(Roughly) Daily

Posts Tagged ‘tesseract

“Men of broader intellect know that there is no sharp distinction betwixt the real and the unreal”*…



Colored cubes — known as “Tesseracts” — as depicted in the frontispiece to Hinton’s The Fourth Dimension (1904)


During the period we now call the fin de siècle, worlds collided. Ideas were being killed off as much as being born. And in a sort of Hegelian logic of thesis/antithesis/synthesis, the most interesting ones arose as the offspring of wildly different parents. In particular, the last gasp of Victorian spirituality infused cutting-edge science with a certain sense of old-school mysticism. Theosophy was all the rage; Huysmans dragged Satan into modern Paris; and eccentric poets and scholars met in the British Museum Reading Room under the aegis of the Golden Dawn for a cup of tea and a spot of demonology. As a result of all this, certain commonly-accepted scientific terms we use today came out of quite weird and wonderful ideas being developed at the turn of the century. Such is the case with space, which fascinated mathematicians, philosophers, and artists with its unfathomable possibilities…

In April 1904, C. H. Hinton published The Fourth Dimension, a popular maths book based on concepts he had been developing since 1880 that sought to establish an additional spatial dimension to the three we know and love. This was not understood to be time as we’re so used to thinking of the fourth dimension nowadays; that idea came a bit later. Hinton was talking about an actual spatial dimension, a new geometry, physically existing, and even possible to see and experience; something that linked us all together and would result in a “New Era of Thought.”…

Hinton’s ideas gradually pervaded the cultural milieu over the next thirty years or so — prominently filtering down to the Cubists and Duchamp. The arts were affected by two distinct interpretations of higher dimensionality: on the one hand, the idea as a spatial, geometric concept is readily apparent in early Cubism’s attempts to visualise all sides of an object at once, while on the other hand, it becomes a kind of all-encompassing mystical codeword used to justify avant-garde experimentation. “This painting doesn’t make sense? Ah, well, it does in the fourth dimension…” It becomes part of a language for artists exploring new ideas and new spaces…

By the late 1920s, Einsteinian Space-Time had more or less replaced the spatial fourth dimension in the minds of the public. It was a cold yet elegant concept that ruthlessly killed off the more romantic idea of strange dimensions and impossible directions. What had once been the playground of spiritualists and artists was all too convincingly explained. As hard science continued to rise in the early decades of the twentieth century, the fin-de-siècle’s more outré ideas continued to decline. Only the Surrealists continued to make reference to it, as an act of rebellion and vindication of the absurd. The idea of a real higher dimension linking us together as One sounded all a bit too dreamy, a bit too old-fashioned for a new century that was picking up speed, especially when such vague and multifarious explanations were trumped by the special theory of relativity. Hinton was as much hyperspace philosopher as scientist and hoped humanity would create a more peaceful and selfless society if only we recognised the unifying implications of the fourth dimension. Instead, the idea was banished to the realms of New Age con-artists, reappearing these days updated and repackaged as the fifth dimension. Its shadow side, however, proved hopelessly alluring to fantasy writers who have seen beyond the veil, and bring back visions of horror from an eldritch land outside of time and space that will haunt our nightmares with its terrible geometry, where tentacles and abominations truly horrible sleep beneath the Pacific Ocean waiting to bring darkness to our world… But still we muddle on through.

Hyperspace, tesseracts, ghosts, and colorful cubesJon Crabb, Editor, British Library Publishing, on the work of Charles Howard Hinton and the cultural history of higher dimensions: “Notes on the Fourth Dimension.”

[TotH to MK]

* H.P. Lovecraft, The Tomb


As we get high(er), we might recall that it was on this date in 1946 that Al Gross went public with his invention of the walkie talkie.  Gross had developed it as a top secret project during World War II; he went on to develop the circuitry that opened the way to personal pocket paging systems, CB radio, and patented precursors of the cell phone and the cordless phone.  Sadly for him, his patents expired before they became commercially viable.  ”Otherwise,” Gross said, after winning the M.I.T. lifetime achievement award, ”I’d be as rich as Bill Gates.”

While Gross himself is almost unknown to the general public, he did achieve one-step-removed notoriety in 1948 when he “gifted” his friend Chester Gould the concept of miniaturized radio transceivers, which Gross had just patented.  Gould put it to use as the two-way wrist radio in his comic strip Dick Tracy.

200px-ALGROS2 source


Written by LW

November 20, 2019 at 1:01 am

“Geometry is not true, it is advantageous”*…



The tesseract is a four dimensional cube. It has 16 edge points v=(a,b,c,d), with a,b,c,d either equal to +1 or -1. Two points are connected, if their distance is 2. Given a projection P(x,y,z,w)=(x,y,z) from four dimensional space to three dimensional space, we can visualize the cube as an object in familar space. The effect of a linear transformation like a rotation

|   1  0    0    0      | 
R(t) =   |   0  1    0    0      |
         |   0  0  cos(t) sin(t) |
         |   0  0 -sin(t) cos(t) |

in 4d space can be visualized in 3D by viewing the points v(t) = P R(t) v in R3.

* Henri Poincare


As we follow the bouncing ball, we might spare a thought for Felix Klein; he died on this date in 1925.  A mathematician of broad gauge, he is best remembered for his work in non-Euclidean geometry, perhaps especially for his work on synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Program, which profoundly influenced mathematics.  He created the Klein bottle, a one-sided closed surface–a non-orientable surface with no inside and no outside– that cannot be constructed in Euclidean space.


A Klein bottle


225px-Felix_Klein source


Written by LW

June 22, 2014 at 1:01 am

%d bloggers like this: