see things differently

Posts tagged with shape

one can basically describe each of the classical geometries (Euclidean, affine, projective,spherical, hyperbolic, Minkowski, etc.) as a homogeneous space for its structure group.

The structure group (or gauge group) of the class of geometric objects arises from isomorphisms of one geometric object to the standard object of its class.

For example,
• the structure group for lengths is ℝ⁺;
the structure group for angles is ℤ/2ℤ;
the structure group for lines is the affine group Aff(ℝ);

the structure group for n-dimensional Euclidean geometry is the Euclidean group E(n);

• the structure group for oriented 2-spheres is the (special) orthogonal group SO(3).

Terence Tao

(I rearranged his text freely.)

(Source: terrytao.wordpress.com)

Angle = Volume

This is trippy, and profound.

The determinant — which tells you the change in size after a matrix transformation 𝓜 — is just an Instance of the Alternating Multilinear Map.

(Alternating meaning it goes + − + − + − + − ……. Multilinear meaning linear in every term, ceteris paribus:

$\begin{matrix} a \; f(\cdots \blacksquare \cdots) + b \; f( \cdots \blacksquare \cdots) \\ = \shortparallel | \ | \\ f( \cdots a \ \blacksquare + b \ \blacksquare \cdots) \end{matrix} \\ \\ \qquad \footnotesize{\bullet f \text{ is the multilinear mapping}} \\ \qquad \bullet a, b \in \text{the underlying number corpus } \mathbb{K} \\ \qquad \bullet \text{above holds for any term } \blacksquare \text{ (if done one-at-a-time)}$ )

Now we trip. The inner product — which tells you the “angle” between 2 things, in a super abstract sense — is also an instantiation of the Alternating Multilinear Map.

In conclusion, mathematics proves that Size is the same kind of thing as Angle

Say whaaaaaat? I’m going to go get high now and watch Koyaanaasqatsi.

Logic, like mathematics, is regarded by many designers with suspicion. Much of it is based on various superstitions about the kind of force logic has in telling us what to do.

First of all, the word “logic” has some currency among designers as a reference to a particularly unpleasing and functionally unprofitable kind of formalism. The so-called logic of Jacques François Blondel or Vignola, for instance, referred to rules according to which the elements of architectural style could be combined. As rules they may be logical. But this gives them no special force unless there is also a legitimate relation between the system of logic and the needs and forces we accept in the real world.

Again, the cold visual “logic” of the steel-skeleton office building seems horribly constrained, and if we take it seriously as an intimation of what logic is likely to do, it is certain to frighten us away from analytical methods. But no one shape can any more be a consequence of the use of logic than any other, and it is nonsense to blame rigid physical form on the rigidity of logic.

Christopher Alexander, Notes on the Synthesis of Form