, it’s meant for illustrative purposes only as regards an individual human’s life, and as such, the indications above aren’t really valid for whole groups of people, the governments of nations and state, companies or industries, and the like—because the indications of table above were generated using that “sub-planet of main-planet” approach only for the scope of an individual human. ZT, given that it is “only a key and not a treatise”, does not give tables for other scopes, but it gives us the means to come up with such tables using the same underlying method as this one.

For instance, say we’re in a battle with some army, and I want to know something about the tactics and strategy I should engage with in order to emerge victorious. Understanding the difference of “strategy” (overall battle plan) and “tactics” (individual steps + logistics), I would want to turn to houses 2 and 5. If we consider the table above, these two houses have the respective meanings of “strength/triumph/glory” and “treasures/gains of all kinds”, which…yeah, kinda work, I guess? But if we look at the planetary considerations, house 2 is both “Mars of the Sun” as well as “Mercury of Mars” (the planning and direction of battle, i.e. strategy), and house 5 is both “Mercury of the Sun” as well as “Mars of Mercury” (the attacks and drives of planning and plotting, i.e. tactics). By doing this, we can expand the indications of each house in the Great Mirror from the scant description given in ZT by understanding the overall method and then extrapolating from it as necessary and as befits a given situation we might be faced with.

Personally? I think this is a really ingenious and elegant system of dividing up a situation into its many different aspects based on particular considerations. Just like with the compound Numbers, a few basic principles are used on general ideas to produce a wide variety of specific ones. Of course, just like with the compound Numbers, this is a lot to take in all at once, or so it’d seem—but the trick behind it is that we don’t need to take it in all at once, but rather just need to understand the method behind the madness. While the table as given above is great for readings at the level of the individual human being, we yet have a method to expand on that to any level or field or context. That said, we’re not done talking about the Great Mirror yet, because there are a few more considerations we have to work through, first.

First, when it comes to drawing tiles to compose the Great Mirror, the process works much as we would expect with Tarot cards or runes: individual tiles are drawn from the Urn without replacement (i.e. a tile can only be drawn a maximum of once), and it is placed in the first available house in the Great Mirror, not skipping to any later house nor replacing the tile in any earlier house. While this makes obvious sense to us modern folk (you don’t take the first Tarot card you draw for a Celtic Cross spread and put it anywhere else but the first position, nor do you take any later card and swap it out with a card in an earlier position), I assume that ZT makes this point explicit because of how new the idea might have been and to reduce any chances of people “making their own fate” by fiddling with the order tiles come out of the Urn and thus how the Great Mirror ought to be composed.

As one reads through ZT, it establishes the rule that, even though there are 112 (or 113) tiles in the whole set used for divination, no more than 37 tiles are to be used in any given reading, because the large hexagram (i.e. the Great Mirror) has only 37 houses. However, that is not technically entirely true, because ZT also has the rule that the two Principles are never used in a Great Mirror. It’s not that they’re separated out from the Urn and can’t be drawn, but if one or both are drawn in the course of composing a Great Mirror, then they’re placed outside it entirely:

Sisamoro (the Good Principle) is placed at the zenith of the Great Mirror, at the top vertex of an equilateral triangle with the leftmost and rightmost corners of the Great Mirror. Senamira, likewise, is placed at the nadir of the Great Mirror, below it in the same sort of arrangement. ZT is, perhaps unsurprisingly, unclear on the exact signification of the Principles if they should appear in a Great Mirror, just that it makes such a divination super notable:

The presence of a Principle, whether one or both, imparts to the Great Mirror superlative properties, the development of which is not the responsibility of a Key. The Pure Spirit then must speak, or the student remains more embarrassed than enlightened by the intervention of these extreme influences; it is even worse when there is conflict [i.e. when both Principles appear]. On the contrary, the true Cabalist is never better served than by those effective extractions where Fate majestically reveals its most admirable decrees.

The only concrete advice that ZT gives us is this, along with what to note when considering when an Intelligence is drawn and put into the Great Mirror as well:

Let us observe at which junction in the laying out of pieces for a Great Mirror where a Principle or Spirit appears.

