Current TOP Formula System v5

Aims of the TOP Formula System

● To make sure the system is comprehensive while having room to grow.

● To make formulae as easily recognizable as possible.

● To help clarify and specify relationships between entities, nearby and distant.

This document is in two parts:

Part 1: The System

Part 2: Explanation of Letters and Numbers

Part 2: Rules and Guidance in Constructing Formulae

Symbol System

Structures: Contents of Domains

Name Symbol Properties Number following
Root R The Root Domain: that contains/emanates the Primary Domains None
Primary P Indicates a structure within the 7 Primary Domains. Superscript numbers indicate whether secondary, tertiary, or quaternary.
Hierarchy H Hierarchy: There are many varieties, indicated by prefixes. 1-7
Level L Level: Creates hierarchies of various sorts. Levels may be numbered using Arabic numerals (1-7, 1-28) or Greek letters (α, β, γ, δ) in 4-level Style Hierarchies.
Grouping G Grouping of Levels: These define requirements in Structural Hierarchies, and specify the Levels in the Tree that is subsequently developed. 1-7
State f State created by the Structural Hierarchy requirement. 1-7

Trees: Dynamics within Domains

Name Symbol Properties Number following
Tree K All holistic hierarchies form Trees. None
Centre O Centres are formed by applying the dynamic duality to Levels. 1-10: Numbered inversely to Levels with 1 at the top and 10 at the bottom.
Channel c Channels defining influence or force that can and should exist between Centres. 1-22

Typologies: Preferences/Controls

Name Symbol Properties Number following
Type T Principal Type found in the Secondary Hierarchy (Principal Typology) which is the source of the Principal Controls. Also used in the Root Domain. 1-7
Q-expansion Q Refers to creation of Arenas in a Domain, and also subsidiary types. 1-7
Subsidiary Type t Subsidiary type defined by the Q-expansion which creates Arenas of Subsidiary Control. 1-7
Style Hierarchy M 4 Level hierarchy which generates the Q-expansion from 7 levels to 28 levels.  
Duality D Bipolar tension of two sorts: simple or complex (i.e. with internal 4:3 structure)  
Typology Essentials Table TET Table created by an Executing Duality, and the basis for drawing a Spiral None

Spirals: Time and Social Control

Name Symbol Properties Number following
Spiral C Time and effort dependent increase in strength of the Means to meet the Need of the Domain. 1-7
Mode μ Context of the Type: but numbering differs from Type No. to reflect the sequence need to strengthen. 1-7
Stage φ New ethos that develops due to a new Mode that emerges. 1-7

Dualities: Tensions

Name Symbol Pole Symbols* Meaning of Poles Internal Structure
Dynamic duality kD S - P Situational (socio-physical environmental-given) v Personal (self-willed-chosen) Two realities of psychosocial existence: no structure.
Executing duality eD X - Y Psychosocial Context of the Field 2 usual Axes of a 2x2 Table: has 4 zones but no internal structure.
Unfolding duality uD G - F Ground state (core) v Force for development (potential) No structure but poles interact.
C at next higher level is fusion of C and G at the lower level.
Approach duality aD a - d Apollonian (bottom left to upper right) v Dionysian (bottom right to upper left) Two diagonal sets of Types:
4 in a; 3 in d.
Internal duality iD b - u Content-Context i.e. Below v Upper Internal levels are a Style Hierarchy: 4 in b; 3 in u.
Oscillating duality oD o - e Odd-numbered vs even-numbered Two sets of things: 4 of o; 3 of e. (reversible in PHs)

