Geometrical Paradox

Curry's Paradox

A Chessboard Paradox

Rectangular Transformation Paradox (4 Pieces)

0

1

2

1

1

2

0

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2

1

+ , | ' , | ' , | ' , | | ' , x1 | ' , | | ' , | x0 ' , | | ' , | | ' , +------ x2-x0 -------+---------- x1+x0 ---------------------+

0

1

2

The classical parameters given by Sam Loyd are (x 0 , x 1 , x 2 , µ) = (3, 5, 8, 1), three consecutive Fibonacci numbers, in which case one rectangle is a square as x 0 + x 1 = x 2 . Other transformations are given in the table below.

x0 x1 x2 µ x1 × x1+x2 x0+x1 × x2 area1 area2 reference 4 5 6 1 5 × 11 9 × 6 55 54 [Walter Lietzmann] 3 5 8 1 5 × 13 8 × 8 65 64 [Sam Loyd] 5 6 7 1 6 × 13 11 × 7 78 77 4 7 12 1 7 × 19 11 × 12 133 132 5 8 13 -1 8 × 21 13 × 13 168 169 2 7 24 1 7 × 31 9 × 24 217 216 [Hermann Schubert] 4 9 16 1 9 × 25 14 × 16 225 224 5 11 24 1 11 × 35 16 × 24 385 384 [Torsten Sillke]

Rectangular Transformation Paradox (6 Pieces)

0

1

2

1

1

2

0

1

2

1

+ , | ' , | | , | | ' , | | ' , x1 | ' , | | ' , | 2*x0 | ' , | | | ' , | | x0 ' , | | | ' , +--- x2-4*x0 ---+------------- x1+2*x0 -------------+------------- x1+2*x0 -------------+

0

1

2

x 0 x 1 x 2 µ x 1 × 2x 1 +x 2 2x 0 +x 1 × x 2 area 1 area 2 reference 2 5 12 1 5 × 22 9 × 12 110 108 [Torsten Sillke] 3 8 21 1 8 × 37 14 × 21 296 294 [Torsten Sillke]

1

2

1

0

1

2

Cassini's Fibonacci identity

n+1

n-1

n

n

n [ 1 1 ] [ F n+1 F n ] n | F n+1 F n | [ 1 0 ] = [ F n F n-1 ] => (-1) = | F n F n-1 |

There is a geometric reasoning if we compare the two rectangles F n+1 × F n-1 and F n × F n . They both contain the rectangle F n × F n-1 . This means

F n+1 × F n-1 = F n × F n-1 + F n-1 × F n-1 and F n × F n = F n × F n-1 + F n × F n-2 . Their difference is

F n+1 × F n-1 - F n × F n = - (F n × F n-2 - F n-1 × F n-1 ).

2

1

0

A combinatorial interpretation of the relation of Cassini was published by N. Werman and D. Zeilberger.

Supplement Rectangles for Curry's Paradox

x x x x x x x x x x x x x x x + x + + x x x + + + x x x x x + x + + + x x + + + + + x x x + + + + + + + + + + + + + + + + x x + + x x x x x + + + x x x x x x x x + . x + + + . x x x + + + + + . x x x x x + + x + + + + + x x + + + + + + + + x x x

x x x x x x x x x x x x x x x x + x x x x + + x x x x x x x . + + . x x + + + + . x x x x + + + + + x + + + + + + + x x + + + + + + + + + + + + + + + x + x x x x x + + x x x x x x x x x x + + x x x x + + + + x x x x x x x + + + + + x x + + + + + + + x x x x + + + + + + x + + + + + + + + + x x

Euclid's Supplement Parallelograms

C-----------F-------------------I / / , / / / , / / / , / / / , / B-----------E-------------------H / , / / / , / / A-----------D-------------------G

If E is not on the diagonal then we have two quadrangles AGIE and ACIE. And area(AGIE) - area(ACIE) = area(DGHE) - area(BEFC) but this is not null. This explains the paradoxes. The 'traingles' are quadrangles.

