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Miscel-

laneous

Hollow Earth Applet

Transformation by Circle Inversion

There are some people believing that we live in the interior of a hollow earth and that the universe itself lies within ("concave hollow-Earth hypothesis"). Cyrus Reed Teed proposed this idea in 1869.
There is indeed a coordinate transformation such that the interior of the Earth becomes "exterior" and the exterior becomes "interior": the transformation by circle inversion.

circle inversion
                      transformation

P' is the inverse of P with respect to the inversion circle, center O, radius R.

OP * OP' = R2

which follows from similar triangles: OP : R = R : OP'

Using rectangular coordinates formulae of the transformation from P(x,y) to P'(x',y') are:

circle inversion
                      transformation fomula


> name="Hollow Earth"



Instructions for use:

earth

To change the center of the inversion circle check the box "Earth Center" and click.

Earth radius
Check the box "Earth Radius" and click to set the radius of the inversion circle.
points
Check the boxes then draw a single point (A), or a line (AB), or a triangle (ABC) by clicking.
trace
Check the box to see the lines connecting the points.
circle
To draw a circle check the box "Circle Center" and click, then check "Circle Radius" and click to change the radius.
details
Select items from the "Details" menu.

motion
                animation
Two points (A, B) or a circle selected will enable the "Motion Options" menu. Choose "Run" to start the animation.

The uniform speed along the original straight line AB and along the circle is set to unity.


Some examples:

line circle
                arc
line > arc of a circle
circle arc
                line
arc of a circle > line
circles
circle <> circle
triangle
triangle
triangle

triangle
sum of angles
                triangle

The transformation is angle preserving.
Select the item "Triangle Angles" from the "Details" menu.
The angle sum (α+β+γ) of the transformed triangle is also 180°:
angles

horizon
The visible region of an observer at P is the white circle,
the circumference (horizon) passing the center of the "hollow Earth".

sky celestial
                sphere

The pink region of a geocentric sphere, as seen by an observer P, is transformed
to the blue region inside the Earth.
radius hollow
                earth
The ratio of the radii r/r is equal to the ratio OM/OM'.
(The image point M' of M is not the center of the image circle!)

line circle

If AB is inside the Earth, the length of the arc A'B' is greater than the length of the straight line AB.

velocity speed
speed
                velocity
The ratio of the velocities of the moving points is inverse to the ratio of their distances from the center O.

hollow earth
          shoe

A solid proof of the the Hollow Earth Theory:

the bend of my shoe sole.


Web Links

Inversion (Wolfram MathWorld)

Inversive geometry (Wikipedia)

Circle Inversion

Hollow Earth (Wikipedia)

Turning the Universe Inside-Out.

What Curiosity in the Structure: The Hollow Earth in Science

Inversion: Reflection in a Circle: What is it?

Turning the Universe Inside-Out

Cyrus Teed (Wikipedia)

Leben wir IN der Erde???

Leben wir auf  der Innenfläche einer Hohlkugelerde? (H. Diehl)

Roman U. Sexl: Die Hohlwelttheorie

Kuriose Theorie: Verkehrte Welt (SPIEGEL ONLINE)

Hohlwelttheorie (W. Lang)

Hohlwelt (EsoWatch)

Pseudowissenschaft entlarvt: Die Hohle Erde Theorie - Leben wir auf einer Kugelschale? (M. Haas)

History of the Hollow Earth Theory

Books and Articles
Johannes Lang: Die Hohlwelttheorie. Schirmer & Mahlau, Frankfurt 1938.

Norbert Treitz: Das Äußerste nach innen gekehrt. Spektrum der Wissenschaft  Dossier 5/10: Physikalische Unterhaltungen. Spektrum der Wissenschaft Verlagsgesellschaft, Heidelberg 2010.

Norbert Treitz: Das Äußerste nach innen gekehrt. Spektrum der Wissenschaft, Heft 2/2008, S. 46-49, Spektrum der Wissenschaft Verlagsgesellschaft, Heidelberg 2008.

Matthias Marschik (Hrsg.): Das Weltall, wie es wirklich ist - Dolan-Theorie, Turia & Kant, Wien 1999.

Dianne Robbins: Messages from the Hollow Earth, Trafford Publishing, 2006, ISBN 978-1412005296.

© 2010-2023 J. Giesen

Modified: 2023, Oct 03