## The figure of the hypercube operates as a design principle underpinning the States and Territories project. A hypercube is a fourth-dimensional object consisting of two interconnected cubes, such that each cube is implicated in the process of becoming the other. Also known as a tesseract, a hypercube provokes the following set of propositions:

## 1.The hypercube both contains and projects itself as a singular object: a becoming-cube whose inside is continuously becoming its outside. A projection of a hypercube is a three-dimensional object in space, while a drawing of a hypercube is a 2D projection of a 3D projection. Rather than an object, a hypercube is an 'objectile' which propels itself across time and space through a kind of autonomous respiration which enacts continuous variations of matter and form (Deleuze, 1992). In this sense, the objectile becomes its own occasion within the unfolding of an event over time.

## 2. A projection of a hypercube can be constructed in two ways:

## a. figure a cube within a cube and connect the vertices to scale (proximal/scale-dependant)

## b. figure two identical cubes and connect the vertices regardless of scale (non-proximal/scale-independant)

## 3. In joining two cubes in the recursive feedback loop of a hypercube, six more cubes are produced: one cube for each of its six faces . A hypercube thus contains/projects a total of eight cubes (the original two plus six more).

## The Hypercube

States and Territories involves the production/projection of twelve hypercubes within an interactive material-semiotic architecture. The twelve hypercubes are linked together to form a wireless network topology that is designed to support context-aware, auto-poeic and intra-active learning environments. Each cube installed on the campus projects a second cube around the vicinity of its lines of site/sight, generating a feedback loop (or fold) between its interior and exterior surfaces. This creates the virtual space that participants can actually walk into, modify and expand through inscriptions, movements and other creative practices of activisation. Six additional cubic surfaces are opened up at each site through this collective process: six surfaces of the twelve cubic archives which map the cartographic becomings of each site. Data analysis then involves transversal re-codings across these surfaces, as initially undertaken by the research team and continued by public audiences once a data topology is established. The project then becomes an ongoing process of anarchiving, in which the archives are continually worked over, through and across to reveal interstitial connections and collectivities which are transveral to the horizontal and vertical axes of the data surfaces. In this way, the cartography is continuously extended into new existential territories through anarchival movements, inscriptions, gestures and constructions over time.

It is helpful to unravel the mathematics of a tesseract, which are the non-coincidental properties of a hypercube as it emerges dimensionally from a singular point of nonlocal space -time. The following proposition has proven to be consistent in allowing us to conceptualise the tesseract as a real object that exists in the fourth dimension and which projects itself, in the third dimension, as a cube within a cube:

The number of cells of dimension i(Ci) in a hypercube of dimension d(HCd) is causally related to the number of cells (of two types) in the hypercube of dimension (d-1)

Cid=2xCi(d-1) + C(i-1), (d-1)

In which: C=# of cells, i=size of cells, d=dimension

Such that: Cid [the number of cells of size i in a hypercube of dimension d] is equal to Ci(d-1) [cells of equal size in a hypercube of one dimension down] times two, plus C(i-1), (d-1) [cells a size lower and also in the next dimension down].

The upshot of this figuration is that the hypercube can theoretically be scaled into infinitely higher dimensions, with the dimension one step lower always a projection of the dimension above. A cube is to a hypercube what a point is to a line, and a line is to a plane. The tesseract beckons us from the threshold of our spatial awareness- we can't quite see it, but we can see its projection if we build a good enough simulation of it. This creates a space in which the virtual potential of an object can be actualised as an occasional event, like an interactive artwork (Massumi, 2011). We can trace the patterns of actualisation to the nth dimension using a conceptual/technical apparatus, but the tesseract is the object beyond which we lose the ability to project and thus visualise subsequent dimensions in spatial terms.

The video below discusses the hypercube in relation to States and Territories, speculative realism and the Anthropocene: