**The disdyakis
triacontahedron**

**The polyhedral counterpart of the Cosmic Tree of
Life**

The table displays the **15** polygons formed by the 60 vertices of the
disdyakis triacontahedron that surround an axis passing through two opposite C vertices (vertices of a
dodecahedron). They are discussed further in #11 and #12 at **Sacred geometry/Polyhedral Tree
of Life**. When they are Type B polygons, the centre of the disdyakis triacontahedron is
surrounded by **76** points that are either corners of triangles inside polygons or these
two C vertices — all of which are unshared with *both* polyhedron and polygons. This shows how
YAHWEH ELOHIM, the Godname of Tiphareth with number value **76**, prescribes this pattern of
points. The six red internal sides of the central hexagon are also sides of other internal triangles generated
by joining the centre of the polyhedron to its vertices. The sharing applies only to this
particular hexagon because none of the other polygons lies in the planes of internal triangles with the
centre of the polyhedron and its vertices as their corners. None of the sides of the **15**
polygons are edges of the disdyakis triacontahedron (see #11 & #12 at **Sacred geometry/Polyhedral Tree
of Life**). The second column indicates the number of sides of the triangles making up each Type B
polygon that are unshared with the polyhedron. The central polygon is a Type B hexagon. It has
30 sides of triangles, of which (30−6=24) are unshared with internal triangles formed by joining the
centre of the disdyakis triacontahedron to its vertices. ELOHA, the Godname of Geburah with number value
**36**, prescribes the set of **15** polygons (also prescribed by YAH, the
Godname of Chokmah) because they have (**76**+294=370=37×10) corners & sides, where 37 is the
**36**th integer after 1. They also have (294+180=**474**) sides & triangles,
where **474** is the number value of Daath. The number of corners/vertices & triangles =
**76** + 180 = 256 = 4^{4}, showing how the Pythagorean Tetrad expresses this
property. The table shows that the centre of the disdyakis triacontahedron is surrounded by 550 geometrical
elements in the **15** polygons and the polyhedron that are *unshared* with one
another. *This number is the number of SLs in the Cosmic Tree of Life!* We discover the remarkable
fact that each unshared geometrical element making up the disdyakis triacontahedron and the
**15** polygons formed by its vertices corresponds to an SL of CTOL! As the root edge of the two
sets of seven polygons making up the inner Tree of Life terminates at the positions of Daath and Tiphareth
projected onto the plane of the (7+7) enfolded polygons, the number 550 = **474** +
**76**, where **474** is both the number value of Daath ("knowledge") (see
here) and the number of sides & triangles and **76** is
both the number value of the Godname of Tiphareth and the number of unshared corners & vertices (see also
#9 at **Maps of reality/Cosmic Tree of
Life**). The number 550, therefore, expresses "knowledge of YAHWEH
ELOHIM (God the Creator)". CTOL is the domain of the manifested Divine Life, and its 550 SLs express the number
of its Sephirothic modifications. EL, the Godname of Chesed with number value **31**, prescribes
the disdyakis triacontahedron because each half has **31** vertices. ELOHA, the Godname of Geburah,
the next Sephirah after Chesed, prescribes it because a complete half of the polyhedron (including the
whole central hexagon) contains **36** vertices, where **36** is the number
value of ELOHA.

That it is not just a coincidence that the number of SLs in CTOL is the same as the number of
unshared geometrical elements in the **15** internal polygons of the disdyakis
triacontahedron is demonstrated by the facts that the eight polygons and C vertex in one half of it consist of 299
geometrical elements (see diagram above), whilst the seven remaining polygons and the other C vertex lying on the
C-C axis comprise 251 geometrical elements. Amazingly, they are the counterparts of the 299 SLs in the
**49**-tree mapping the cosmic physical plane and the 251 SLs in the remaining 42 Trees that map
the 42 subplanes of the six cosmic superphysical planes (see **Maps of reality/Cosmic Tree of
Life**). Just as we saw in that section that one half of the inner Tree of Life encodes
the **49**-tree and that the five largest polygons in its other half encode the remaining 42 Trees
of the Cosmic Tree of Life, so we find here that — in terms of the correspondence between geometrical
elements and SLs —a physical half of the polyhedron maps the cosmic physical plane, whilst its remainder maps
the six cosmic superphysical planes! Each unshared, geometrical element making up the **15**
polygons, whose planes are perpendicular to a C-C axis, corresponds to one of the 550 SLs of CTOL. *The
disdyakis triacontahedron is the single, polyhedral representation of the physical and spiritual cosmos.*
Any two diametrically opposite C vertices, which are north and south poles of the polyhedron, serve in this
representation as the nadir and zenith of CTOL, namely, its first and last SLs.

The encoding of CTOL in the disdyakis triacontahedron is discussed also in #7 at **Miscellaneous
wonders**.

The 550 geometrical elements that are unshared between the **15** polygons and
the disdyakis triacontahedron are composed of 180 triangles with **76** points and 294
lines, i.e., 370 points & lines. This exactly corresponds to the geometrical composition of the
**50** faces of the five Platonic solids because their 180 sectors have 370 corners & sides (see
the table here). This 180:370 division manifests in the 550 hexagonal yods composing the
dodecahedron as the 180 hexagonal yods within its 12 pentagonal faces and the 370 hexagonal yods that are either
on its edges or in its interior (see here). The fact that three different examples of "sacred geometry" have
geometrical compositions characterised by not only the *same* number but also its *same*
division into two other numbers strongly refutes the possibility of coincidence. Clearly, it is extremely
implausible that a number appearing supposedly by chance in three different contexts should display
*also by accident* the same pattern of division in each case!

Established here for the first time is an indisputable, mathematical connection between the
geometry of the disdyakis triacontahedron, the Theosophical doctrine of the seven cosmic planes of consciousness
and their division into the **49** subplanes of the cosmic physical planes and the 42 subplanes of the
six cosmic superphysical planes. Sceptics must ask themselves: how can coincidence (the only
alternative, conventional explanation available to them) be a plausible reason for why the very
*same* number 550 is *naturally* embodied in six sacred geometries, as demonstrated in this
section of the website and as summarised in the picture shown below?: