THE MOEBIUS AND THE KLEIN

The Möbius Strip is a good introduction to how strange and odd topology is. You can make a Möbius strip by taking a simple length of paper, let’s say 2 cm wide by 20 cm long, and twisting one end one half turn then joining the two ends together. The intriguing result is that the strip has lost one of its two edges and one of its two surfaces. Trace your finger around the surface and you will see that it is now a 2-dimensional creature with one side and one edge sitting very comfortably in our 3-dimensional world. Very cool. Now make another with the opposite twist and join them together along their edges to make a Klein bottle. Oops ….impossible in our world, you need to go to 4-space, the curious and enticing hinterland of topology.

Looking for 4-Space (Copy) — Looking for 4-Space

Möbius strips emerge from a 2D plane. Gracefully and sinuously they reach out through our 3 dimensional world seeking 4-space where they can unite. Two are seen starting to join to form a Klein bottle.
HB, 7B pencil 60 x 80 cm

Chopped in 4-space ( I will put up a better photo of this eventually!) (Copy) — Chopped in 4-space ( I will put up a better photo of this eventually!)

Here Möbius strips are trying to push their way through a barrier to reach 4-space. The barrier is shredding them. Chopped into pieces they drift off into an unknown dimension.
(The barrier is the triangulation of a Klein bottle, a rectangular instruction on how to form a Klein bottle.)
HB, 7 B and 9B pencil 60 x 80 cm

Seven Planes (Copy) — Seven Planes

Two pairs of Möbius bands drift in 3-space. Seven planes pass through them.
HB and 7B pencil 51.5 x 64 cm

Klein Half (Copy) — Klein Half

This sculpture is a single Möbius strip that is slowly deforming itself trying to form the Klein bottle….. Without its counterpart, its life partner, its goal is destined to remain unfulfilled.
The sculpture is a metre tall and is currently in plaster awaiting casting. The foundry can make it any size a purchaser may want and can cast in one of several choices of material. Please leave me a message through the Information Page if you are interested..

Half Klein (Copy) — Half Klein

This sculpture is a single Möbius strip that is slowly deforming itself to form the Klein bottle it believes it should be. Without its counterpart, its life partner, its goal is destined to remain unfulfilled.
The sculpture is a metre tall and is currently in plaster awaiting casting. The foundry can make it any size a purchaser may want and can cast it one of several choices of material. Please leave me a message through the Information Page if you are interested..

Unraveling (Copy) — Unraveling

A Klein bottle gently unraveling.
HB and 7B pencil on paper 30 x 40 cm

Seven Addled Klein Contours (Copy) — Seven Addled Klein Contours

This was drawn as if the pencil was taking a joyful tour around a Klein bottle in what ever direction that took its fancy. The 7 contours are the exact same shape to a topologist. Any shape that can be stretched and distorted in to another is the same thing as far as they are concerned. So a cube is the same thing as a sphere and a circle the same thing as a square.
9B pencil 30 x 30 cm

Just about to Join (Copy) — Just about to Join

Two Moebius strips with opposite twist can join along their edge and form a Klein bottle. But only if they have the 4-space to do it in!

Closing (Copy) — Closing

Is this the sensuous closing of two Möbius strips. They will need to be negotiate 4-space to complete the Klein bottle.
HB pencil 30 xx 20 cm

Beginning to Close (Copy) — Beginning to Close

Here two Möbius strips are closing to form a Klein bottle
HB pencil 30 x 20 cm

Dance (Copy) — Dance

Two inseparable Klein halves dance.
HB pencil 40 x 20 cm

Gentle Klein Doodle (Copy) — Gentle Klein Doodle

A gentle stroll around the surface of a Klein bottle with a soft pencil.
20 x 20 cm 9B pencil

Addled Klein Contour 3 (Copy) — Addled Klein Contour 3

Using pen and ink wash I took another tour about the surface of a very small Klein bottle.
8 x 8 cm

Addled Klein Contour 1 (Copy) — Addled Klein Contour 1

Pen and ink take a tour about the surface of a very small Klein bottle.
8 x 8 cm

Addled Klein Contour 5 (Copy) — Addled Klein Contour 5

The pen and ink take a tour about the surface of another very small Klein bottle.
8 x 8 cm

Gestating Swirl (Copy) — Gestating Swirl

Somewhere in the swirl is a Klein bottle forming.
Pencil and coloured pencil 20 x 30 cm

Inseparable (Copy) — Inseparable

This pen drawing is an outline of two inseparable halves of a Klein bottle .
Pen and ink 30 x 20 cm