Area of a Parallelogram

References

W. W. Rouse Ball, H. S. M. Coxeter;

Mathematical Recreations & Essays, Cambridge 1974, 12th edition

(1st ed., 1892, pp. 34-36. 8 x 8 to 5 x 13 and generalizations)

Chapter 3.2 Geometrical Paradoxes, 3rd Paradox

- 4 piece dissection showing 8*8 = 5*13 and Fibonacci generalization; (continued fraction convergents)

Mathematical Recreations & Essays, Cambridge 1974, 12th edition (1st ed., 1892, pp. 34-36. 8 x 8 to 5 x 13 and generalizations) Chapter 3.2 Geometrical Paradoxes, 3rd Paradox - 4 piece dissection showing 8*8 = 5*13 and Fibonacci generalization; (continued fraction convergents) Karl Ferdinand Braun;

Der junge Mathematiker und Naturforscher: Einführung in die Geheimnisse der Zahl und Wunder der Rechenkunst, Otto Spamer Verlag, Leipzig 1876

(Reprint: Geheimnisse der Zahl und Wunder der Rechenkunst, Rowohlt Verlag, Reinbeck 2000, ISBN 3-499-60808-1, p72-76)

Chapter 3

Der junge Mathematiker und Naturforscher: Einführung in die Geheimnisse der Zahl und Wunder der Rechenkunst, Otto Spamer Verlag, Leipzig 1876 (Reprint: Geheimnisse der Zahl und Wunder der Rechenkunst, Rowohlt Verlag, Reinbeck 2000, ISBN 3-499-60808-1, p72-76) Chapter 3 M. Busch;

Über einen geometrischen Trugschluß, Mathematisch-Naturwissenschaftliche Blätter 13 (1916) 89.

Über einen geometrischen Trugschluß, Mathematisch-Naturwissenschaftliche Blätter 13 (1916) 89. Stuart Dodgson Collingwood;

The Lewis Carroll Picture Book, T. Fisher Unwin, 1899.

(Reprint: Dover, 1961, with new title: Diversions and Digressions of Lewis Carroll)

The Lewis Carroll Picture Book, T. Fisher Unwin, 1899. (Reprint: Dover, 1961, with new title: Diversions and Digressions of Lewis Carroll) T. Devendran (editor);

Neues aus dem Mathematischen Kabinett, Hugendubel Verlag, München 1985

Section II.2: Fibonacci und die Bienen (Thomas Hartkopf, Siegfried Rösch)

(continued fraction convergents A/B and C/D with the constraint A+B=D) Shows Schubert's 7/24 to 9/31 example.

Neues aus dem Mathematischen Kabinett, Hugendubel Verlag, München 1985 Section II.2: Fibonacci und die Bienen (Thomas Hartkopf, Siegfried Rösch) (continued fraction convergents A/B and C/D with the constraint A+B=D) Shows Schubert's 7/24 to 9/31 example. Henry E. Dudeney;

Amusements in Mathematics, Thomas Nelson and Sons, London 1917, p141, 247

(Reprint: Dover, 1958, ISBN 0-486-20473-1)

problem 413: A Chessboard Fallacy

- 3 piece dissection showing n*n = (n+1)*(n-1)

Amusements in Mathematics, Thomas Nelson and Sons, London 1917, p141, 247 (Reprint: Dover, 1958, ISBN 0-486-20473-1) problem 413: A Chessboard Fallacy - 3 piece dissection showing n*n = (n+1)*(n-1) John Fisher;

The Magic of Lewis Carroll, Thomas Nelson and Sons, 1973

(german: Alice im Wunderland der Rätsel, Hugendubel, 1985, ISBN 3-88034-268-7, p66-67)

- 4 piece dissection showing 8*8 = 5*13

The Magic of Lewis Carroll, Thomas Nelson and Sons, 1973 (german: Alice im Wunderland der Rätsel, Hugendubel, 1985, ISBN 3-88034-268-7, p66-67) - 4 piece dissection showing 8*8 = 5*13 Greg. N. Frederickson;

Dissections: Plane & Fancy, Cambridge Univ. Press, 1997

- chapter 23

Dissections: Plane & Fancy, Cambridge Univ. Press, 1997 - chapter 23 Greg. N. Frederickson;

Geometric dissections on the web

Geometric dissections on the web David Gale;

Tracking the Automatic Ant, Springer Verlag, 1998

Chaper 16.Addendum: Jigsaw Paradoxes, 128-130.