*The pole is always placed as a subscript after the Duality symbol

Letter - Symbols

Letter Position Meaning
a prefix approach, qualifying a duality
a subscript Apollonian pole within an approach duality : lower right to upper left diagonal.
α suffix first (lowest) level within a Style Hierarchy
B subscript balanced, qualifying a centre and implying fusion of a dynamic duality
b subscript Bottom/below: content pole of the internal duality
β suffix second level within a Style Hierarchy
C   spiral trajectory of growth
c   channel, in a tree
D   duality
d subscript Dionysian pole within an approach duality : lower right to upper left diagonal.
δ suffix fourth (top) level within a Style Hierarchy
e prefix executing, qualifying a duality
e subscript even-numbered level quality in an oscillating duality.
F subscript force for development pole (potential) of an unfolding duality
f   State : created by requirements in a structural hierarchy
G   grouping (i.e. level) forming a structural hierarchy
g   group (i.e. level), stratifying a grouping and defining a quality.
G subscript ground state  pole (core) of an unfolding duality
γ suffix third level within a Style Hierarchy
H   hierarchy
i prefix internal, qualifying the 3:4 division (duality) of a Tree or Spiral
i subscript intermediate:  qualifying μ1 at the end of Cycle-1 in a spiral
K   to indicate a tree created from a holistic hierarchy or a dynamic tree level (KL) rather than a static hierarchy level (plain L)
κ   kappa: to indicate a Tree Level in any Domain (may not be required)
k   dynamic (kinetic), qualifying a duality
L   level in a hierarchy, but in a structural hierarchy use G
M   indicates a Style Hierarchy or Style Level
m subscript maximum, qualifying first phase in a spiral at the end of Cycle-2.
μ   Mode version of a Type and key feature in a spiral.
O   centre, an element of a tree and derived from a level
o prefix oscillating, qualifying a duality
o subscript odd-numbered level quality in an oscillating duality
P   primary domain, and qualifying entities in the domain (e.g. Hierarchy, Spiral)
P subscript personal: a dynamic duality pole and forming a centre
φ   Stage in a Spiral trajectory: also refers to ethos i.e. the cumulation of modes at a particular Stage
Q prefix modal expansion of a typology hierarchy, qualifying spiral, typology, hierarchy.
q suffix quality of an entity (eg duality diagonal; or internal level in groupings)
R   root domain, and qualifying many entities within the Root domain.
S subscript social -environmental: a dynamic duality pole and forming a centre
s subscript spontaneous starting point: minimum automatic form of the first Mode of a Spiral. (This may not be needed.)
s prefix structural, qualifying a hierarchy
T   principal type in a secondary hierarchy.
t   subsidiary type, emerging from Q-expansion of a typology level or root hierarchy level.
U Subscript Upper : Context pole of the internal duality
u prefix unfolding (or progressive), qualifying a duality
v   between duality poles = versus
X subscript X-axis: pole of executing duality
Y subscript Y axis: pole of executing duality
Z   matrix

Numbers & Number Sets


Identifier Position Meaning
Cardinal suffix = level in a hierarchy
Greek suffix = level in a mode
Cardinal superscript = group within a grouping; or order of hierarchy.
Cardinal subscript = level within a group
After P or R Superscript Numbers indicate the nesting pattern i.e. 2 = secondary hierarchy; 3 = tertiary hierarchy; 4 = quaternary hierarchy.
After PH or RH
(website only)
Superscript ' = secondary hierarchy; " = tertiary hierarchy.


Group Size Location in THEE
1’s The Root; any Framework on its own; Heptads in structural hierarchies.
2’s Dualities of all varieties; Hexads in structural hierarchies
3’s Pentads in structural hierarchies
4’s Tetrads in structural hierarchies; Modal expansion of each level/system in a typology
5’s Triads in structural hierarchies
6’s Dyads in structural hierarchies
7’s Key hierarchies (ie root, primary, structural, Q-, tertiary); Typologies; Stages in spirals
10’s Centres in Trees
1 - 22 Channels between centres in trees

Rules and Guidance in Constructing Formulae

■ Abbreviate formulae for simplicity when the context makes the meaning clear.

e.g.Closed during study of PH5, instead of using PH5L1 constantly, L1 may be sufficient; primes can be brought forward, so PH'5L4 would become L'4, and RH"L3 might become R"L3 or simply L"3.

■ Use hyphens to improve readability of formulae or to make an emphasis within them.

e.g.Closed PsH1 = P-sH1 = PsH-1 = structural hierarchy defined by the 1st primary hierarchy; similarly PH’1QC3-F2 is identical to PH’1Q-C3F2 and PH’1QC3F2, but the two formulae draw attention to the components differently.

Note: It is rarely necessary to use more than one hyphen.

■ Use “ • ” as a wildcard to represent the full range of possibilities.

e.g. Closed PH•L2 stands for 2nd level in all 7 primary hierarchies; Q•1 stands for all Q-structures of the 1st (lowest) form in an expansion of a typology hierarchy i.e. typology (Qt1), spiral (QC1), hierarchies (QH1 and QsH1) and trees (QHK1, QsHK1). Two wildcards are possible e.g. PH'•QC•μ2 = (all) Mode-2 entities in all Q Spirals of all Principal Typologies.

■ Indicate hierarchy by an “H”.

e.g.Closed RH = root hierarchy, sH = structural hierarchy, MH = Style hierarchy

Note:  Hierarchies usually have 7 levels, but MH has 4 levels. Q-expansion has 28 levels.

■ Place numbers after letters i.e. numbers qualify the letters immediately preceding them.

e.g.Closed L3 and not 3L; in PH2L4, the 2 refers to the PH and the 4 to the L.

■ Indicate contents of the Root Domain by an “R” prefix.

e.g. Closed RH = root hierarchy, RL3 = third level in root hierarchy, RM = Root modes.

Note:  R on its own refers to the Root as the entity (or cell) from which everything else unfolds. All formulae start with either R or P

■ Indicate levels of 7-level hierarchies (except structural hierarchies) by an “L” prefix

e.g. Closed RL2 = RHL2 = second level in root hierarchy; L6 = sixth level in a hierarchy; PH4-L1 = first level in the 4th primary hierarchy; PH"6L3 = third level of the 6th tertiary hierarchy (or the tertiary hierarchy in the 6th Domain).

Note:  Indicators of levels in other hierarchies (e.g. modal, structural) have different prefixes.

■ Indicate contents of the Primary Domains by “P” prefix.

e.g. Closed PH = primary hierarchy; PH26 = 6th Principal Typology.

Note: PH3 has the same formal name as RL3 , but the PH3 formula specifies a 7-level structured entity, while the RL3 formula specifies a particular level (domain) within the root hierarchy (root or will domain).

■ Indicate structural hierarchy by “s” prefix.

e.g. Closed sH = structural hierarchy; sPH3 = structural hierarchy derived from the 3rd primary hierarchy; RsH" = Tertiary Root Structural Hierarchy

Note:s” is commonly placed prior to the H it qualifies e.g. RsH = Root Structural Hierarchy; •QsH2 - structural hierarchy of the 2nd Q-Hierarchy. sPH• is replacing PsH• to harmonize but both forms are currently in use.

■ Indicate level in structural hierarchies or their Trees by “G” prefix (for grouping)

e.g. Closed G1 = first level/grouping in a structural hierarchy; PsH4G2 = second level/grouping in the structural hierarchy based on the 4th primary hierarchy.

■ Indicate grouping, i.e. number of adjacent levels combined to generate a structural hierarchy, by “G” prefix.

e.g. Closed G4 = the Tetrads (grouping of 4 adjacent levels) in a structural hierarchy; PsH2-G6 = grouping of six adjacent levels in the structural hierarchy based on the 2nd  primary hierarchy; RsHG2 = grouping of two adjacent levels in the structural hierarchy developed from the root hierarchy.

Note: Greek-based labels are used for groups of adjacent levels within a structural hierarchy e.g. Monad = level taken singly (i.e. 7 in all), Dyad = group of 2 levels (6 in all), Triad = group of 3 levels (5 in all), Tetrad = group of 4 levels (4 in all), Pentad = group of five levels (3 in all), Hexad = group of six levels (2 in all), Heptad = group of 7 levels (only one).

■ Indicate a particular group in a structural hierarchy by a superscript number placed after the grouping numeral.

e.g. Closed G-34 = 4th Triad;  PsH2-G61 = lower Hexad in the structural hierarchy formed from the 2nd primary hierarchy.

Note: Groups are counted in alignment with levels i.e. lower groups contain lower levels.

■ Indicate level within a particular group by a subscript number following the superscript in the group formula.

e.g.Closed G-342  = 2nd  level in the fourth Triad of the hierarchy under consideration.

Note: The same level in a hierarchy may have different formulae reflecting subtly different qualities of the entity when existing as part of a more complex entity 

e.g. Closed L3 is identified by all of the following:  G-131, G-222, G-313, G-322, G‑331, G-413, G-422, G‑431, G-513, G‑522, G-531, G-613, G-622, G-713.

■ Refer to a particular level within a number of groupings with a "g" prefix.

e.g. Closed g4 = fourth level in groupings within a structural hierarchy; it defines a common quality which requires distinctive identification for both practical purposes and formal testing.

Note: "g•" does not refer to a taxonomic entity because it specifies the quality of a variable number of cells within a structural hierarchy.

e.g. Closed g4 would exist in 10 groupings out of a total of 28, because there cannot be a 4th level in the 7 monads, 6 dyads or 5 triads. g4 refers to the 4th level within each of the 4 Tetrads, the 4th level within each of the 3 Pentads, the 4th level within each of the 2 Hexads, and the 4th level within the Heptad.

■ Indicate typology hierarchy (also called a ‘principal typology’, 'secondary hierarchy' or ‘secondary nested hierarchy’) by a superscript 2 after the P and before the H (handled in .xls files using a prime mark “ ' ”, after the L or PH).

e.g. Closed P2H2-L1 = first level in a nested hierarchy within the second primary hierarchy.

