4 types of differentiations seem in force for nouns - at least in IE-languages as Swedish and English: case - gender / class - number - definite/indefinite form. (Fewer then than for verbs, perhaps possible to see as a consequence of nouns being of lower d-degree than verbs in the interpretations here?)
   Can these differentiations be related to different d-degrees? From some aspects such connections could look like this:

Cases:

Gender/class - number and definite/indefinite form are categories applied to the noun in itself, already separated from the sentence as a whole. Cases on the other hand, as nominative, accusative, genitive, dative, instrumental, locative etc. concern the functions of nouns in the whole sentence, designing their grammatical relations as subject or object, receiver or actor, position or instrument etc.

Case forms are in this sense derivable from an underlying level or higher d-degree than the other determinations. They are expressions for "syntax" as structure of the whole sentence. Syntax as step 4 - 3* operating on next level of word classes, here the nouns.
   Case marking includes relations between verbs and nouns, directions in relations as subject - object and functions within attributive and adverbial determinations.
   In the case forms there is more left of a "geometrical feeling" as some linguist wrote, and cases have been called a "deep structure". It's reason here to remind of the geometrical analysis of cases by Roman Jakobson, mentioned in first file.

* For a discussion on this view, see Syntax.

Regarding the differentiation in the roles of nouns as steps of displacements in a dimension chain, a development could be sketched from subjects to nouns as objects to more and more secondary roles in the sentences:

a) Primary subjects out of the situation of speech, speaker - addressee as poles 0 — 00 of d-degree 4 from step 5→ 4. Subject as a clause element.

b) With the geometry developed through step 4 → 3 the subjects are displaced to the nouns, a word class.

c) Through step 3 → 2 a further degrading of the nouns to objects and other oblique cases.

d) In following steps -2 →1 → 0/00 the roles of nouns get features of adjectival and adverbial, attributive roles, nouns as tools, means or pure material - or as nouns in time and space determinations.

The more and more complex geometry in the sentence expressed through cases and/or later through prepositions.

There are further differentiations within a certain category of cases - as a 3rd "fractal" level of development - which explicit features of underlying geometries:
   Uralic had originally 6 cases, divided 3 - 3 (BC,u): 3 for primary roles of the subject or noun as nominative, accusative or genitive, 3 for locatives. Compare in the dimension model d-degree 3 polarized 3a/3b into the complementary poles Mass and Space on the physical level.
   Further, in Uralic the 3 locative cases are polarized in 3 for something inner / 3 for something outer, which simultaneously have features of the opposition Mass-Space and corresponds to poles 2b/2a of d-degree 2 in our model, one geometrical definition of these being inside / outside.

The 3 locative cases of Primitive Uralic are described as below (d-degree figures added here)>

4 Translative - for direction…
3 Essive - for location, position; lasting, presence…
2 Partitive - for separated part of something or removal…

It's categories that easily may be related to d-degrees 4 - 3 - 2 respectively in a semantic sense, even if the use of those cases are said to have much more varying meanings.

Number of cases:
As for the number of cases in different languages, it looks like they originated from a dimension chain in the form 2x2 (!) - also the number series behind electron shells of atoms in the Periodic system.

25* in Hungarian, 16 in Finnish (from the 6 in Old Uralic),
2 in Indo-European, which became 6 in Latin, 5 in Old Greek, 4 in Old Swedish.
*(Statements about number in Hungarian vary: according to BC the number was 21 of which 18 were newer.)

Gender and Class adjuncts:

Class and gender determinations have a roughly adjectival character or function but of more fundamental kind than other adjectives, still built-in, not debranched. Hence here regarded as following from step d-degree 3 →2 → in this secondary dimension chain of nouns.
   They imply segmentation within nouns as such and may leave their marks on the word class adjectives as well. It's said that they also may affect verbs in some languages. Cf. verbs as radii, representing one pole of d-degree 3.

Bantu languages are said to have 12 class prefixes for living / not living nouns, different categories of animals etc. etc.
   The division of nouns according to geometrical forms in Chinese has been mentioned before: as in spatial, flat, spread out, long and narrow things etc., as of d-degree 3 - 2 - 1... respectively (!).