(Reprint of: Mathematical Intelligenzer 17:1 (1995) 25-26 and the correction 17:2 (1995) 39) - Jean Brette's multiple reorderings of a right 9x16 triangle

Tracking the Automatic Ant, Springer Verlag, 1998 Chaper 16.Addendum: Jigsaw Paradoxes, 128-130. (Reprint of: Mathematical Intelligenzer 17:1 (1995) 25-26 and the correction 17:2 (1995) 39) - Jean Brette's multiple reorderings of a right 9x16 triangle Martin Gardner;

Mathematics, Magic and Mystery, New York, Dover Publ. 1956

(german: Mathematik und Magie, DuMont, 1981 Tb 106, ISBN 3-7701-1048-X and Mathemagische Tricks, Vieweg 1981) Capter 8: Geometrical Vanishing Parts II, Best extensive discussion of the subject and its history.

- Paradox of William Hooper (1794)

- Paradox of Langman (variation of William Hooper)

- Paradox of Paul Curry (1953)

- Zig Zag Dissection of an 'Rectangle' into two pieces



Mathematics, Magic and Mystery, New York, Dover Publ. 1956 (german: Mathematik und Magie, DuMont, 1981 Tb 106, ISBN 3-7701-1048-X and Mathemagische Tricks, Vieweg 1981) Capter 8: Geometrical Vanishing Parts II, Best extensive discussion of the subject and its history. - Paradox of William Hooper (1794) - Paradox of Langman (variation of William Hooper) - Paradox of Paul Curry (1953) - Zig Zag Dissection of an 'Rectangle' into two pieces Martin Gardner (Ed.);

Best Mathematical Puzzles of Sam Loyd, New York, Dover Publ., 1959

(german: Sam Loyd, Martin Gardner; Mathematische Rätsel und Spiele, Dumont, 1978)

problem 24: The Gold Brick problem

- 3 piece dissection showing n*n = (n+1)*(n-1)

Best Mathematical Puzzles of Sam Loyd, New York, Dover Publ., 1959 (german: Sam Loyd, Martin Gardner; Mathematische Rätsel und Spiele, Dumont, 1978) problem 24: The Gold Brick problem - 3 piece dissection showing n*n = (n+1)*(n-1) Martin Gardner;

The Scientific American Book of Mathematical Puzzles and Diversions, Simon & Schuster (1959)

Chapter 14: Fallacies (the Curry triangle)

The Scientific American Book of Mathematical Puzzles and Diversions, Simon & Schuster (1959) Chapter 14: Fallacies (the Curry triangle) Martin Gardner;

The Second Scientific American Book of Mathematical Puzzles and Diversions, Simon & Schuster (1961)

Chapter 8: Phi: The Golden Ratio - 4 piece dissection showing (a+b)*(a+b) = b*(a+2b). See Fig. 44

The Second Scientific American Book of Mathematical Puzzles and Diversions, Simon & Schuster (1961) Chapter 8: Phi: The Golden Ratio - 4 piece dissection showing (a+b)*(a+b) = b*(a+2b). See Fig. 44 Martin Gardner;

New Mathematical Diversions from Scientific American, Simon & Schuster (1966)

Chapter 11: Mr. Apollinax Visits New York

- Figure 50 is a missing hole configuration (also shown in [Gardner 1998])

New Mathematical Diversions from Scientific American, Simon & Schuster (1966) Chapter 11: Mr. Apollinax Visits New York - Figure 50 is a missing hole configuration (also shown in [Gardner 1998]) Martin Gardner;

aha! Gotcha Paradoxes to puzzle and delight, Freeman, 1982, New York-San Francisco

