Techno-Art of Selariu SuperMathematics Functions

Techno-Art of Selariu SuperMathematics Functions

A e r o n a u t i c s C a p s u l e

2007

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Techno-Art of Selariu SuperMathematics Functions

Editor: Florentin Smarandache

ARP

2007

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Front cover art image, called Aeronautics Capsule, by Mircea Eugen elariu. This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann Arbor MI 48106-1346, USA Tel.: 1-800-521-0600 (Customer Service)

http://wwwlib.umi.com/bod/basic Copyright 2007 by American Research Press, Editor and the Authors for their SuperMathematics Functions and Images. Many books can be downloaded from the following Digital Library of Arts & Letters: http://www.gallup.unm.edu/~smarandache/eBooksLiterature.htm Peer Reviewers: 1. Prof. Mihly Bencze, University of Braov, Romania. 2. Prof. Valentin Boju, Ph. D. Officer of the Order Cultural Merit, Category Scientific Research MontrealTech - Institut de Technologie de Montral Director, MontrealTech Press P. O. Box 78574, Station Wilderton Montral, Qubec, H3S 2W9, Canada (ISBN-10): 1-59973-037-5 (ISBN-13): 978-1-59973-037-0 (EAN): 9781599730370 Printed in the United States of America

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C-o-n-t-e-n-t-s

A e r o n a u t i c s c a p s u l e first cover FOREWORD (FOR SUPERMATHEMATICS FUNCTIONS) 6-16

Selariu SuperMathematics Functions & Other SuperMathematics Functions 17

D O U B L E C L E P S Y D R A 18 S U P E R M A T H E M A T I C S F L O W E R S 19 The Ballet of the Functions 20 J a c u z z i 21 D a m a g e d p a r t o f T I T A N I C 22 K A Z A T C I O K (Russian Popular Dance) 23 F l y i n g B i r d 1 24 F l y i n g B i r d 2 25 M U L T I C O L O R E D S U N 26 R e d S u n 27 T h e d o u b l e N o z z l e f o r N A S A 28 T h e s u p e r m a t h e m a t i c s C o m e t 29 The Lake of Swans 30

+ The Dance of Swords 30 + The Nut Cracker 30 + The Decease of Swan 30

T h e F l o w e r i n g 31 The supermathematics ring surface 32 The ex-centric sphere 33 T h e s u p e r m a t h e m a t i c s Screw Surface 34 T h e T r o j a n H o r s e 35 T h e A m p h o r a s 36 B U D D H A 37 J E T P L A N E 38 T R O U B L E D L A N D

o r D O U B L E A N A L Y T I C A L E X C E N T R I C F U N C T I O N 39

S E L F - P I E R C E B O D Y 40 H I L L S a n d V A L L E Y S 41 B e r n o u l l i s L e m n i s c a t e, C a s s i n i s O v a l s a n d o t h e r s 42 Continuous transforming of a circle into a haystack 43 C y c l i c a l S y m m e t r y 44 S m a r a n d a c h e S t e p p e d F u n c t i o n s 45 S C R I B B L I N G S W I T H . . . H E A D A N D T A L E 46 Q U A D R I P O D 47 E X C E N T R I C S Y M M E T R Y 48

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D R A C U L A S C A S T L E 49 T U N I N G F O R K 50 d e x O I D S + r e x O I D S 51 E x c e n t r i c g e o m e t r y , p r i s m a t i c s o l i d s 52 V a s e 53 H E X A G O N A L T O R U S 54 O p e n s q u a r e t o r u s 55 D o u b l e s q u a r e t o r u s 56 S i n u o u s C o r r u g a t e W a s h e r s

o r N a n o - P e r i s t a l t i c E n g i n e 57 I G L O O + T h e m a g i c c a r p e t 58 Multiple Ex Centric Circular SuperMathematics Functions 59 E X C E N T R I C T O R U S R I N G 60 H Y P E R S O N I C J E T A I R P L A N E 61 P L U M P V A S E 62 A R R O W S 63 H Y P E R B O L I C Q U A D R A T I C C Y L I N D E R 1 64 H Y P E R B O L I C Q U A D R A T I C C Y L I N D E R 2 65 E X - C E N T R I C F U L L S P R I N G 66 E X - C E N T R I C E M P T Y S P R I N G 67 E X C E N T R I C P E N T A G O N H E L I X 68 C Y L I N D E R S w i t h C O L L A R S 69 U n i c u r s a l S u p e r m a t h e m a t i c s F u n c t i o n s 1

+ O l d w o m a n f r o m C a r p a t i M o u n t a i n ( R o m a n i a ) 70 U n i c u r s a l S u p e r m a t h e m a t i c s F u n c t i o n s 2 71 U n i c u r s a l Su p e r m a t h e m a t i c s F u n c t i o n s 3 + W a l k i n g P i n g u i n s 72 T o D o u b l e a n d S i m p l e C a n o e 73 R o m a n i a n f o l k d a n c e 74 P i l l o w 75 T e r r a w i t h M a r k e d M e r i d i a n s 76 S u p e r m a t h e m a t i c s C o l u m n s 77 A e r o d y n a m i c S o l i d 78 H a l v i n g C u r v e 79 The crook lines (s 0) - a generalization of straight lines ( s = 0 ) 80-82 A R A B E S Q U E S 1 83 S T A R S 84 H y s t e r e t i c C u r v e s 1 85 H y s t e r e t i c C u r v e s 2 86 H y s t e r e t i c C u r v e s 3 87 E x C e n t r i c C i r c u l a r C u r v e s w i t h E x C e n t r i c V a r i a b l e 88 F i l i g r e e 1 89 F I L I G R E E 2 90 U s h a p e d C u r v e s i n 2 D a n d 3 D 91 E x p l o s i o n s 92 S p a t i a l F i g u r e 93 P l a n e t s a n d s t a r s 94 E x C e n t r i c C i r c l e ( n = 1 ) a n d A s t e r o i d ( n = 2, 4 , 6 ) 95 E x C e n t r i c A s t e r o i d ( n = 3, 5 ,7 ,9) 96 E x C e n t r i c L e m n i s c a t e s 97 B u t t e r f l y w i t h S y m m e t r i c a l C e n t e r 1 98 B u t t e r f l y w i t h S y m m e t r i c a l C e n t e r 2 99 B u t t e r f l y w i t h S y m m e t r i c a l C e n t e r 3 100 B u t t e r f l y R a p i d l y F l a p p i n g t h e W i n g s 101 F l o w e r w i t h F o u r P e t a l e s 102