Let us pay great attention to the quality of two numbers by which an Intelligence, drawn from the Urn, follows and precedes, and also how, in the Great Mirror, such an Intelligence is surrounded, and whether it forms a full orbit in its placement or a truncated one.

That latter point is especially interesting when it comes to the Intellligences. If an Intelligence is drawn, then it has a meaning just like any Number tile, but it also forms a sort of incidental planetary house of its own, and thus the houses that surround it form a sort of accidental orbit—but if such an Intelligence appears in the outer belt of the Great Mirror, such an orbit will necessarily be “truncated” and, thus, incomplete. If such an accidental orbit is a full/complete one, then that might give an extra planetary consideration to each of the houses according to its “fractal directions”; if such an orbit is a truncated/incomplete one, then not all planets would get to be represented in such a way. It’s a really neat idea to play with.

Astute readers will note that I’ve avoided talking about the inclusion of the signs of the Zodiac in the Great Mirror. For the most part, the signs of the Zodiac don’t matter all that much for the overall indications of the houses. However—and we’ll get to this more in a later post—the signs of the Zodiac are used by ZT to relate to the various stages of life that one undergoes, starting with Aries as birth and ending with Pisces as death. Each of the sides of the zodiacal belt relate to one of the “six divisions of life” according to ZT (childhood, youth, adulthood, middle age, old age, senility), and so the signs of the Zodiac within them correspond to particular aspects of that growth (which is why house 20, given to Aries, also has “birth” in its indications, 21 “infancy”, 22 “puberty”, and so forth). Beyond that, however, ZT doesn’t really do a whole lot with the Zodiac here, although that doesn’t say that one couldn’t feasibly find some way to work it into the system (even if ZT might discourage doing so, given its anti-astrology bias).

One last topic to round out this discussion on the Great Mirror. Although ZT says that the planets are all equal and aren’t ranked among themselves in the planetary belt of the Great Mirror (the only planet with primacy being the Sun in the center), ZT also notes that it doesn’t have an account for why the planets are positioned on the Great Mirror the way they are: it notes that it does not appear to have anything necessarily astronomical about it nor anything that is particular astrological, either, just that it’s something that (it claims) is “of such antiquity sunk deepest into the darkness of the past; sub judice lis est [the case is still before the judge]”. Admittedly, I’m not sure where ZT might have gotten this planetary arrangement from, either. When it comes to hexagonal arrangements of the planets, one might be more inclined to recall the planetary hexagram…

…which is, of course, a development from the qabbalistic Tree of Life, like that of Athanasius Kircher in his Œdipus Ægyptiacus from 1652, and later used for any number of Hermetic or Western occultists who make use of the so-called “Kircher Tree”:

Of course, given how distant ZT’s own “Great Cabala” is from anything properly seen in kabbala of any sort, to say nothing of how much it would caustically say about established traditions anyway, I somehow doubt that this would have been an influence here along these lines. However, if we compare the qabbalistic planetary hexagram with the hexagram formed by the Great Mirror, we see the same planetary triangles (Saturn-Mercury-Venus, Mars-Jupiter-Moon), just with a different rotation/reflection applied.

That being said, wherever the pattern of planets here came from, there is a logic and order in it. If we proceed through pairs of the planets counterclockwise around the Great Mirror, we see two kinds of patterns arising of similar pairs and dissimilar pairs:

Similar pairs arise between Venus-Jupiter (the benefics), Mercury-Moon (the neutrals), and Saturn-Mars (the malefics). This has the result of making the horizontal rows of the Great Mirror form pairs as well: the middle row (Saturn-Jupiter) represents the greater planets (the greater malefic and benefic), the lower row (Mars-Venus) the lesser planets (the lesser malefic and benefic), and the upper row (Moon-Mercury) the neutral small planets.
Dissimilar pairs arise between Mars-Venus (male/female), Jupiter-Mercury (king/servant or philosopher/sophist), and Moon-Saturn (creator/destroyer or youth/elder). This suggests an awareness of the opposition of particular zodiac signs and extending that to the planets, e.g. how Mars rules Aries and Scorpio, which are in opposition to Venus-ruled Libra and Taurus.

As of this writing, I’m not familiar with any source that arranges the planets in the way ZT does; while ZT definitely has a logic that suggests a good awareness of basic astrological principles and zodiacal correspondences, I’m not sure if that’s enough to trace it to any particular origin, especially when such arrangements have usually been more magical than astrological. This is another of those unanswered questions I have, and it may be that this arrangement is unique to ZT. If you have any notion of where such an arrangement might have an antecedent or any similar leads for further research, dear reader, or if you spy any other insights or patterns in this arrangement, do let me know in the comments!

Clarifications on Terms for Symbols

October 29, 2015 By polyphanes in occult, philosophy, whining Tags: alphabet, correction, design, geometry, hexagram, pentacle, pentagram, runes, seal, sigil, symbol, terminology, tetragrammaton, writing 2 Comments

It’s a bit of a pet peeve of mine when people badly use terms in an occult context. To be fair, different traditions may use certain terms in particular ways that are specific to that particular tradition, which may or may not differ from normal use. Other groups treat some terms completely interchangeably when, strictly speaking, the terms signify different things. Generally, however, there’s not much rigor in how people use specific terms, and end up misusing them (through their own ignorance and confusion) or abusing them (to intentionally mislead or annoy others). I’d like to clear up a few things and offer some of my definitions for particular terms used in an occult context, this time focusing specifically on terms used for different types of symbols.

For any of these terms, “symbol” is the highest-level term I can think of for any of these following terms. If you’re not sure what kind of symbol a particular thing is, just say “symbol”. Everyone understands that. Not everyone understands what a particular person means by “sigil” or “rune”, however. Granted, these words are given with my personal definitions, and again, may not be those used by other traditions. However, for the sake of having a regular inventory of words with specific, unambiguous meanings, here’s how I use these particular things.

Glyphs are symbols used to indicate a basic thought or sound. In other words, a glyph is much like a written-down word. Individual letters communicate sounds; individual numerals communicate numbers; individual Chinese characters communicate sounds or concepts or words; the glyphs for the planets, zodiac signs, elements, and alchemical concepts communicate those things and only those things. Glyphs are essentially a generalized notion of a letter in an alphabet; they are characters in a writing system that includes letters, numbers, punctuation, labels, and so forth. Glyphs may or may not be used in an occult context; for instance, these words you’re reading right now are composed of glyphs (letters and punctuation of the English alphabet), but so is an astrological chart (the symbols used to denote the planets and Zodiac signs) or a computer science textbook (punctuation and numerals and diagrams to indicate logical connections or mathematical operations). Glyphs may be used one at a time (using the symbol for the Sun) or in combination with other glyphs (multiple letters to spell out a name).

Seals are symbols that are invented as a complete unit or are received from a spirit. Seals cannot be decomposed into more basic things, but are a whole unto themselves. They are symbols that are not generated according to a particular rule or composed according to sacred geometry. They are simply abstract symbols that refer to something. Importantly, especially in my own work, seals are “revealed” or given unto someone by a spirit or person to refer to themselves; seals are an abstract “body” to give an idea a graphical or visual form. Consider the symbols used to refer to spirits in the Lemegeton Goetia; these are not composed of more base units or other symbols, but are whole things unto themselves. These are seals, and often have no origin besides “this is what I was shown to use and has no rhyme or reason beyond that”. Seals are to constructed diagrams what barbarous words of power are to words in the dictionary; they may not have any communicable meaning that us humans can understand, but they work.