Number: Singular - Dual - Trial - Plural:

This kind of differentiation can mark both Pronouns, Verbs, Nouns and Adjectives.
   D-degree 1 makes up the foundation for numbers, one reason to regard them as a differentiation in last step of the dimension chain.
    On the other hand they reflect the fundamental principle between first poles 0 and 00 as the polarity units — multitudes in the model, which give its character to poles of all d-degrees. Further, d-degree 1 is represented by each step in the dimension chain, which may be a cause for numbers differentiating forms of all the four mentioned word classes.
   The dual may be regarded as an expression for the polarization principle as such, into two poles in the model.

In inflecting languages there are vowel exchanges in some words between singular and plural forms of nouns, as English foot - feet, Swedish bok - böcker, in similarity with tense changes in verbs.
    The common geometry could be identified as inward ~ backward direction towards past tense in verbs, and inward, convergent direction towards one unit (a 0-pole) in numbers; outward direction for present tense in verbs and outward, divergent direction towards multitudes (the 00-pole) in plural forms. (In the mentioned two examples the vowel for the unit is also an inner one, with tongue drawn inwards, the plural form an outer vowel.)

Repetition as expression for plurals replaces inflections or affixes in some languages, also sometimes acting as substitute for the conjunction and. In our model its hypothezised that the straight way 5 → 4 → 3 → 2 →1 → 0/00 could lead to repetitions (poles 1a/1b of d-degree 0/00 as oscillating motions "to and from each other"), while the loop model with opposite directions meeting in step 3-2 could be regarded as the way leading to development of new levels.

Ordinal numbers as first, secondary…are numbers appearing as adjectival attributes or in adverbials. This connects also numbers with the step 2 — 1 in the underlying dimension chain of word classes. (D-degree 2 leading to d-degree 1 is the only step in the dimension chain implying a mathematical halving.)

Dual forms disappear "with increasing soul cultivation" (sic!) according to Björn Collinder. Why not rather with a decreasing one?

Articles, definite - indefinite forms:

This differentiation can be regarded as another aspect on the polarity between 0- and 00-poles, the singular, individual thing or noun pointed out and the more diffuse, generalized concept as connected with multitudes. In terms of our model: "motions towards each other" (pole 1a) define a new centre or 0'-pole, "motions from each other" (pole 1b) define a new anticentre, a 00'-pole.
   In Uralic for instance the plural can in certain contexts be replaced by a kind of collective singular: a multitude and indefinite collective (as a circular enclosing concept; a circle as circumference and 00-pole in relation to the origin).

In a language as Swedish both article and definite form (coupled) express at the same time class (2 in Swedish) in singular. Hence, all the last three differentiation steps of nouns, class - number - definite / indefinite form, are combined, but only in singular and definite form (and article). Why?    Probably because class represents a deeper level, a higher d-degree, and as such a differentiation in direction from the 0-pole and therefore gets applied to units only in singular and definite form in direction outwards the 00-pole.

The articles or affixes for definite - indefinite originate always from demonstrative pronouns, implying something individual pointed out, in the development from inflecting towards analytical languages (PW). Hence the are coupled to pronouns as word class. (See about pronouns here.)
   There is implicitly a vector character in definite articles or affixes, and we have the 0- and 00-poles as "outer poles" or partial structures from the polarization 5 → 4, defining d-degree 4 and physically vectors in the model.

To unite all these aspects on the word classes with application of the dimension model, each step in a primary dimension chain is assumed as developed to secondary dimension chains - in similarity with fractals - and the result becomes a level chain.
   Word classes become levels in this chain.

An outline of such a level chain, with many exclusions:

Fig 1: The principle:

(Surely 3 and perhaps more coordinate axes would be needed to illustrate all the differentiations.)

Fig. 2: With differentiations abbreviated:

The scheme could also elucidate why many differentiations that seem to be connected with only a certain word class, also influence others - as vertically connected, as gender also may influence verbs and adjectives, cases both pronouns, nouns and adjectives and such things.

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To Pronouns - Prepositions