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# Penroses’s triangle

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### Penroses’s triangle

1. 1. PENROSES’S TRIANGLE OR IMPOSSIBLE TRIANGLE                                           ESTEFANÍA  CASAL  CAMPOS.  1260902      -­‐    Üniversitat  Alpen  Adria.  Visual  Illusions  as  a  tool  to  investigate  the  Brain   Processes  (Simone  Gori).
2. 2.   2     CONTENTS:         • IMPOSSIBLE  FIGURES  (Origin)  ……………………………………………………….3     • PENROSE..……………………………………………………………………………………..4     • HANS  RIJK  SOLUTION…………………………………………………………………….5     • 3D  PRINTED  PENROSE’S  TRIANGLE………………………………………………..6     • HOW  TO  MAKE  IT………………………………………………………………………….8     • CURIOSITIES………………………………………………………………………………….10     • LITERATURE…………………………………………………………………………………..11
3. 3.   3       • IMPOSSIBLE  FIGURES:     The  origin  of  the  impossible  figures  study  seems  that  took  place  in  1934.  In   that  year  the  artist  Oscar  Reutersvärd  was  just  a  student,  bored  in  the  classes   of  Latin,  figures  filled  the  margins  of  books  with  figures.       One  of  his  favourite  pastimes  was  to  draw  multi-­‐pointed  star  as  regular  as   possible.  One  day  he  tried  to  draw  a  6-­‐pointed  star  surrounding  it  with  cubes   (Fig  1).  When  he  did,  he  discovered  that  the  cubes  were  a  strange  figure.       Indeed,  that  is:  the  cubes  are  3  rows:  1  and  2,  3  and  4  and  5   and  6  (Fig.  2).  The  first  row  and  the  third  are  horizontal  while   the  second  is  vertical.  Cubes  1  and  6,  on  the  one  hand,  appear   on  the  same  level  while  between  cubes  2  and  5  there  is  a   difference  in  height,  which  is  absurd  because  the  rows  to  1  and   2  and  5  and  6  are  horizontal.       FIG  1               However,  the  intuition  of  Reutersvärd  led  him  to  place  three  new  cubes  in  the  corners   so  that  they  formed  a  perfect  triangle  ...  and  impossible.         FIG  2
4. 4.   4       • PENROSE:     In  1956  L.  S.  and  Roger  Penrose  published  the  article:  "Impossible  figures:  a   special  kind  of  Visual  Illusions."  It  introduced  figures  as  the  "tribar"   impossible  triangle  formed  by  three  bars  (left)  or  the  endless  staircase   (right).  Also  showed  the  photo  of  another  ladder,  on  the  chosen  angle,   looked  like  impossible.                                   Roger  Penrose  confessed  years  later  that  much  of  the  inspiration  for  the   article  came  to  visit  an  exhibition  of  the  work  of  Escher.  By  then,  the  work   of  the  Dutch  painter  did  not  include  impossible  elements,  but  the  nature  of   their  work  was  an  inspiration  to  Penrose,  Many  of  Escher's  drawings  have   so  absurd  or  impossible  situations  which  strike  our  senses  or  our  habitual   way  of  seeing  the  world.  In  any  case,  for  the  dates  of  publication  of  the   article,  Escher  was  already  working  with  impossible  figures  (like  this  cube   that  appeared  in  his  Belvedere)  while  the  text  of  the  Penrose  turn  served  as   inspiration  for  new  creations.
5. 5.   5       • HANS  RIJK  SOLUTION:     The  most  usual  explanation  of  Penrose’s  tribar  or  impossible  triangle  that  I   have  found  was  the  one  offered  for  the  mathematical  Hans  Rijk,  friend  of   Escher.       Obviously  the  Hans  solution  is  projection  or  perspective,  so  that  the  three-­‐ dimensional  solid  figure  under  a  specific  angle  view  build  the  "Impossible   Triangle"  this  is  an  illusion  because  it  is  not  a  solid  as  such  because  it  is   impossible  to  create.     When  Roger  Penrose  rediscovered,  he  described  it  as  “impossibility  in  its   purest  form”.       Tribar  is  formed  by  the  three  sides  of  square  section  which  are  joined  at   right  angles  at  the  vertices  creating  a  triangle  and  hence  the  impossibility  as   actual  geometric  solid  shape.       The  Penrose  triangle  shown  below  is  seen  from  a  perspective  on  the   isometric  projection  usually  used  in  technical  drawing.  The  following  image   shows  the  Tribar  seen  from  the  right  angle  while  the  figure  shows  observed   from  the  back  in  the  same  isometric.  In  the  next  view  of  the  figure  can  be   seen  that  this  is  the  paradox  of  optical  perception,  we  see  that  there  are   three  rods  attached  to  each  other  at  90  °  each  with  respect  to  each  other   and  separated,  one  of  these  arms  is  bevelled  and  is  the  element  that  create   the  perfect  illusion  when  the  solid  body  is  viewed  from  the  proper   perspective.
