But the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed. —Albert Einstein

These  revolutionary  ideas  seem  strange  at  first  because  we  take  for granted  that  our  everyday  world  has  three  dimensions.  As  the  late  physicist  Heinz  Pagels  noted,  "One  feature  of  our  physical  world  is  so  obvious that  most  people  are  not  even  puzzled  by it. The fact  that  space  is  three dimensional."  Almost  by  instinct  alone,  we  know  that  any  object  can  be described  by  giving  its  height,  width,  and  depth.  By  giving  three  numbers,  we  can  locate  any  position  in  space.  If  we  want  to  meet  someone for  lunch  in  New  York,  we  say,  "Meet  me  on  the  twenty-fourth  floor  of the  building  at  the  corner  of  Forty-second  Street  and  First  Avenue."  Two numbers  provide  us  the  street  corner;  and  the  third,  the  height  off  the ground. Airplane  pilots,  too,  know  exactly  where  they  are  with  three  numbers—their  altitude  and  two  coordinates  that  locate  their  position  on  a grid  or  map.  In  fact,  specifying  these  three  numbers  can  pinpoint  any location  in  our  world,  from  the  tip  of  our  nose  to  the  ends  of  the  visible universe.  Even  babies  understand  this:  Tests  with  infants  have  shown  that they  will  crawl  to  the  edge  of  a  cliff,  peer  over  the  edge,  and  crawl  back. In  addition  to  understanding  "left"  and  "right"  and  "forward"  and "backward"  instinctively,  babies  instinctively  understand  "up"  and "down."  Thus  the  intuitive  concept  of  three  dimensions  is  firmly  embedded  in  our  brains  from  an  early  age. Einstein  extended  this  concept  to  include  time  as  the  fourth  dimension.  For  example,  to  meet  that  someone  for  lunch,  we  must  specify  that we  should  meet  at,  say,  12:30  P.M.  in  Manhattan;  that  is,  to  specify  an event,  we  also  need  to  describe  its  fourth  dimension,  the  time  at  which the  event  takes  place. Scientists  today  are  interested  in  going  beyond  Einstein's  conception of  the  fourth  dimension.  Current  scientific  interest  centers  on  the  fifth dimension  (the  spatial  dimension  beyond  time  and  the  three  dimensions  of  space)  and  beyond.  (To  avoid  confusion,  throughout  this  book I  have  bowed  to  custom  and  called  the  fourth  dimension  the  spatial dimension  beyond  length,  breadth,  and  width.  Physicists  actually  refer to  this  as  the  fifth  dimension,  but  I  will  follow  historical  precedent.  We will  call  time  the  fourth  temporal  dimension.) How  do  we  see  the  fourth  spatial  dimension? The  problem  is,  we  can't.  Higher-dimensional  spaces  are  impossible to  visualize;  so  it  is  futile  even  to  try.  The  prominent  German  physicist Hermann  von  Helmholtz  compared  the  inability  to  "see"  the  fourth dimension  with  the  inability  of  a  blind  man  to  conceive  of  the  concept of  color.  No  matter  how  eloquently  we  describe  "red"  to  a  blind  person, words  fail  to  impart  the  meaning  of  anything  as  rich  in  meaning  as  color. Even  experienced  mathematicians  and  theoretical  physicists  who  have worked  with  higher-dimensional  spaces  for  years  admit  that  they  cannot visualize  them.  Instead,  they  retreat  into  the  world  of  mathematical  equations.  But  while  mathematicians,  physicists,  and  computers  have  no problem  solving  equations  in  multidimensional  space,  humans  find  it impossible  to  visualize  universes  beyond  their  own. 

 At  best,  we  can  use  a  variety  of  mathematical  tricks,  devised  by  mathematician  and  mystic  Charles  Hinton  at  the  turn  of  the  century,  to  visualize  shadows  of  higher-dimensional  objects.  Other  mathematicians,  like Thomas  Banchoff,  chairman  of  the  mathematics  department  at  Brown University,  have  written  computer  programs  that  allow  us  to  manipulate higher-dimensional  objects  by  projecting  their  shadows  onto  flat,  twodimensional  computer  screens.  Like  the  Greek  philosopher  Plato,  who said  that  we  are  like  cave  dwellers  condemned  to  see  only  the  dim,  gray shadows  of  the  rich  life  outside  our  caves,  Banchoff's  computers  allow only  a  glimpse  of  the  shadows  of  higher-dimensional  objects.  (Actually, we  cannot  visualize  higher  dimensions  because  of  an  accident  of  evolution.  Our  brains  have  evolved  to  handle  myriad  emergencies  in  three dimensions.  Instantly,  without  stopping  to  think,  we  can  recognize  and react  to  a  leaping  lion  or  a  charging  elephant.  In  fact,  those  humans who  could  better  visualize  how  objects  move,  turn,  and  twist  in  three dimensions  had  a  distinct  survival  advantage  over  those  who  could  not. Unfortunately,  there  was  no  selection  pressure  placed  on  humans  to master  motion  in  four  spatial  dimensions.  Being  able  to  see  the  fourth spatial  dimension  certainly  did  not  help  someone  fend  off  a  charging saber-toothed  tiger.  Lions  and  tigers  do  not  lunge  at  us  through  the fourth  dimension.)