■ Indicate a Principal Typology Level as T for Type.

e.g. Closed P2H2-T1 = first Type in a nested hierarchy within the second primary hierarchy.

■ Indicate tertiary hierarchy by superscript 3 after the P and before the H (handled in .xls files using a double prime" ” after the L or PH).

e.g. Closed P3H1 = first level in a tertiary hierarchy; or 3L-1 if PH can be assumed; sP3H4-G3 = third grouping of a structural hierarchy formed from a tertiary hierarchy based on PH4.

■ Indicate quaternary hierarchy by superscript 4 after the P and before the H: probably only applies to P4H6 (handled in .xls files using “ º ”, after the L or PH).

■ Indicate a 4-level Style Hierarchy (as formed by expansion of a Root Hierarchy level or a level in a typology hierarchy into its modes) by “ M ”.

e.g. Closed RL3M = style hierarchy (4 levels or types) within 3rd level of the Root Hierarchy; 

P2H6-L4MH = style hierarchy in the 4th type or level of the 6th typology hierarchy.

Note: The root hierarchy is unique in that its levels are both holistic (i.e. 7 combinable levels) and discrete (i.e. each forms 7 levels) and capable of a modal expansions. RL6 also contains a nested typology hierarchy. The style hierarchy expansion (or Q-expansion) of a nested typology hierarchy generates 28 levels.

■ Number the four styles (i.e. levels in a Style Hierarchy) using lower case Greek alphabet: α, β, γ, δ.

e.g. Closed Ma = Style α ; RL5β = the 2nd style of the 5th level in the Root Hierarchy; L4γ = the 3rd style of the 4th system; P2H3L4Mα = 1st style of the 4th level/type of the principal typology within the 3rd primary domain.

■ Indicate a subsidiary type in the modal expansion of the primary typology, by "t". t is used (rather than L or M) when the reference is to a distinctive human identity or theoretical approach as distinct from the underlying function.

e.g. Closed P2H2QH5t6 = 6th subsidiary type in the fifth hierarchy within the modal expansion of the 2nd principal typology: so it corresponds to:  P2H2QH5L6Mβ

■ Label all 7-element structures (typologies, spirals and hierarchies) formed from the full modal (i.e. 28 level) expansion of a typology hierarchy with “Q” (for quasi-). The Q structures (Qt, QC, QH) are formed by taking 4 levels from one mode and adding 3 more consecutive levels from the next mode. At the 7th mode, this means adding levels from the 1st mode.

e.g. Closed P2H7QH4 = 4th hierarchy formed from the 7th primary hierarchy; PH21Qt6 =  6th subsidiary typology in the modal expansion of the nested typology within the 1st primary hierarchy/domain (i.e. includes MH6α-δ + MH7α-γ).  P2H3QC5 = 5th spiral formed from modal expansion of the typology within the 3rd primary hierarchy.

Note: Q hierarchies (QH) can have their levels influencing each other, leading to formation of trees and to derived structural hierarchies based on grouping adjacent levels i.e. they are holistic.

■ Indicate spiral of growth by “ C ”. C is for 'coil' because S has too many other associations (self, social)

e.g. Closed P2H1C = spiral derived from the 1st principal typology.

■ Indicate modes (contexts of the Types) in the spirals of growth by “μ”.

e.g. ClosedP2H4C-μ3 = third mode in a spiral from the 4th principal typology (primary domain).

■ Indicate stages or phases in the spirals of growth by “φ”.

e.g. ClosedP2H4C-φ3 = third phase in a spiral from the 4th principal typology (primary domain).

Note: Phase 1 is re-entered twice and hence requires 3 descriptors. These are indicated by subscripts: a for automatic, i for intermediate, and m for maximum  e.g. PH'4C-φ1i

■ Indicate the ethos (cumulation of modes) in the spirals of growth by “ε” (but usually labeled the same as the Stage (φ).

e.g. ClosedP2H4C-ε3 = third ethos in a spiral from the 4th principal typology (primary domain).

■ Indicate a duality by “D”: all D’s require qualification to indicate the type of duality. Sometimes, the duality has distinctive components as well. Qualify dualities as follows:

■ “oD” for oscillating dualities, whose poles are the odd versus the even levels in hierarchies.

e.g. Closed oD. If identification is required, use suffix o for odd and suffix e for even. So PH5oD = the odd-even alterations in the 5th Primary Hierarchy. PH5oDo = odd levels, PH5oDe = even levels.

■ “eD” for executing dualities, whose poles are the X and Y axes used to plot types, so X & Y are placed as subscripts.

e.g. Closed eD with its poles:  aDX vs  eDY . So RH’CeDX= X-axis of the Root Hierarchy Typology spiral.

