Unlocking the Observatory: Ideal Triangles in the Great Mirror

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:

  1. 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.
  2. Let us appreciate what such a triplicity might mean, whether for good or ill.
  3. 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.
  4. 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)

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

Downwards (27 total)

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

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

  1. 2–20–22 (Sun-orbit, Saturn-orbit, Mars-orbit)
  2. 2–22–24 (Sun-orbit, Mars-orbit, Venus-orbit)
  3. 2–24–12 (Sun-orbit, Venus-orbit, Jupiter-orbit)
  4. 2–12–5 (Sun-orbit, Jupiter-orbit, Mercury-orbit)
  5. 2–5–18 (Sun-orbit, Mercury-orbit, Moon-orbit)
  6. 2–18–20 (Sun-orbit, Moon-orbit, Saturn-orbit)

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

  1. 3–8–23 (Sun-orbit, Saturn-orbit, Mars-orbit)
  2. 3–23–25 (Sun-orbit, Mars-orbit, Venus-orbit)
  3. 3–25–27 (Sun-orbit, Venus-orbit, Jupiter-orbit)
  4. 3–27–14 (Sun-orbit, Jupiter-orbit, Mercury-orbit)
  5. 3–14–6 (Sun-orbit, Mercury-orbit, Moon-orbit)
  6. 3–6–8 (Sun-orbit, Moon-orbit, Saturn-orbit)

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

  1. 4–7–10 (Sun-orbit, Saturn-orbit, Mars-orbit)
  2. 4–10–26 (Sun-orbit, Mars-orbit, Venus-orbit)
  3. 4–26–28 (Sun-orbit, Venus-orbit, Jupiter-orbit)
  4. 4–28–30 (Sun-orbit, Jupiter-orbit, Mercury-orbit)
  5. 4–30–16 (Sun-orbit, Mercury-orbit, Moon-orbit)
  6. 4–16–7 (Sun-orbit, Moon-orbit, Saturn-orbit)

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

  1. 5–18–2 (Sun-orbit, Saturn-orbit, Mars-orbit)
  2. 5–2–12 (Sun-orbit, Mars-orbit, Venus-orbit)
  3. 5–12–29 (Sun-orbit, Venus-orbit, Jupiter-orbit)
  4. 5–29–31 (Sun-orbit, Jupiter-orbit, Mercury-orbit)
  5. 5–31–33 (Sun-orbit, Mercury-orbit, Moon-orbit)
  6. 5–33–18 (Sun-orbit, Moon-orbit, Saturn-orbit)

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

  1. 6–36–8 (Sun-orbit, Saturn-orbit, Mars-orbit)
  2. 6–8–3 (Sun-orbit, Mars-orbit, Venus-orbit)
  3. 6–3–14 (Sun-orbit, Venus-orbit, Jupiter-orbit)
  4. 6–14–32 (Sun-orbit, Jupiter-orbit, Mercury-orbit)
  5. 6–32–34 (Sun-orbit, Mercury-orbit, Moon-orbit)
  6. 6–34–36 (Sun-orbit, Moon-orbit, Saturn-orbit)

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

  1. 7–37–21 (Sun-orbit, Saturn-orbit, Mars-orbit)
  2. 7–21–10 (Sun-orbit, Mars-orbit, Venus-orbit)
  3. 7–10–4 (Sun-orbit, Venus-orbit, Jupiter-orbit)
  4. 7–4–16 (Sun-orbit, Jupiter-orbit, Mercury-orbit)
  5. 7–16–35 (Sun-orbit, Mercury-orbit, Moon-orbit)
  6. 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)

  1. 8–12–16 (upwards)
  2. 18–14–10 (downwards)

Fully external (6 total)

  1. 1–37–22 (downwards, First Division, Aries/Taurus)
  2. 1–22–25 (upwards, Second Division, Gemini/Cancer)
  3. 1–25–28 (downwards, Third Division, Leo/Virgo)
  4. 1–28–31 (upwards, Fourth Division, Libra/Scorpio)
  5. 1–31–34 (downwards, Fifth Division, Sagittarius/Capricorn)
  6. 1–34–37 (upwards, Sixth Division, Aquarius/Pisces)

Partially external (12 total)

  1. 20–3–17 (upwards, Aries/Moon)
  2. 21–11–6 (upwards, Taurus/Venus)
  3. 23–4–19 (downwards, Gemini/Saturn)
  4. 24–13–7 (downwards, Cancer/Jupiter)
  5. 26–5–9 (upwards, Leo/Mars)
  6. 27–15–2 (upwards, Virgo/Mercury)
  7. 29–11–6 (downwards, Libra/Venus)
  8. 30–3–17 (downwards, Scorpio/Moon)
  9. 32–13–7 (upwards, Sagittarius/Jupiter)
  10. 33–4–19 (upwards, Capricorn/Saturn)
  11. 35–15–2 (downwards, Aquarius/Mercury)
  12. 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)

  1. 21–26–16 (Taurus/Leo)
  2. 20–12–33 (Aries/Capricorn)
  3. 8–27–32 (Virgo/Sagittarius)

Downwards (3 total)

  1. 35–30–10 (Scorpio/Aquarius)
  2. 36–14–23 (Gemini/Pisces)
  3. 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)