6. 6.   6     • PRINTED  3D  PENROSE  TRIANGLE:     Until  2  years  ago,  one  of  the  most  popular  illusions  it  was  only  that,  an   illusion  that  cannot  be  possible  in  the  reality.  But  maybe  we  are  wrong.       After  Reading  several  articles  about  the  same,  the  triangle  of  Penrose  could   be  created  as  a  solid  with  the  help  of  the  3D  technology.     Ulrich  Schwanitz,  a  young  designer  based  in  the  Netherlands,  has  used   rapid  prototyping  service  to  produce  what  he  says  is  the  Penrose  triangle  in   3-­‐D.  Mathematically  this  is  impossible,  because,  as  I  mentioned  before,  the   Penrose  triangle  is  formed  by  three  straight  sections  of  square  section,   which  are  joined  at  right  angles  at  the  ends  of  the  triangle.  This   combination  of  properties  cannot  be  adopted  for  any  three-­‐dimensional   figure  in  Euclidean  geometric  space.     As  maximum,  we  can  manufacture  solid  three-­‐dimensional  objects,  when   viewed  from  the  proper  angle;  appear  to  be  Penrose’s  triangles  (as  this   sculpture  at  the  Museum  of  Perth).
7. 7.   7     Schwanitz  insists  he  has  not  resorted  to  the  usual  tricks  of  perspective  and   have  given  a  new  approach  to  a  Penrose  triangle  impossible  in  the  real   world.  "We  have  designed  a  truly  three-­‐dimensional  object,  unlike  other   attempts,  it  has  no  opening  or  hidden  rotations,  is  no  necessary  look  at  a   certain  position".     The  object,  which  is  12  cm  high,  is  made  of  plastic  foil  with  printing  systems   3-­‐D  of  the  Shapeways  Dutch  company,  which  also  sells.  Judging  by  the   pictures  appears  to  meet  on  three  sides,  with  three  real  angles  as  shown   hand  reference  doll.     However,  many  people  argue  about  a  doll  does  not  eliminate  the  possibility   of  it  being  a  visual  trick.  Schwanitz  refuses  to  share  the  3-­‐D  model  that  has   followed  to  conform  it  and  has  given  the  media  only  two  images  of  the   object  (from  the  same  perspective).                                       As  Ulrich  preferred  to  let  people  guess,  weeks  later,  Tchoukanov  Artur,  a   Shapeways  ex  internal  found  the  way  for  reproducing  the  achievement  of   Schwanitz.  Artur  decided  to  upload  to  Thingiverse  the  instructions  for   others  to  emulate  his  achievement.  The  story  caught  the  attention  on   BoingBoing,  where  the  news  spread.  That  same  day,  Schwanitz  sent  a   DMCA  notice  to  remove  the  content  for  violation  of  intellectual  property.   Of  this  way  born:  the  first  lawsuit  against  3D  printing  technology.
8. 8.   8     • HOW  TO  MAKE  IT:     To  finish  understanding  the  triangle  impossible  at  all,  I  decided  to  try   making  one  by  myself.       It  was  easy  get  the  necessary  information  to  start  with  the  project.  Several   schemes  are  available  online  from  programs  like  Autocad  for  the  creation  of   the  Tribar.       Following  the  plan  bellow,  I  start  with  my  own:
9. 9.   9     1. The  first  step  was  printing  the  previous  document  in  the  correct  and  original   size.       2. Then,  the  figure  cut  ready  for  being  built.  In  this  case,  I  chose  following  this   plan  because  makes  the  work  easier,  but  it  is  possible,  get  other  schemes  with   the  size  of  every  side  of  each  piece  and  build  it  from  the  beginning  or  with   different  materials.                                     3. Folding  the  dashed  lines.     4. The  Tribar  ready.  Being  from  the  right  perspective  and  other  random  one.
10. 10.   10     • CURIOSITIES:                  The  tribar  of  Penrose  is  visible  in  paintings  and  even  as  main  argument  in  the   film  “Inception”  by  Christopher  Nolan.   Also,  other  authors  felt  curious  about  this  famous  figure  and  its  variants  as   Goo-­‐Shung  Wan,  who  filmed  a  funny  short  film  “Halluci”  of  3:44  min.  In  this   case,  the  argument  focuses  on  the  impossible  stairs.       http://www.youtube.com/watch?feature=player_embedded&v=hhfhgbmZe9s
11. 11.   11       LITERATURE:       Penrose  L.S.,  Penrose  R.  (1958).  Impossible  Objects:  a  special  type  of  visual  illusion.  The  British  Journal   of  Psychology.  49.   Taschen,  Ernst  B.  (1997).  Optical  Illusions.   New  World  Encyclopedia,  Penrose  triangle.  Web:   http://www.newworldencyclopedia.org/entry/Penrose_triangle   Impossible  World,  Impossible  triangle.  Web:  http://im-­‐ possible.info/english/articles/triangle/triangle.html   Cool  optical  illusions.  Web:  http://www.coolopticalillusions.com/illusions/coolprinttriangle.pdf   i.materialise.  J.  Peels  (2011)  Web  http://i.materialise.com/blog/entry/impossible-­‐3d-­‐printed-­‐penrose-­‐ triangle-­‐solved   Co.Design  S.  Labarre.  Web:  http://www.fastcodesign.com/1663226/impossible-­‐penrose-­‐triangle-­‐now-­‐ possible-­‐with-­‐3-­‐d-­‐printing-­‐updated