Invisible Solids |

Invisibility is one of mans ancient dreams. A mysterious property which is found in many books and movies. Invisibility has been firmly placed in the world of fiction, but a certain limited kind of invisibility has now moved into the real world. I have discovered what I call "Invisible Solids", and have found 3 different types which are described in detail on this page.

What is an invisible solid? To the mathematicians here is a definition:

An invisible solid L is defined by
• L is a solid in R ^{3}
• • with volume >0 • • with a surface S = S _{F} ∪ S_{E} so that
• • • S _{F} (the faces) is regular
• • • S _{E} (the edges) has area =0
# Imagine L with parts of the S _{F} surface mirror coated.
# The special property L has to have to be invisible is... • there exists a vector v in R ^{3}, v ≠0
• so every light-ray that moves in direction v, and that hits L on a S _{F} surface, is reflected around on L and leaves L in direction v, so that the light-ray will coincide with one and the same straight line before and after interacting with L
# The rest of this page is rather informal, but the mathematics behind it is straight forward, so it should be easy to figure it out. |

To all of you that think this is jabberwocky (mathematicians included) - see the pictures below.

And yes, it makes sense to show pictures of invisible solids :-)

The type 1 solids are the simplest type. It is the only type where all the mirrors are plane surfaces. So if you want to build an invisible solid for real, this type is probably the best to begin with.

To the right of the picture are buttons to start some small videos showing simulations of the type 1 solid. There are two videos with mirrors where you can see how visibility and invisibility changes with point of view. When the invisibility works the best only the shadow can be seen! There are two videos without mirrors, to show the contrast.

To understand the principle of the type 1 solid, take the two prisms and view them from above - then they look like the drawing below. In the drawing the light-rays are horizontal. Those that hit the solid are reflected four times. After the last reflection they proceed as if they hadn't hit anything. If you cover the middle of the drawing with a piece of paper the light-rays could as well be straight lines all of them.

Imagine a straight line parallel to the light-rays. To an observer on this line, infinitely far away, the invisible solid will be... invisible!... I admit, that was rather theoretical. So imagine instead the line, parallel to the light-rays, that goes through the center of the invisible solid. The points that are close to this line, and not too close to the solid, are marked green in this picture.

To an observer in the green domain the solid will be practically invisible. Any background, even one that moves, will be seen through the solid. There will only be small unnoticeable deviances from the real background. The farther from the solid and the closer to the line, the smaller the deviances.

The invisible solids of type 2 and type 3 has similar green domains.

The formulas I have used to construct the type 1 solid in the pictures and videos can be seen in this graph.

The mirrors of the type 2 solid are parabolas.

A horizontal light-ray hits a parabola and is sent through the focal point to another smaller parabola. From there it is sent horizontally on to two parabolas that are the mirror image of the two first. The drawing shows how it works.

The formulas I have used to construct the type 2 solid in the pictures and videos can be seen in this graph.

If you put a vertical plane through the center of the type 2 solid, all the horizontal light-rays will be going horizontally through that plane. Because of that you can put in an extension.

If it is all dimensioned right you can put a smaller copy of the solid in the extended part. And a still smaller copy in that... and so on. In that way you can construct a fractal-like invisible solid. In practice however it will be unnecessarily complicated. Besides that every layer will add four reflections, and you really want to minimize the number of reflections.

Both the type 1 and type 2 solids used four reflections. The type 3 solid uses only three.

Judging from the simulations it is type 3 that gives the best invisibility. It is the type with the biggest "green domain".

A horizontal light-ray hits a parabola that sends it to a focal point on a plane mirror. From there it is sent on to a parabola which is a mirror image of the first. The drawing shows how it works.

The formulas I have used to construct the type 3 solid in the pictures and videos can be seen in this graph.

It is easy to make asymmetrical versions of the type 3 solids.

All the invisible solids, shown so far, has been in two parts. That of course is very awkward if you want to construct one for real. How do you change it to be one connected solid?

1) One method is to assemble the two parts with an invisible material, for example glass. (OK, that was cheap)

2) Another method is to assemble 4 invisible solids into one, as shown in the picture below.

It can be extended from a 2×2-grid to a larger grid, that doesn't even have to be rectangular.

3) You can also rotate your curve around the x-axis, and get a tubular invisible solid.

4) If you use type 2 solids, you can take two, extend the one and make a boxlike invisible solid.

I haven't built an invisible solid yet, so I don't know how well they work in real life; required tolerances, quality of mirrors, and the like.

**NB!** When I simulate methods 3) and 4) the invisibility doesn't work. But I am sure they should do. I am very interested to hear if anybody know why the simulations doesn't work.

I know of only one other invisibility technology. It is currently being developed at Tokyo University, in a laboratory under professor Susumu Tachi. It is called Transparent Cloak. It consists of a screen (or cloak) that is set in front of the object to be hidden. A picture of the background is then projected onto the screen.

As is evident, this technology differs completely from the invisible solids.

## Transparent Cloak |
## Invisible solids |

Uses a lot of complicated electronics. Requires electricity, expertise, faultlessness, maintenance and capital. | Is very simple. |

Makes it possible to modify the picture of the background. Information can be put in or objects can be elucidated. | Shows only the background. |

The observer has to be near a point for the invisibility to work. | The observer has to be near a line (but not too close to the invisible solid) for the invisibility to work. |

If the background is far from the screen the invisibility will be destroyed when the observer moves just a little bit. | The distance of the background doesn't matter. |

Anything covered by the screen is invisible. | The object that is to be invisible has to be put into some awkward two-piece containers. |

The screen (or the cloak) can be made out of a flexible material. | The mirrors and the parts that hold them in place has to be rigid and nonflexible. |

The invisibility is limited to the few frequencies that are projected onto the screen. If you see the screen through a filter, that blocks these frequencies, the screen will not be invisible. | The invisibility consists of all the frequencies that the mirrors reflect. This is a broad spectrum and not a few narrow bands. Every reflection "removes" a bit of light however, so the fewest reflections gives the best invisibility. |

Here you can read a more formal mathematical treatment of invisible solids.

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