Octahedron

Eight *lines *in *three-dimensional *terms references a Hexahedron. (Platonic Solids and other geometries when of equivalent status will be capitalized in the singular.) Consider lines to be relationships point-to-point and it would seem the Hexahedron relates to eight in a slightly indirect way if we think of points as more basic elements than lines.

Hexahedron

Eight *points *in *two-dimensional *terms references an Octahedron. Should we happen to think lines more basic than points, this would seem slightly indirect.

Oxahedron

For a variety of reasons, for now left unexplained, we shall not consider it a given that the point is the single basic unit. It is also so that whether we would choose Hexahedron or Octahedron as the geometry corresponding distinctly with eight, we would be able to construct one from the other, meaning neither corresponds *distinctly *with eight. For this reason, let us consider them as a group of two geometries which could be transformed into eight. Let us call this group the Oxahedron.

Let us consider Hexahedron/Octahedron the limits of eight and the Oxahedron to be eight proper.

OxaSphere; x^3

As a form an Oxahedron would be *represented *by a sphere, but a sphere in a particular process involving the limits Hexahedron and Octahedron.

x^4

At which point we need to revisit our friend four again. If the OxaSphere is involved in a process, we have introduced a fourth constraint beyond the spatial ones of geometry. The fourth constraint — ’the fourth dimension’ — applied is usually said to be time.

So, a self-similar geometry based upon Octahedron and Hexahedron and undergoing a process would need to be considered spherical. Further conclusions would be that it would have to be considered to be, at any point in time or space, *either *Hexahedral or Octahedral. We can construct either from the other, but the construction can *not *be continuous.

Presumably we should assume the sphere a perceptual mean value appearing where Octahedron and Hexahedron overlap so much that any differentiation falls into the realm of petite perceptions.

4×1 = 1×4

The conclusion that the Oxahedron is a mean value, the sphere an appearance, would imply that the ’fourth dimension’, time, resolves into a set of two identical-but-different two-dimensional geometries, presumably self-similar or even self-dual. The only known such kind is the self-dual Tetrahedron.

4 = 4

In four the square and the Tetrahedron are not distinct if we go by the number of points, similar to how in eight Hexahedron and Octahedron were impossible to tell apart. As we want to establish self-similarity and recursion, we shall see where it is maximized.

4×1

The square has four points and four lines connecting these four points. It does, however, only have one plane.

2×2 = 2+2

The Tetrahedron seems more promising. It has four points and four lines *and *four planes, and the center would seem to be in a self-similar relationship to the vertices.

1×4 = 4×1

*However*, the price for this is that the Tetrahedron is somewhat un-unique in that it has a self-dual Tetrahedron mirroring it. Defining one we have defined the other. We are in a situation similar to the one that forced us to create the Oxahedron liminal group and its corresponding OxaSphere. We have not eliminated the limits we sought to collapse in the Oxagroup, we have just moved them about.

Same same but notsame.same

One Tetrahedron equals another Tetrahedron. Identical in form, identical in center, but their limits — vertices and lines — are un-identical. Two constraints equal, two constraints differ. If we read only the product — or the sum — ’4’ they would look identical, *are *identical. But where one chops apples, the other shops oranges.

4≠4

Four does not equal four. The Tetrahedron is a group of two (limits) just as the Oxahedron was. It, too, would correspond with a sphere, the TetraSphere. As Tetrahedron *qua *group would be an indistinct designation, let us call the liminal group *TetraDuon.*

We are forced to accept what Aristotle could not accept.

A ≠ A

At this point it might be tempting to investigate whether not we could push the duality in the TetraDuon elsewhere. What about the point? If lines and bodies and planes do not work, what about points? The short answer is that if we push to the point, we would need to consider it *zero *numerically but *one *geometrically, *the *point being *one *geometrical object. If A does not equal A, no problem, but the conclusion will nevertheless once more have to be that moving a constraint is not to eliminate it.

An alternative like zero numerically and zero geometrically could be found in the Cartesian coordinate system, but we cannot accept this solution unless we also admit that this zero is defined as X orthogonal to Y, meaning we have only moved the limit onto the map. It is not gone.

Terra Incognita

We also would probably easily miss that not only is X orthogonal to Y, the opposite is also the case. That we consider X to be ’first’ is a convention, meaning the constraint is halfway off the map already. If we go further, we go off the map entirely, in which case its intersection of X and Y will be relatively irrelevant.

It is also very easy to see when looking upon the Cartesian coordinate system that it falls apart into four basic categories in a way corresponding quite well to how we see eight to be *either *a duplication of ’two times two’ *or *of ’two plus two’, the four domains XX, XY, YY and YX. The Tetrahedra of the Tetraduon would be described by different coordinates even if you disregarded their threedimensionality. They are absolutely incollapsible into one another.