(german: Gotcha, Pradoxien für den Homo Ludens, Hugendubel 1985)

- Randi's carpet, p72-74

aha! Gotcha Paradoxes to puzzle and delight, Freeman, 1982, New York-San Francisco (german: Gotcha, Pradoxien für den Homo Ludens, Hugendubel 1985) - Randi's carpet, p72-74 Martin Gardner;

A Quarter-Century of Recreational Mathematics, Scientific American. (Aug 1998) v. 279(2) p. 68-75

Zbl 1999f.04107 (Ein Vierteljahrhundert Unterhaltungsmathematik, Spektrum der Wissenschaft, (Nov 1998), 112-120)

- Das Paradox vom verschwundenen Loch, p117, 120 [Gardner 1966]

A Quarter-Century of Recreational Mathematics, Scientific American. (Aug 1998) v. 279(2) p. 68-75 Zbl 1999f.04107 (Ein Vierteljahrhundert Unterhaltungsmathematik, Spektrum der Wissenschaft, (Nov 1998), 112-120) - Das Paradox vom verschwundenen Loch, p117, 120 [Gardner 1966] Ron L. Graham, Donald E. Knuth, Oren Pataschnik;

Concrete Mathematics, Addison Wessley, Reading, 1994, 2nd Ed.

Chapter 6.6 Fibonacci Numbers, Excercise 6.75

- 4 piece dissection showing 8*8 = 5*13

Concrete Mathematics, Addison Wessley, Reading, 1994, 2nd Ed. Chapter 6.6 Fibonacci Numbers, Excercise 6.75 - 4 piece dissection showing 8*8 = 5*13 Hjärnbruk;

swedish page www.fof.se/main/hjarnbruk/ showing some dissections found in Martin Gardner MM&M [1956]. Trollar bort rutor Currys triangle Schackkongressen 1858 Lösningarna

swedish page www.fof.se/main/hjarnbruk/ showing some dissections found in Martin Gardner MM&M [1956]. William Hooper;

Rational Recreations. 1774. Vol. 4,

pp. 286-287: Recreation CVI _ The geometric money.

3 x 10 cut into four pieces which make a 2 x 6 and a 4 x 5. (The diagram is shown in Gardner, MM&M [1956], pp. 131-132.)

Rational Recreations. 1774. Vol. 4, pp. 286-287: Recreation CVI _ The geometric money. 3 x 10 cut into four pieces which make a 2 x 6 and a 4 x 5. (The diagram is shown in Gardner, MM&M [1956], pp. 131-132.) E. I. Ignatjew;

Mathematische Spielereien, 1982, 2nd Ed. ISBN 3-87144-646-7

first russion edition is from 1908.

Chap 8: Geometrische Sophismen und Paradoxa

- 4 piece dissection showing 8*8 = 5*13

Mathematische Spielereien, 1982, 2nd Ed. ISBN 3-87144-646-7 first russion edition is from 1908. Chap 8: Geometrische Sophismen und Paradoxa - 4 piece dissection showing 8*8 = 5*13 Johnson;

Fibonacci Quarterly 41 (2003) B-960, pg 182.

F(a)F(b) - F(c)F(d) = (-1) r ( F(a - r)F(b - r) - F(c - r)F(d - r) )

5F(a)F(b) - L(c)L(d) = (-1) r ( 5F(a - r)F(b - r) - L(c - r)L(d - r) )

with a+b=c+d for any integers a,b,c,d,r



Fibonacci Quarterly 41 (2003) B-960, pg 182. F(a)F(b) - F(c)F(d) = (-1) ( F(a - r)F(b - r) - F(c - r)F(d - r) ) 5F(a)F(b) - L(c)L(d) = (-1) ( 5F(a - r)F(b - r) - L(c - r)L(d - r) ) with a+b=c+d for any integers a,b,c,d,r Ron Knott;

Harder Fibonacci Puzzles shows

- A Fibonacci Jigsaw puzzle or How to Prove 64=65

- The same puzzle but losing a square or How to Prove 64=63!!

