Repository: UIHistories Project: Mathematical Models Construction and Use of Mathematical Models (Fehr & Hildebrandt) [PAGE 14]

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E v o r y student in the class can m a k e h i s own m o d e l f r o m o r d i n a r y g r a p h p a p e r . Bend a p i e c e of g r a p h p a p e r l e n g t h w i s e . Select the origin and d e t e r m i n e the b o n d p o i n t v a l u e of x rnd slit the p a p e r along the c r e a s e . t o this b o n d p o i n t v a l u e . C r e a s e e a c h slit p a r t i n t o three equal p a r t s , fold the two i n t e r i o r sections of e a c h p a r t together p l a c i n g the two third sections t o g e t h e r and in the same p l a n e as the u n s l i t p a r t . The two m i d d l e sections c o m p r i s e the c o m p l e x p l a n e . Label the real, and i m a g i n a r y axis and g r a p h the curve on all v i s i b l e s i d e s . L o o k i n g t o w a r d s the f r o n t only the r e a l p a r t of the curve is seen; f r o m the sid only t h e c o m p l e x p a r t . In a glass m o d e l b o t h p a r t s are r e a d i l y v i s i b l e . H«, The same structure can b o carried to cubic and h i g h e r equations. The m o d e l ' i s similar except that the cubic h a s / its c o m p l e x values in two h y p e r b o l i c c o n c a v e "nN / surfaces instead of a p l a n e . (See P l a t e 7JE >' n o . I ) . O n such a g l a s s m o d e l , all the n r o o t s , r e a l or c o m p l e x , arc r e p r e s e n t e d . / 4. Gonic S e c t i o n s : All the conic s e c t i o n s can easily b o cut f r o m a cone (so • P l a t o I I I , nof . 6 ) . A d o u b l e n a p p e d cone is b e s t and can b e joined at the v e r t i c e s by a counter sunk p i n . T h e conic sections can also b e m a d e f r o m paper b y c u t t i n g f r o m circles of r a d i u s , r, three s e c t o r s w h o s o arc l e n g t h s arc less t h a n , equal t o , and greater than ff r \J2 r e s p e c t i v e l y . Parting the two edges t o g e t h e r there areses an acute a n g l e d , r i g h t a n gled and obtuse angle-.-) cone r e s p e c t i v e l y * C u t t i n g t h r o u g h the cones p e r p e n d i c u l a r to a sl.ont h e i g h t , the cut w i l l r e s u l t in an e l l i p s e , P a r a b o l a and h y p e r b o l a , r e s p e c t i v e l y . / N 5. Ellipse: Around two round headed thumb'; t a c k s , on a d r a w i n g b o a r d , p l a c e a l o o p e d , n o n - fczZ elastic c o r d . P l a c i n g a p e n c i l inside the loop and m o v i n g the p o i n t so as to k e e p the cord t a u t , _ an ellipse isjb^acod. As the thumb tacks arc p l a c e d closer gother^ the ellipse a p p r o a c h e s a c i r c l e .

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6. Ruled Surface: Between a double napped cone and a. cylinder there are i n t e r v e n i n g f i g u r e s . (See P l a t o I I I , n o . 5 and 8.) T h e t r a n s i t i o n s can b e shown b y the f o l l o w i n g m o d e l . S e p a r a t e two blocks, on w h i c h h a v e b e e n d r a w n c i r c l e s of d i a m e t e r 5 i n c h e s , b y an eight inch b a r , fixed to the center, of one circle and c l a m p e d a1' the other c e n t e r . P e r f o r a t e the circumfor?onces of the circles w i t h S m a l l h o l e s a b o u t one h a l f inch apart* L a c e w i t h elastic b a n d s so that e a c h b a n d is v o r t i c a l , thus f o r m i n g a c y l i n d r i c a l sur•i face. As the loose b l o c k is t w i s t e d ^ t h e cylinder 1i is changed to single sheeted h y p o r b o l o i d s and fin<k r t tf 1 allv to a d o u b l e n a p p e d c o n e .