With a far-shooting association back to the beginning, we noted then that there was a difference between numerator and denominator. For the denominator we also noted that it represented *either *multiplication *or *division. Then we took off into unknown lands, a journey that led us to the discovery that four is not four but which really means the opposite, that four is *four.*

Why is that? Was Aristotle right after all?

Apple-Maps

That four is, after all, *four *means this:

• Give me two apples two times and I have four apples.

• Give me two apples one time and two oranges the next and I have four *fruits.*

This is certainly trivial, but it is nevertheless essential. What it means in this context is that in four, categories collapse. Four has a generic ’quality’ to it. It is essentially meaningless to distinguish between apples and oranges for groups of fruit equal to or smaller than four. Groups of X — where X need not be identical but are relatively similar — will have an extremely limited number of combinations available.

Let us look at a nice fruit-list.

**Zero**

- No apple
- No orange
- No apples
- No oranges
- No apple oranges-mix
- No oranges apples-mix

Let us compare nice fruit-list one with nice fruit-list two, looking like this:

**One**

- One apple
- One orange
*No apples**No oranges**No apple oranges-mix**No oranges apples-mix*

It should be easily seen that list *Zero *and list *One *overlap in four of six possible manifestations. If we allow ourselves to define things by what they are not, which would seem permissible at least if zero is involved, absence/absences being how *zero *is defined, then it would seem that they are both non-plurals. It is also immediately obvious that there can be no other number non-plural.

From the point of view of singulars, letting singulars be the group of non-plural objects, it would seem not impermissible to consider the idea that the group of singulars could or even should contain only a single object. It would seem to be what the definition of *singular *tells us.

We may thus reconstruct the fruit-lists above in more generic terms, more *mathematical *terms, and in a singular unified list:

**Singulars**

- One non-plural
- One singular
- No plurals
- No combination of singular with plural

**One Zero, Zero Ones**

If the group of singulars contain two terms only, it would seem that we know that we have defined either by the other. We know that every singular digit in a binary string must be either zero or one. As the word singular is yet too easily understood as a reference to what is one in number, we shall think of the group under a name that hints at the possibly non-plural status of an individual singular. We shall think of this group as the *Zingular.*

**One or the Other**

If *either *is defined by *the other*, we can also think of either as *zero *and the other as *one. *It is always one *or *the other. If it is *one *or *the other*, and if *either is defined by the other*, it would seem that in relation to the other, meaning the ’generic other, that which is non-self’, both one and zero can oppose *the other *by being *either *zero or one.

Before deteriorating into word-games, let us conclude with something. This something is that just as as the Zingulars are either one or zero, they are also either specific or non-specific, never both. The group of Zingulars is a binary.

It is time for another list.

**’A’ versus ’The’**

- One zero, any zero
- One zero,
*this*one - One one, any one
- One one,
*this*one

This list contains as many items as the most recent one above, *Singulars*, where one and zero were defined against the plurals. *Singulars *and *Zingulars *contain the *same number *of objects and describe *the same objects*, suggesting we may posit a relationship between *number *and *object *for these lists.

These lists also seem to indicate that when you make a definition, any definition, you create a binary. Either the definition defines a something or other, or the definition defines a nothing. If we draw a square, we have marked out an area A, but we have also, implicitly, marked an area Everything *But *A, everything not mapped by A. The map idea now back on the map, let us return to maps, apple-maps and apples.

If *A *corresponds with *Everything But A *it is immaterial what A is, if *area *or *apple. *We are defining the nature of definition here, mapping its *meaning. *In this, all A are equal. All A equal A.

A = A

The *non-*apple map will define the infinite vistas beyond A; all apricots, bananas, cherries, dates, elderberries, figs, grapes, honey (hunny), guava, jujube, kiwi… and on so forth until we reach the yams, which are found in areas where the alphabet of fruit ends, the countably finite limit for fruit, beyond which is the uncountable, the infinite realm of z-fruits, uncountable, unreachable, indefinite, seemingly empty but possibly overflowing.

That which is non-A is thus not only not A in some mushy generic sense, it is also *everything specific, singular or plural, specifically *not A. It is the absence indicated by the presence of A.

But neither A, nor non-A, can stake any claims for what is beyond the map, beyond *all *maps, in the realm where the zitron, the zoonut (possibly not a fruit proper, but that would probably be true for all the z-fruits anyhow), the zebraberry and the zerozerofruit hang low from the trees.

**Fruits Mapped**

From the above it would seem that when we are talking about the Zingulars, we are not talking about the zitrons or the zebraberries. No definition can include among the binary Zingular fruits one single zitron, nor exclude one single zebraberry, or any other amount of them. Whether there be many or few of them we will never know, meaning also they are neither singular nor plural.

Nevertheless, if we want to speak of that which we do not know one whit, we must invent the words and the names that we need, together with others or on our own. How would we know whether else there would be anything worth to know among the *presently *unknown unknowns?

Without wit, without leaving unturned no stone, with *kein Stein umgewandt*, the precious one stone, *Das Ein Stein*, would be left in the unworlds, undiscovered, *un*unturning then the light-speed revolution of science and evolution of mankind towards ever higher, ever nobler, ever grander vistas and greater challenges.