- Yet another Fibonacci Jigsaw Puzzle!



Harder Fibonacci Puzzles shows - A Fibonacci Jigsaw puzzle or How to Prove 64=65 - The same puzzle but losing a square or How to Prove 64=63!! - Yet another Fibonacci Jigsaw Puzzle! Boris A. Kordemsky;

The Moscow Puzzles, 1972, problem 357: a paradox

(german: Köpfchen muss man haben, 1975, problem 320)

- 4 piece dissection showing 8*8 = 5*13

The Moscow Puzzles, 1972, problem 357: a paradox (german: Köpfchen muss man haben, 1975, problem 320) - 4 piece dissection showing 8*8 = 5*13 W. Lietzmann, V. Trier;

Wo steckt der Fehler? Trugschlüsse und Schülerfehler, B. G. Teubner, Leipzig, 1913, p17

Series: Mathematische Bibliothek Band 10

Trugschluss 22: 63 = 64 = 65

- 4 piece dissection showing 8*8 = 5*13

Wo steckt der Fehler? Trugschlüsse und Schülerfehler, B. G. Teubner, Leipzig, 1913, p17 Series: Mathematische Bibliothek Band 10 Trugschluss 22: 63 = 64 = 65 - 4 piece dissection showing 8*8 = 5*13 Walter Lietzmann;

Wo steckt der Fehler? B. G. Teubner, Stuttgart, 1972, 6. Auflage, p92-93, ISBN 3-519-02603-1

Trugschluss V.9: 64 = 65

- 4 piece dissection showing 8*8 = 5*13

- general case b*c and a*d rectangles with the constraints d=a+b and |b*c-a*d| = 1.

Wo steckt der Fehler? B. G. Teubner, Stuttgart, 1972, 6. Auflage, p92-93, ISBN 3-519-02603-1 Trugschluss V.9: 64 = 65 - 4 piece dissection showing 8*8 = 5*13 - general case b*c and a*d rectangles with the constraints d=a+b and |b*c-a*d| = 1. Sam Loyd;

The 8th Book of Tan, New York, Loyd & Co. Publ. 1903

(reprint: New York, Dover Publ, 1968)

- Tangram paradoxes

The 8th Book of Tan, New York, Loyd & Co. Publ. 1903 (reprint: New York, Dover Publ, 1968) - Tangram paradoxes Sam Loyd;

Cyclopedia of Puzzles, Franklin Bigelow Corporation, Morningside Press, New York, 1914

pages 288 and 378: Puzzleland Chess Boards

- 4 piece dissection showing 63 = 8*8 = 5*13

- He writes that he invented this dissection and presented it one the first American Chess congress 1858.

page 32: The Gold Brick Problem

- 3 piece dissection showing n*n = (n+1)*(n-1)

Cyclopedia of Puzzles, Franklin Bigelow Corporation, Morningside Press, New York, 1914 pages 288 and 378: Puzzleland Chess Boards - 4 piece dissection showing 63 = 8*8 = 5*13 - He writes that he invented this dissection and presented it one the first American Chess congress 1858. page 32: The Gold Brick Problem - 3 piece dissection showing n*n = (n+1)*(n-1) M. Edouard Lucas;

Récréations Mathématiques II, Gauthier-Villars, Paris, 1883

Section: Un Paradoxe Géométrique, p152-154

- 8*8 = 5*13 dissection and Fibonacci generalization;

Récréations Mathématiques II, Gauthier-Villars, Paris, 1883 Section: Un Paradoxe Géométrique, p152-154 - 8*8 = 5*13 dissection and Fibonacci generalization; mathworld.wolfram.com Triangle Dissection Paradox Dissection Fallacy Curry Triangle Tangram Paradox

Mittenzwey;