■ “iD” for internal (or immovable/static) dualities which are found to split Trees or Spirals (but not hierarchies) into the above 3 levels (called "A") which are a controlling context for the lower 4 levels of content (called "B" for below).

e.g. Closed PH1KiDA = KL5, KL6, KL7 in the 1st Primary Hierarchy Tree. The Levels within the duality are related to the Style Hierarachy and are therefore labeled with the Greeks. e.g. PH1KiDAβ refers to KL6.

iD in primary spiral hierarchies turn out to be adjacent Root levels. In all Spirals, it corresponds to the Cycles i.e. Cycle-1 = B (Below), and Cycle-2 is A (Above). P2H1C-iDA refers to Cycle-1 in the 1st Primary Spiral, and P2H1C-iDAμ2 refers to the second step in Cycle-1 of the 1st Primary Spiral.

■ “aD” for approach dualities, whose poles are the diagonals which result from plotting types on an appropriate 2x2 graph defined by an executing duality. The distinction is between Apollonian and Dionysian orientations. So the poles are A-subscript for the diagonal going from near the origin to top right, and D-subscript for the diagonal running from top left to bottom right.

e.g. Closed aD  with its poles:  aDA  v  aDD. So P2H3-aDA =  diagonal running from bottom left to top right in the 2x2 table generated by the 3rd Principal Typology.  P2H1QH6 aDD = diagonal running from bottom right to top left in the 2x2 table generated by the 6th typology hierarchy formed combining Style Hierarchies of P2H1-L6 and P2H1-L7.

Each diagonal specifies a set of Types: principal or subsidiary: so P2H3-aDDT or P2H1QH6aDAt refers to those.

■ Types on each approach duality change in a particular quality as the diagonal is ascended or descended. By convention the ascending quality is chosen for definition. This quality is referred to by “q.

e.g. ClosedP2H3QC2aDRq

■ “uD” for unfolding dualities, whose poles are either the core (C-subscript) or the growth potential (G-subscript)

e.g. Closed uD  with its poles:  uDC  vs  uDG. So P2H3uD = unfolding dualities in the Nested Typology of the 3rd Primary Hierarchy. P2H3uD4C = core element of the duality of 4th Type in the Principal Typology of the 3rd Primary Domain.

■ “kD” for dynamic dualities which create distinct centres within levels 3, 5 & 6 in the major hierarchies: the generic sign for the dominant centre (shown on the right in diagrams) is M-subscript; and F-subscript is used for the other centre (shown on the left in diagrams).

e.g. Closed sPH7kD = dynamic duality for creating the tree in the Structural Hierarchy of the 7th Primary Hierarchy.
P2H4sQH2kD is the dynamic duality to create the tree from the 2nd structural Q hierarchy formed from the modal expansion of the nested typology of the 4th Primary Hierarchy. Dualities have a different feel at each level: so the two forms of the duality in L-6 of this entity would be shown as follows: P2H4sQH2-L6S  and PH24sQH2-L6P

■ Indicate a tree by “K

e.g. Closed PH2K is the tree generated by the second primary hierarchy; P2H4sQH2K is the tree defined by the structural hierarchy developed from the 2nd hierarchy revealed by the full modal expansion of the nested typology within the 4th primary hierarchy.

■ Indicate a centre on a tree by “O

e.g. Closed PH2KO1 is the single centre at 7th level of the Tree created from the second primary hierarchy.


i) The centres are counted from 1-10 starting with 1 in the single centre in the 7th level down to 10 which is the single centre in the 1st level i.e. 01 = L7.

ii) Where levels are already pre-defined there is an equivalence: so PH'5-sQH6KO5 is equivalent to PH'5-sQH6-L5P i.e. the less dominant form of the 5th level of the 6th structural hierarchy &c.

■ Indicate a channel joining tree centres by “c

e.g. ClosedP3H2Kc1 is the first channel from •O1 to •O2 in the tree P3H2K; and P3H2Kc22 is the last (i.e. 22nd) channel running from •O9 to •O10.

■ Indicate a matrix by “Z

e.g. Closed a matrix of entities is formed by plotting the entities revealed by the Q hierarchies against each other.

Note: possibly use conventional mathematical notation for this set.

Originally posted: August 2009; Last updated 12-Feb-2016.

All posted material is part of a scientific project and should be regarded as provisional. Visitors are encouraged to think through the topics and propositions for themselves. Copyright © Warren Kinston 2009-2016.
All Rights Reserved.

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