**Dogma:**

Whereof one cannot speak, thereof one mustbe silent.not

Did we not talk about the unknowns, converse on the unicorns or Ur-Unicorn choosing between visiting the Un-Unicorn and the Uni-Unicorn — and did the funny bunny bring any hunny or is the bunny unhunny and unfunny? — there would be neither poesy nor philosophy, no mirrorful metaphorismeries, no necessary neologismeries, no phantasmagoric mesmeries, no revolutionary revelries. As the limit of the countably infinite is forever moving the borderland between countable fruits and uncountable fruits is never fixated. It is not *Terra Incognita *but *Mare Incognitum*, its lines written in the sand of endless seashores, islands and continents continually rising from it, returning to it.

The universe of discourse is unending, always expanding, or it is not *discourse. *An unexpanding universe of ’discourse’ would be a stiff and boring self-*identical *monologue, repeating itself over and over and over and over and over and over… *ad infinitum, *and — really — *ad nauseam. *Not even one of the infinitely many Nietzsches would be able to stand *that *eternal return, even less embrace it with love; *amor fati.*

On the contrary, the indistinct borderland invites us to travel beyond the known, to go off the map, to explore and invent and create, to build Crete, to speak creole, to weave the rainbows of tomorrow that we may find the golden pot of honey at the end of it, to chant for unicorns and be Kekules dreaming of benzene rings, to invent ukuleles or the calculus, to be a child turning stones and casting marbles, building castles in the sand upon the shore, moving ever forward, having a fun run for the hunny bun. That is;

To *play.*

We would not be *homo sapiens *were we not *homo ludens*, and if we were not allowed to speak about the beyonds, we would cease to be what we are; evolving towards the *future *unknown unknowns, which will be different than the present ones. Where now the maps are marked *hic sunt dracones*, scaring us that there may be dragons, monsters and heffalumps in the uncharted dephts there may instead be in the undeep waters heffalumps, undragons and unmonsters unfighting.

There may even be unicorns uniting.

And *hunny*.

Marks the SpotX

**One **X might on its own be no more than a generic point, in which case it is like X and Y intersecting on zero in a Cartesian coordinate center, meaning its value is zero on its own. Zero is *anywhere and anytime.*

Pointing on a zero-point will make the zero and anywhere-point one unit particular. One is *here. *If we go off the maps metaphor, one is either *here *or *now*, or alternatively *here*/*now. *To mark the point by pointing at it is to add a constraint, to *make *it one. Zero, thus, is in a one-to-one relationship with one. Or, if you prefer, a zero-to-one relationship. Center-zero opens the axes X *and *Y. *In toto*, one zero opens into two.

Why?X

**Two **X will show a line between two points or indicate a midpoint in the exceptional case, meaning two opens *more *than two doors. Two points as a duo is one thing, establishing a line between them introduces a new object, a midpoint upon that line is a third thing. Already in the line is an indefinite orthogonal line implied the midpoint of that line. And if one can point out the zero-zero coordinates of here, it is possible to assume pointing away from zero-zero, letting the pointing-constraint be direction; *there. *Anagrammatically, *there *becomes *three.*

The Power ofX

**Three **X will make triangulation possible. Where triangulation is possible, the world gets bigger than *here *and *there. *Three related points open an everincreasing number of points. Currently, as of the time of this writing, the *Encyclopedia of Triangle Centers *lists in excess of 5555 centers, and the number seems to be increasing with minimally one per day (mean value). Let these triangular centerpoints inter-reference one another and they shall soon escalate with 5555 to the power of X. Fittingly, the abbreviation for *Encyclopedia of Triangle Centers *is ETC.

In short, three opens the door to the *countably *infinite, to *everywhere.*

Fromto EternityX

**Four **X closes the three-door by showing us that there is a limited and exact number of *ultimate *limits. It is the point that closes the three-lettered abbreviation *etc. *The number of the maximal-minimal liminal is 4. It is also implied in the multiplying triangulated centers as the fourth point. Where there is a group of three, four will be its limit. The integer numerical recursion happens between these two numbers, meaning their common domain defines (defies?) all differentiation.

Four may be *center*, but it may also be *nowhere. *It may be here, or there or now or past or future also, the point here is that whatever the form it takes, that form represents limit. Zero is the open or opening door, the four is the closing or closed door. Three being the countably infinite, which can be reached at least in principle, four is the end of countability and the beginning of indefinite and noncountable infinities.

Four brings us back to the family of Tetrahedra.

IncognitaTetra

One other thing to note about the Tetraduon is that an individual Tetrahedron is not really or necessarily a reflected by a *particular *other Tetrahedron. The reflection can happen through any of its faces, meaning there are three reflections for one Tetrahedron. They are all distinct and we shall for that reason give them the needed distinct designations.

The numerological series continues *here*.