Mathematische Kurzweil, Leipzig und Wien, 1880

Problem 299: - 4 piece dissection showing 8*8 = 5*13

Mathematische Kurzweil, Leipzig und Wien, 1880 Problem 299: - 4 piece dissection showing 8*8 = 5*13 William R. Ransom;

One Hundred Mathematical Curiosities, Weston Walch, Publisher, Portland, Maine, 1955

Section: Gaining or Losing a Unit of Area, p29-33

- 4 piece dissection showing 63 = 8*8 = 5*13

- Fibonacci generalization, golden section

One Hundred Mathematical Curiosities, Weston Walch, Publisher, Portland, Maine, 1955 Section: Gaining or Losing a Unit of Area, p29-33 - 4 piece dissection showing 63 = 8*8 = 5*13 - Fibonacci generalization, golden section Gianni A. Sarcone;

Paradoxical Tangram and Vanishing Puzzles, Journal of Recreational Puzzles 29:2 (1998) 132-133

Problem 2424

Paradoxical Tangram and Vanishing Puzzles, Journal of Recreational Puzzles 29:2 (1998) 132-133 Problem 2424 Gianni A. Sarcone;

Paradoxical Tangram, Cubism For Fun 49 (June 1999) 17

- Tangram paradoxes (part of [Sarcone 1998])

Paradoxical Tangram, Cubism For Fun 49 (June 1999) 17 - Tangram paradoxes (part of [Sarcone 1998]) V. Schlegel;

Verallgemeinerung eines geometrischen Paradoxons. Zeitschrift für Mathematik und Physik 24 (1879) 123-128 & Plate I.

- 8 x 8 to 5 x 13 and generalizations.

Verallgemeinerung eines geometrischen Paradoxons. Zeitschrift für Mathematik und Physik 24 (1879) 123-128 & Plate I. - 8 x 8 to 5 x 13 and generalizations. O. Schlömilch;

Ein geometrisches Paradoxon, Zeitschrift für Mathematik und Physik 13 (1868) 162

- 4 piece dissection showing 8*8 = 5*13

Ein geometrisches Paradoxon, Zeitschrift für Mathematik und Physik 13 (1868) 162 - 4 piece dissection showing 8*8 = 5*13 Hermann Schubert;

Mathematische Mußestunden, (kleine Ausgabe) G. J. Göschen'sche Verlagshandlung, Leipzig 1904, 2. Auflage, p141-144

(Walter de Gruyter, Berlin 1941, 7. Auflage, p111-114)

- Chapter: Trugschlüsse, 8*8 = 5*13 dissection and Fibonacci generalization; shows an 9*24 = 7*31 example; (continued fraction convergents)

Mathematische Mußestunden, (kleine Ausgabe) G. J. Göschen'sche Verlagshandlung, Leipzig 1904, 2. Auflage, p141-144 (Walter de Gruyter, Berlin 1941, 7. Auflage, p111-114) - Chapter: Trugschlüsse, 8*8 = 5*13 dissection and Fibonacci generalization; shows an 9*24 = 7*31 example; (continued fraction convergents) David Singmaster;

Sources in Recreational Mathematics,

An Annotated Bibliography, 7th Pre. Ed., Oct. 1999

- Part II, Sect. 6.P Geometric Vanishing

Sect. 6.P.1 paradoxical dissections of the chessboard based on Fibonacci numbers

Sources in Recreational Mathematics, An Annotated Bibliography, 7th Pre. Ed., Oct. 1999 - Part II, Sect. 6.P Geometric Vanishing Sect. 6.P.1 paradoxical dissections of the chessboard based on Fibonacci numbers Jerry (= G. K.) Slocum, Jack Botermans;

Puzzles Old & New _ How to Make and Solve Them,

Univ. of Washington Press, Seattle, 1986

(german: Geduldspiele der Welt, Hugendubel, 1987)

- p144: vanishing puzzles - 4 piece dissection showing 8*8 = 5*13 = 63 made from wood ca. 1900

Puzzles Old & New _ How to Make and Solve Them, Univ. of Washington Press, Seattle, 1986 (german: Geduldspiele der Welt, Hugendubel, 1987) - p144: vanishing puzzles - 4 piece dissection showing 8*8 = 5*13 = 63 made from wood ca. 1900 Warren Weaver;

Lewis Carroll and a geometrical paradox, American Mathematical Monthly 45 (1938) 234-236

- Describes unpublished work in which Carroll obtained (in some way) the generalizations of the 8 x 8 to 5 x 13 in about 1890-1893.

