The  Truth  About  In-inity  QUEST  Summer  2014  Jenny  Johnson,  jejohnson@ksu.edu    

“The  infinite!  No  other  quesBon  has  ever  moved  so  profoundly  the  spirit  of  the  man;    no  other  idea  has  so  fruiIully  sBmulated  his  intellect;    yet  no  other  concept  stands  in  greater  need  of  clarificaBon  than  that  of  the  infinite.”      -­‐  David  Hilbert  

In-inity  seems  to  come  up  not  just  in  mathematics,  but  in  science,  art,  and  religion.    

M.  C.  Escher  

Mathematics        

•  EsBmaBng  the  circumference  of  a  circle  by  inscribing  polygons.    

•  Build  up  to  conBnuously  compounding  interest.    

Fractals  

Sierpinski  Triangle  The  Sierpinski  Triangle  is  made  by  repeatedly  removing  the  middle  triangle  from  the  prior  generaBon.  The  number  of  colored  triangles  increases  by  a  factor  of  3  at  each  step.    

Koch  Curve    The  Koch  curve  is  made  by  repeatedly  replacing  each  segment  of  a  generator  shape  with  a  smaller  copy  of  the  generator.    At  each  step,  the  total  length  of  the  curve  gets  longer,  eventually  approaching  infinity.    

The  Koch  Snow-lake    

A  curve  of  infinite  length  that  encloses  a  finite  area.        

Draw  a  circle  around  your  snowflake,  will  the  snowflake  ever  grow  outside  your  circle?    

Geometric  Series  Sum  infinitely  many  posiBve  numbers:              

What  is  the  sum  of  these  infinitely  many  posiBve  numbers?    Use  a  number  line  to  show  that  the  sum  is  finite.