#AtoZChallenge2015: Orthogonal

Orthogonal projectionThe word orthogonal conjures up memories of wonderful maths classes, many years ago, as well as more obscure readings, much later, of absconse topological subjects. Geometry was one of the great pleasures of my youth: yes, we are all different!

I quote from: Barile, Margherita. “Orthogonal.” From MathWorld–A Wolfram Web Resource, created by Eric W. Weissteinhttp://mathworld.wolfram.com/Orthogonal.html

“In elementary geometry, orthogonal is the same as perpendicular. Two lines or curves are orthogonal if they are perpendicular at their point of intersection. Two vectors v and w of the real plane R^2 or the real space R^3 are orthogonal iff their dot product v·w=0. This condition has been exploited to define orthogonality in the more abstract context of the n-dimensional real space R^n.

More generally, two elements v and w of an inner product space E are called orthogonal if the inner product of v and w is 0. Two subspaces V and W of E are called orthogonal if every element of V is orthogonal to every element of W. The same definitions can be applied to any symmetric or differential k-form and to any Hermitian form.”

For those of my readers so inclined, Wikipedia has an interesting article on matrices!

“In linear algebra, an orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e.

Q^\mathrm{T} Q = Q Q^\mathrm{T} = I,

where I is the identity matrix.

This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse:

Q^\mathrm{T}=Q^{-1}, \,

An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q*) and therefore normal (Q*Q = QQ*) in the reals. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the dot product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation or reflection. In other words, it is a unitary transformation.

The set of n × n orthogonal matrices forms a group O(n), known as the orthogonal group. The subgroup SO(n) consisting of orthogonal matrices with determinant +1 is called the special orthogonal group, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation.

The complex analogue of an orthogonal matrix is a unitary matrix.”

Image: orthogonal projection at http://english.rejbrand.se/algosim/visualisation.asp?id=orthogonal_projection

From the Klein Bottle to Shibari, and back again

  As a lifelong student of Topology it is tempting for me to claim that my recent discovery of the ancient art of knots and ropes stemmed from the same mathematical interest, the link being the gracious curves of the rope as it is shaped into pentagrams, and other lovely sinuosities. This would be a shameful lie, and I am not enough of a “faux cul”, as we used to say at college, to sully this – mostly – honest blog.

Topology is a magical (the contradiction here, is but all superficial) branch of pure mathematics, with wonderful real world applications, and some surprising constructs. Take the Klein Bottle. This, goes the definition, is a two-dimensional manifold – as you may have guessed already. Well… it’s always looked pretty much 3-dimensional to me, but then, combinatorial topology proves me wrong with such ease… Topology is the art of continuous deformation in the plane: that’s a better definition.

The maths on all of this is far from trivial – at least to this blogger. However you may go back to the classical Möbius ring to get my meaning… They say that the edge of the ring is topologically equivalent to a circle: what could be simpler?

But I have to come clean: what inspired me to do a bit of research, as one does, on Shibari, was not, initially, the abstruse, intricate and beautiful way knots can be tied, but the sheer eroticism of Japanese damsels in distress, whose pictures ornate specialised art galleries and, inevitably, afficionados’s blogs. Shibari is merely the preferred western name for Kinbaku, the “beauty of tight-binding”.

According to Wikipedia – in this, as in most things, an inexhaustible source of priceless – and thus free – information, “The aesthetics of the bound person’s position is important: in particular, Japanese bondage is distinguished by its use of specific katas (forms) and aesthetic rules”…

 I have to admit to a particular fascination with this genre. The use of soft ropes and bamboo sticks, the artful, eerie suspension of roped, naked and endlessly desirable creatures, appear to me such a blend of medieval barbarity and exquisite delicacy, that it titillates my writing imagination (I hear your laughter, dear reader!) Seeing a master at work is a great visual pleasure, in the slow, unravelling demonstration of skills, the helpless submission of the victim (?), the explicit or semi-hidden nakedness.

Rooted in 16th century Hojojutsu, and ancient Japanese martial art, itself part of the Budo school of unarmed combat, Kinbaku is a relatively recent art form, revived by Seiu Ito, a Japanese painter, born in 1882.

Now armed with (some) knowledge of the subject matter, I am building my own Klein Bottle full of wonderful knots and ropes, I have even started pinning!