Lewis Carroll and a geometrical paradox, American Mathematical Monthly 45 (1938) 234-236 - Describes unpublished work in which Carroll obtained (in some way) the generalizations of the 8 x 8 to 5 x 13 in about 1890-1893. N. Werman, D. Zeilberger;

A bijective proof of Cassini's Fibonacci identity, Discrete Mathematics 58:1 (1986) 109

Zbl. 578.05004

Cassini's Fibonacci identity is F n+1 × F n-1 - F n × F n = (-1) n .

A bijective proof of Cassini's Fibonacci identity, Discrete Mathematics 58:1 (1986) 109 Zbl. 578.05004 Cassini's Fibonacci identity is F × F - F × F = (-1) . William F. White;

A Scrap-Book of Elementary Mathematics. Open Court, 1908.

[The 4th ed., 1942, is identical in content and pagination, omitting only the Frontispiece and the publisher's catalogue.]

- Geometric puzzles, pp. 109-117

8*8 = 5*13 = 63

A Scrap-Book of Elementary Mathematics. Open Court, 1908. [The 4th ed., 1942, is identical in content and pagination, omitting only the Frontispiece and the publisher's catalogue.] - Geometric puzzles, pp. 109-117 8*8 = 5*13 = 63 Alex Bogomolny;

A Faulty Dissection: What Is Wrong? and its solution Fibonacci Bamboozlement - an applet showing an 4 piece dissection using the relation F n × F n - F n-1 × F n+1 = (-1) n+1 where F n is the n-th Fibonacci number.

A Faulty Dissection: What Is Wrong? and its solution Fibonacci Bamboozlement - an applet showing an 4 piece dissection using the relation F × F - F × F = (-1) where F is the n-th Fibonacci number. Alex Bogomolny;

Curry's Paradox: How Is It Possible? - an applet for Curry's Paradox, Cassini's Fibonacci identity.

Curry's Paradox: How Is It Possible? - an applet for Curry's Paradox, Cassini's Fibonacci identity. Alex Bogomolny;

Eye Opener Series - an applet showing an 2 piece dissection using the relation n × n versus (n+1) × (n-1).

Eye Opener Series - an applet showing an 2 piece dissection using the relation n × n versus (n+1) × (n-1). Alex Bogomolny;

Technologies: Past and Future - The WWW didn't help to solve a 'Geometric Fallacy'

Technologies: Past and Future - The WWW didn't help to solve a 'Geometric Fallacy' Markus Götz;

Rätsel - 0021: Flächentreue Abbildung?

explains Curry's Paradox

Rätsel - 0021: Flächentreue Abbildung? explains Curry's Paradox Norbert Treitz;

Pflastersteine - Fibonacci Bamboozlement shown at the MNU meeting

Hardware

Geometrex from Rex Games, Inc

GEOMETREX. Incredible Puzzles create a simple way to learn the natural sequences of numbers, as well as, the Fibonacci and Lucas sequences. The sequences of natural numbers and related sequences have astonishing properties, which can be used in architecture or for creating paradoxical puzzles.

GEOMETREX. Incredible Puzzles create a simple way to learn the natural sequences of numbers, as well as, the Fibonacci and Lucas sequences. The sequences of natural numbers and related sequences have astonishing properties, which can be used in architecture or for creating paradoxical puzzles. TangraMagic from Tessellations

This is the tangram paradox of [Gianni A. Sarcone].

This is the tangram paradox of [Gianni A. Sarcone]. Quadrix from Archimedes' Lab puzzles

(follow the link Math curiosities->The golden number.)

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Last Update: 2004-10-13