  1. 34–18–16 (all in orbit of the Moon, touching upper left corner)
  2. 8–22–10 (all in orbit of Mars, touching lower left corner)
  3. 14–12–28 (all in orbit of Jupiter, touching right corner)
  4. 35–19–6 (Aquarius and Saturn)
  5. 19–21–2 (Taurus and Saturn)
  6. 2–23–11 (Gemini and Venus)
  7. 4–11–27 (Virgo and Venus)
  8. 15–4–29 (Libra and Mercury)
  9. 33–6–15 (Capricorn and Mercury)
  10. 36–20–7 (Pisces/Aries in orbit of Saturn)
  11. 3–24–26 (Cancer/Leo in orbit of Venus)
  12. 32–5–30 (Sagittarius/Scorpio in orbit of Mercury)
  13. 17–7–5 (Moon)
  14. 7–9–3 (Mars)
  15. 5–3–15 (Jupiter)
  16. 18–8–1 (Sun)
  17. 1–10–12 (Sun)
  18. 16–1–14 (Sun)
  19. 6–2–4 (all in orbit of the Sun, surrounding the center)

Leftwards (19 total)

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

  1. 35–21–4 (Taurus/Aquarius)
  2. 23–27–6 (Gemini/Virgo)
  3. 33–29–2 (Libra/Capricorn)
  4. 34–8–14 (touching upper left corner)
  5. 22–12–18 (touching lower left corner)
  6. 10–28–16 (touching right corner)
  7. 17–13–9 (Moon/Jupiter/Mars)

Leftwards (7 total)

  1. 24–5–20 (Aries/Cancer)
  2. 30–26–7 (Leo/Scorpio)
  3. 32–3–36 (Sagittarius/Pisces)
  4. 31–12–18 (touching upper right corner)
  5. 25–14–8 (touching lower right corner)
  6. 10–16–37 (touching left corner)
  7. 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).

  1. 22–28–34 (rightwards, extremes of Mercury/Venus/Saturn)
  2. 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)

  1. 17–14–2 (Moon)
  2. 17–8–4 (Moon)
  3. 19–16–3 (Saturn)
  4. 19–10–5 (Saturn)
  5. 9–18–4 (Mars)
  6. 9–12–6 (Mars)
  7. 11–8–5 (Venus)
  8. 11–14–7 (Venus)
  9. 13–10–6 (Jupiter)
  10. 13–16–2 (Jupiter)
  11. 15–12–7 (Mercury)
  12. 15–18–3 (Mercury)

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

  1. 17–31–4 (Moon)
  2. 17–37–2 (Moon)
  3. 19–34–5 (Saturn)
  4. 19–22–3 (Saturn)
  5. 9–37–6 (Mars)
  6. 9–25–4 (Mars)
  7. 11–22–7 (Venus)
  8. 11–28–5 (Venus)
  9. 13–25–2 (Jupiter)
  10. 13–31–6 (Jupiter)
  11. 15–28–3 (Mercury)
  12. 15–34–7 (Mercury)

Sun + Two Zodiac Signs (12 total)

  1. 1–20–23 (Sun, Aries/Gemini)
  2. 1–21–24 (Sun, Taurus/Cancer)
  3. 1–23–26 (Sun, Gemini/Leo)
  4. 1–24–27 (Sun, Cancer/Virgo)
  5. 1–26–29 (Sun, Leo/Libra)
  6. 1–27–30 (Sun, Virgo/Scorpio)
  7. 1–29–32 (Sun, Libra/Sagittarius)
  8. 1–30–33 (Sun, Scorpio/Capricorn)
  9. 1–32–35 (Sun, Sagittarius/Aquarius)
  10. 1–33–36 (Sun, Capricorn/Pisces)
  11. 1–35–20 (Sun, Aquarius/Aries)
  12. 1–36–21 (Sun, Pisces/Taurus)

Small Single Zodiac (12 total)

  1. 35–8–5 (Aquarius)
  2. 36–16–2 (Pisces)
  3. 20–20–6 (Aries)
  4. 21–18–3 (Taurus)
  5. 23–12–7 (Gemini)
  6. 24–8–4 (Cancer)
  7. 26–14–2 (Leo)
  8. 27–10–5 (Virgo)
  9. 29–16–3 (Libra)
  10. 30–12–6 (Scorpio)
  11. 32–18–4 (Sagittarius)
  12. 33–14–7 (Capricorn)

Ideal Three-Move Skewed Triangles

Two Zodiac Signs (6 total)

  1. 20–24–5 (Aries/Cancer)
  2. 23–27–6 (Gemini/Virgo)
  3. 26–30–7 (Leo/Scorpio)
  4. 29–33–2 (Libra/Capricorn)
  5. 32–36–3 (Sagittarius/Pisces)
  6. 35–21–4 (Aquarius/Taurus)

Small Zodiac-Planet (12 total)

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

Large Single Zodiac (12 total)

  1. 35–31–3 (Aquarius)
  2. 36–22–4 (Pisces)
  3. 20–34–4 (Aries)
  4. 21–25–5 (Taurus)
  5. 23–37–5 (Gemini)
  6. 24–28–6 (Cancer)
  7. 26–22–6 (Leo)
  8. 27–31–7 (Virgo)
  9. 29–25–7 (Libra)
  10. 30–34–2 (Scorpio)
  11. 32–28–2 (Sagittarius)
  12. 33–37–3 (Capricorn)

Ideal Four-Move Skewed Triangles

Elemental Zodiac Groups (4 total)

  1. 32–26–20 (Aries/Leo/Sagittarius, i.e. fire signs)
  2. 33–27–21 (Taurus/Virgo/Capricorn, i.e. earth signs)
  3. 35–23–29 (Gemini/Libra/Aquarius, i.e. air signs)
  4. 36–24–30 (Cancer/Scorpio/Pisces, i.e. water signs)

Large Zodiac-Planet (12 total)

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

2 responses

  1. Pingback: Unlocking the Observatory: the Great Dial and Determining Times/Options « The Digital Ambler

  2. Pingback: Unlocking the Observatory: Summary and Recap « The Digital Ambler

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