Sigils are symbols that are constructed according to a particular algorithm. Think of the standard way of creating a letter-based sigil according to Agrippa (book III, chapter 30) or as used in modern chaos magic, or like with my own shorthand system. Alternatively, consider the sigils used for the planets with their planetary intelligences and spirits from Agrippa (book II, chapter 22), which are lines drawn over the qameas of particular planets and playing connect-the-dots with the gematria values of individual letters of a name or word. Sigils are symbols created according to a defined set of rules (combine these letters, connect these numbers on this qamea, etc.). They are not always artistically made, although the algorithms used to generate a sigil may have some leeway for style and innovation. A painting may incorporate sigils, but a sigil is not made of pictures; a sigil is a geometric, abstract form composed or generated from glyphs.

Runes are letters of the writing systems used for Germanic languages prior to the introduction of the Roman script. In other words, runes are no more than letters of a particularly old style of European alphabet. These can be classified, generally speaking, into two families: the Scandinavian futhark (both Elder and Younger, together used between the 2nd and 11th centuries) and the Anglo-Saxon futhorc. There were medieval runes used in some astrological contexts, but generally runes stayed out of Hermetic and Western ceremonial stuff. However, a particular alphabet known as Darlecarlian runes was in use until the 20th century in a small province in Sweden, but this was certainly the exception to the historical abandonment of runic writing. There are other systems of writing and symbols that are runiform, such as Old Turkic and Old Hungarian, but these bear only a superficial resemblance to Germanic runes, and are not technically runes on their own as they belong to a different writing system, culture, and geographic area.

Pentagrams are five-pointed stars. That’s it. Nothing more than that. You can only really draw a pentagram one way, regardless of orientation.

Hexagrams are six-pointed stars . Again, nothing special here, but there’s a bit more complexity. The Star of David is nothing more than a hexagram composed of two overlapping equilateral triangles, which is what’s usually meant by “hexagram”. The unicursal hexagram is another type, though it’s not original to Crowley by any means; the mathematician Blaise Pascal depicts it in one of his works from 1639. The “elemental hexagrams” shown in the Key of Solomon (book I, chapter 3) are not, strictly speaking, hexagrams (with the exception of one); they are configurations of two triangles each that do not, necessary, combine to form a proper hexagon.

Pentacles are not stars. They are not necessarily pentagrams, nor are they necessarily hexagrams. Pentacles are more of a system of symbols that work together in unison for a particular goal; they are something usually, but not always, more elaborate than a sigil and are not necessarily combined in an algorithmic way. Consider the pentacles from the Key of Solomon (book I, chapter 18), or the Elemental Weapon of the Earth as used in the Golden Dawn, or the protective lamen with the pentagram and extra symbols used in the Lemegeton Goetia, or that used in the Heptameron of Pietro d’Abano. Pentacles are, essentially, the physical version of a graphic design composed of one or more symbols, often including letters and names, and arranged in a method more akin to sacred geometry than algorithmic combining or tracing. Pentacles are tangible objects, things you can hold and touch and wear. All pentacles are talismans, although not all talismans are pentacles. For instance, a talisman engraved in a circular stone may have the design of a fish surrounded by Hebrew words can be considered a pentacle, but a talisman of a stone fish with words engraved on it is not a pentacle. Pentacles are generally round, flat objects such as a circular piece of paper or a metal disc that have a design engraved, painted, drawn, or otherwise inscribed upon it as a graphic design of a system of symbols. Pentacles are not oddly-shaped things like carved statues or rings or wands, despite its talismanic properties or designs on them. Although the words “pentacle” and “pentagram” are related and were originally used interchangeably, the word “pentacle” started to be used for any magical talisman in the form of a pentagram or hexagram starting in medieval French. An alternate etymology combines this with an older French word for pendant, pentacol or pendacol, or something worn around the neck. Indeed, most pentacles are typically worn around the neck as lamens, which is probably the most correct use of this word in my opinion, but can easily be expanded to other (typically circular and flat) objects with a system of magical symbols inscribed upon it.

Tetragrammaton (more properly the Tetragrammaton) is another word for the four-letter name of God, Yod-Heh-Vav-Heh or Yahweh or Jehovah or whatnot. The word is Greek and literally means “the thing of four letters”. It is a title to refer to the sacred name of God, akin to the Hebrew haShem “the Name”, but is often used in Hermetic and Solomonic work as itself as a sacred name of God. However, this is nothing more than a word composed of individual letters; the word “Tetragrammaton” does not refer to any pentacle or other occult design.

