2. V
Unlimited

3. Without end, but every point an be reached

6. Without end, and not every point can be reached?

7. Parallel axiom or parallel postulate
If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended infinitely, meet on that side on which the angles sum to less than two right angles.
Euclid
( )?
Plato ( ): Light beams are straight lines.

8. August Ferdinand Möbius (1790-1868)
German mathematician and astronomer
1816 Professor at Leipzig
1844 Director of the observatory
Started topology: which propeties of a surface remain unchanged when the surface is deformed?
1865 Möbius-band: a single-sided surface.

11. Max Bill

12. two Möbius bands
cut Möbius band = no Möbius band

13. Inversion of a circle
r = 1
b = 1/a
1/0 = 
1/ = 0

14. M.C. Escher

15. Giuseppe Peano ( )
1890 Professor at Turin
Inroduced the symbol 
Latino Sine Flexione (artificial language)
Peano-Axioms of natural numbers
Peano-curve (1890)
Infinite length, to every point the distance is less than e.

16. Dimension N = 1/R2 N = 1/R N = 1/Rdim = R-dim logN = -dimlogR
dim = -logN/logR
for R  0
dim = -log3/log(1/3)
dim = -log9/log(1/3)
= log3/log3 = 1
= 2·log3/log3 = 2

17. Helge v. Koch ( )
Swedish mathematician
Snowflake-curve: first fractal (1904)
Factor 4/3. Length between two points grows infinitely.

18. Helge v. Koch ( )
Swedish mathematician
Snowflake-curve: first fractal (1904)
Fractional dimension:
dim = -logN/logR for R  0
dim = -log4/log(1/3)
= log4/log3
 1,262
Factor 4/3. Length between two points grows infinitely.

19. Girard Desargues ( )
French engineer and mathematician
1639: Projektive geometry
All parallels converge to a point in the line of infinity.

21. Trinity College

22. Pietro Perugino: Fresco at the Sistine Chapel, 1482

23. Meteors

24. Hyperbola: y = 1/x
Sir Winston Churchill
( )
I saw, as one might see the transit of Venus, a quantity passing through infinity and changing its sign from plus to minus. I saw exactly how it happened and why the tergiversation was inevitable ...
but it was after dinner and I let it go!

25. Evangelista Torricelli
Paradox of the rotational hyperboloid y = 1/x
Evangelista Torricelli
( )

26. Volume:
Length: (ds)2 = (dx)2 + (dy)2 = (dx)2 [1 + (dy)2/(dx)2)] 
Surface: dA = 2prds > 2prdx

27. Paradox of the rotating hyperbola :
Infinite surface, finite volume
Paint outside is infinite, Paint inside is finite,
unless thin enough. forced to be thin enough.
Thomas Hobbes ( ), English philosopher and geometer, author of "Leviathan: the ideal state", and teacher of the Prince of Wales:
"To understand this for sense, it is not required that a man should be a geometrician or logician, but that he should be mad."

28. The Eternity Machine 1 revolution per second 16 gear wheels
gear ratio 1/7
716 = 33,232,930,569,601
1 revolution per MY

30. Appendix

31. Johann Schultz ( ), theologian and mathematician, professor at Königsberg, personal friend of Kant, proved the parallel postulate by means of infinite angular areas. Such infinite magnitudes were generally accepted at the end of the 18th century, in contrast to infinitely small quantities.
Sylvestre Franc Lacroix ( ), in an influential text book of 1799, qualified this method as the only strict one.
Adrien Marie Legendre ( ) delivered in 1833 a review of proofs of the parallel postulate for the French Académie des Sciences. He listed six, according to his opinion, strict proofs three of which used infinite areas.