#boundary
AT THE BOUNDARY OF THIS AND THE OTHER WORLD by Fuco Ueda is printed in “Ephemeral - Territory of Girls” 2015 exhibition catalog. 28 pages of all female artists. Available for purchase at akatako.
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Making matrices better: Geometry and topology of polar and singular value decomposition
- Dennis DeTurck
- Amora Elsaify
- Herman Gluck
- Benjamin Grossmann
- Joseph Hoisington
- Anusha M.Krishnan
- Jianru Zhang
2017
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You don’t want to view matrices as 9 entries in an array. So you learn the inner product, rank, and the determinant. Then how do matrices look?
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The two components of O(3) appear as real projective 3-spaces in the 8-sphere, each the core of a open neighborhood of nonsingular matrices, whose cross-sectional fibres are triangular 5-dimensional cells lying on great5-spheres. The common boundary of these two neighborhoods is the 7-dimensional algebraic variety V⁷ of singular matrices.
This variety fails to be a submanifold precisely along the 4-manifold M⁴ of matrices of rank 1. The complement V⁷−M⁴, consisting of matrices of rank 2, is a large tubular neighborhood of a core 5-manifold M⁵ consisting of the “best matrices of rank 2”, namely those which are orthogonal on a 2-plane through the origin and zero on its orthogonal complement.
V⁷ is filled by geodesics, each an eighth of a great circle on the 8-sphere, which run between points of M⁵ and M⁴ with no overlap along their interiors. A circle’s worth of these geodesics originate from each point of M⁵, leaving it orthogonally, and a 2-torus’s worth of these geodesics arrive at each point of M⁴, also orthogonally.
Trix’s full family; only some appear in the comic and all very sparingly, but I love them all.
From left to right, stepbrother Steven, mom Jan, stepdad Jim, Trix, twins Dillon and Jaden, dog Manny.
The dog never appears in-comic, which is a tragedy frankly.
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2ha #13 is now available online.
Issue 13 considers the physical, legal, economic and symbolic borders which bind our everyday definition of suburban life. Three essays outline the contested nature of this space and the multiple means of separation made for the benefit of some, to the exclusion of others.
James O'Leary (Architect and Lecturer, UCL Bartlett) describes the origins of Belfast’s ‘Peace walls’, the shaping of the residential areas through which they run, and their continued impact on the development of the city.
Therese Kenna (Lecturer, UCC) discusses the shifting methods of boundary-making since the emergence of the modern suburb and how trends in design, urban governance, and the law are undermining the possibility of a shared metropolitan future.
Walter Greason (Dean of the Honors School, Monmouth University) highlights the conflagration of race, racism, urbanism, and economic development which persists, and is in some cases maintained by the chaotic application of zoning principles in the United States’ ex-urban edge.
For more information see 2ha.ie.
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