Mathetic Exercise: Light-bringing Breath

September 20, 2015 By polyphanes in mathesis, occult Tags: apollo, breath, exercise, geometry, iamblichus, light, meditation, qabbalah 1 Comment

Today, while making an offering to Apollo, he (if you’ll forgive the pun) shed some light on a bit of mathetic practice. I’m still dusting off some of the tools and prayers I was working on, but he’s given me some ideas to work with. While he’s still pretty stoic and detached in his approach to me, he’s generously helped me begin the process of refining and applying some of the things in mathesis I’ve been wanting to develop. Of this, in addition to the usual daily mathetic stuff I would be doing, he’s given me a breathing exercise to do. I don’t know if you’ve seen this one particular GIF around the Internet recently, but it suddenly popped in my mind when I asked him what should be done.

So, I took that idea, hashed it out a bit with Apollo, and applied it. It fills a need I wasn’t aware I needed, but it makes complete sense in retrospect, especially with some ideas I’ve gotten from the late Neoplatonic philosopher and theurgist Iamblichus as of late.

Before we get into the exercise, though, let’s go over a bit of geometry. We all know the Tetractys, right? We all know the Tetractys.

One of the many mathematical interpretations of the Tetractys is that it can represent the first four dimensions of geometry, starting with with the zeroth dimension:

A single point, with neither length nor breadth nor depth. No measure, only location.
Two points, forming a straight line segment with length. With a line segment, we can identify an infinite line extending in two directions: forward and backward.
Three points, defining a triangle with length and breadth, together known as area. With a triangle, we can identify an infinite area (a plane) extending in four directions: forward and backward, left and right.
Four points, defining a tetrahedron with length and breadth and depth, together known as volume. With a tetrahedron, we can identify an infinite volume (a space) extending in six directions: forward and backward, left and right, up and down.

Dion Fortune in her Mystical Qabalah (chap. 28, para. 25) says as much, in more sephirotic terms:

The point is assigned to Kether;
the line to Chokmah;
the two-dimensional plane to Binah;
consequently the three-dimensional solid naturally falls to Chesed.

We can see this using the ten sephiroth of the Tree of Life by dividing it up into four groups of points: one (Kether), two (Chokmah and Binah), three (Chesed, Geburah, Tiphareth), and four (Netzach, Hod, Yesod, Malkuth). Such a diagram illustrates this idea of emanation in both a geometric way as well as a qabbalistic way.

In a Hermetic, Neoplatonic, Iamblichian, Pythagorean, or whatever sense, the Monad (a.k.a. Kether, the Source, whatever) is the fundamental principle that defines and underlies everything that exists. (Whether it’s a distinct entity/non-entity/process is something of a debate in the blogosphere and I don’t presume to get into it here.) Looking at the Tetractys, the Monad is the first step in the process of manifestation: from the Monad comes the Dyad, from the Dyad the Triad, and from the Triad the Tetrad. The Tetrad is what gives us body and form, but it comes from a higher principle, and that principle comes from a higher one, and so forth. If we really want to bring change from above down to here, we have to give it form in some sense, or we have to align some part of our being with the process of bringing power from the Source to us.

However, although four points identifies the minimal solid there can exist, we are far more than just four points. Yes, we are a combination of fire and air and water and earth, but not in equal measures, nor in a regular fashion. Our bodies are animal, but our spirit partakes in something of the Divine; in order to better make our lives and bodies more appropriate to interacting with the divine, we should try to induce a slightly different body in ourselves that makes ourselves more divine. For the Neoplatonic Iamblichus, this is the form of the sphere, the most ideal solid there can be, and the body of the heavenly entities. A sphere is not a tetrahedron, but they are both bodies. We don’t want to be content with a tetrahedron, as we’re already far too complex to abide in it, but we want to get to a sphere. In one sense, going from a tetrahedron to a sphere is nothing, after having gone and passed through the point, line, and shape in order to get a form; in another, going from a tetrahedron to a sphere is the most daunting thing of all, as we go from one point to two to three to four is one thing, but to go from four to an infinite number of points is daunting, to say the least.

In addition to all this, it’s known that part of the theurgic practices of Iamblichus involved a process of “light” and filling oneself up with it, which we can also see in other theurgic rituals, like that of the Mithras Liturgy from PGM IV.475-834. In that, we find the following:

Draw in breath from the rays [of the Sun], drawing up 3 times as much as you can, and you will see yourself being lifted up and ascending to the height, so that you seem to be in midair. You will hear nothing either of man or of any other living thing, nor in that hour will you see anything of mortal affairs on earth, but rather you will see all immortal things. For in that day and hour you will see the divine order of the skies…

…So stand still and at once draw breath from the divine into yourself, while you look intently…

The whole Mithras Liturgy is a spiritual astral travel-type of initiation, where one ascends into the heavens and deals directly with the gods and guards of heaven. However, important to this ritual is an act of ritualized breathing, where one breathes in rays of light or the breath of the divine, and in doing so changes or alters one’s nature or consciousness. This is also similar to the Howl of Orpheus rite I found a bit ago, with its own special type of breathing and bellowing. Breathing in divine light is not just the light of a particular planet or a star or fire, but to breathe in the light of the Divine itself, that of the Monad, the fundamental essence that undergirds all things that exist.

So, let’s put this all together into a coherent ritual, shall we?

While sitting or standing, breathe out completely, from the head to the toes, completely exhaling all breath from the lungs. Make a popping sound to expel all breath once the normal exhale is done.
Breathe in deeply from the toes to the head, picturing a point of Light in your heart. Hold the breath for four counts, then exhale completely, this time with a hissing sound towards the end. Maintain the point of Light in your heart as you keep your lungs empty for a few counts.
Breathe in deeply, picturing the point of Light in your heart splitting into two points, one at the crown of your head and one at the soles of your feet, connected by a line of Light rising from the feet, through the spine, to the head. Hold the breath for four counts, then exhale completely with a hissing sound. Maintain the line of Light in your body as you keep your lungs empty for a few counts.
Breathe in deeply, picturing the line of Light extending by another point from its middle into a triangle that envelops your body, aligned side-to-side through your body, with its base at your feet and its apex at your head. Hold the breath for four counts, then exhale completely with a hissing sound. Maintain the triangle of Light through your body as you keep your lungs empty for a few counts.
Breathe in deeply, picturing the triangle of Light extending by another point from its center into a tetrahedron, with its base at your feet and its apex at your head, completely enveloping you inside. Hold the breath for four counts, then exhale completely with a hissing sound. Maintain the tetrahedron of Light through your body as you keep your lungs empty for a few counts.
Breathe in deeply, maintaining the pyramid of Light around you, as you hold the breath, exhale completely, and hold your lungs empty several times. Silently call out to the Divine Source until you can form some sort of connection, until you can sense the Source of the Light that has been forming within and around you this whole time. Repeat this step until you have sensed it and formed a connection with it, then continue.
With the lungs completely empty, breathe in deeply, but this time, breathe in the Light from the Source. As you do so, picture the pyramid around you swelling up slowly, bulging at its sides, until it becomes the shape of a perfect sphere that completely surrounds you. Repeat this step until you can form a stable, perfect sphere of Light.
Once you’ve formed a stable sphere of Light that surrounds you completely, let your image of yourself dissolve and merge into the sphere, becoming one with it, letting the sphere become your entire body. Maintain this mental state as a form of meditation as long as desired.
When finished with the meditative sphere of Light, let the image of your body form from the Light within the sphere, maintaining the boundary of the sphere around you as a shield or shell. Exhale slowly with another popping sound to finish.

The Digital Ambler

Always Forward Between Heaven and Earth

Tag Archives: geometry

Unlocking the Observatory: Figures, Mirrors, and the Great Mirror

Clarifications on Terms for Symbols

Mathetic Exercise: Light-bringing Breath