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E c C e n t r i c P y r a m i d 103 A e r o d y n a m i c P r o f i l e

w i t h S u p e r m a t h e m a t i c s F u n c t i o n s 1 104 A e r o d y n a m i c P r o f i l e

w i t h S u p e r m a t h e m a t i c s F u n c t i o n s 2 105 S A L T C e l l a r 106 S U P E R M A T H E M A T I C S T O W E R 107 THE CONTINUOUS TRANSFORMATION OF A RIGHT TRIANGLE INTO ITS HYPOTENUSE + THE CONTINUOUS TRANSFORMATION OF QUADRATE (Rx = Ry)

OR RECTANGLE (Rx Ry) INTO ITS DIAGONAL 108 Mo d i f i e d c e x a n d s e x 109 S i n u o u s S u r f a c e w i t h A n a l i t i c a l S u p e r m a t h e m a t i c s

F u n c t i o n s 110 S u p e r m a t h e m a t i c s S p i r a l

+ S u p e r m a t h e m a t i c s P a r a b l e s 111 A R A B E S Q U E S 2 112 S u p e r m a t h e m a t i c s f u n c t i o n s cex xy and sex xy 113 S u p e r m a t h e m a t i c s f u n c t i o n s rex xy and dex xy 114 E x C e n t r i c F o l k l o r e C a r p e t 1 115 E x C e n t r i c F o l k l o r e C a r p e t 2 116 E x C e n t r i c F o l k l o r e C a r p e t 3 117 W A T E R F A L L I N G 118 S I N G L E and D O U B L E K C Y L I N D E R 119 S U P E R M A T H E M A T I C A L K N O T S H A P E D B R E A D

a n d O N E C R A C K N E L ( P R E T Z E L ) 120 S i x C o n o p y r a m i d s 121 F O U R C O N O P Y R A M I D S V I E W E D F R O M A B O V E 122 D O U B L E C O N O P Y R A M I D

or the transformation of circle into a square with circular ex-centric supermathematics function dex or quadrilobic functions cosq and sinq v 123

E X - C E N T R I C S a n d V A L E R I U A L A C I C U A D R O L O B S 124 P E R F E C T C U B E 125 E x c e n t r i c c i r c u l a r c u r v e s 1 126 E x c e n t r i c c i r c u l a r c u r v e s 2 o r E x c e n t r i c L i s s a j o u s c u r v e s 127 References in SuperMathematics 128 Postface back cover

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FOREWORD

(FOR SUPER-MATHEMATICS FUNCTIONS)

In this album we include the so called Super-Mathematics functions (SMF), which constitute the base for, most often, generating, technical, neo-geometrical objects, therefore less artistic.

These functions are the results of 38 years of research, which began at University of Stuttgart in 1969. Since then, 42 related works have been published, written by over 19 authors, as shown in the References.

The name was given by the regretted mathematician Professor Emeritus Doctor Engineer Gheorghe Silas who, at the presentation of the very first work in this domain, during the First National Conference of Vibrations in Machine Constructions, Timioara, Romania, 1978, named CIRCULAR EX-CENTRIC FUNCTIONS, declared: Young man, you just discovered not only some functions, but a new mathematics, a supermathematics! I was glad, at my age of 40, like a teenager. And I proudly found that he might be right!

The prefix super is justified today, to point out the birth of the new complements in mathematics, joined together under the name of Ex-centric Mathematics (EM), with much more important and infinitely more numerous entities than the existing ones in the actual mathematics, which we are obliged to call it Centric Mathematics (CM.)

To each entity from CM corresponds an infinity of similar entities in EM, therefore the Supermathematics (SM) is the reunion of the two domains: SM = CM EM, where CM is a particular case of null ex-centricity of EM. Namely, CM = SM(e = 0). To each known function in CM corresponds an infinite family of functions in EM, and in addition, a series of new functions appear, with a wide range of applications in mathematics and technology.

In this way, to x = cos corresponds the family of functions x = cex = cex (, s, ) where s = e/R and are the polar coordinates of the ex-center S(s,), which corresponds to the unity/trigonometric circle or E(e, ), which corresponds to a certain circle of radius R, considered as pole of a straight line d, which rotates around E or S with the position angle , generating in this way the ex-centric trigonometric functions, or ex-centric circular supermathematics functions (EC-SMF), by intersecting d with the unity circle (see.Fig.1). Amongst them the ex-centric cosine of , denoted cex = x, where x is the projection of the point W, which is the intersection of the straight line with the trigonometric circle C(1,O), or the Cartesian coordinates of the point W. Because a straight line, passing through S, interior to the circle (s 1 e < R), intersects the circle in two points W1 and W2, which can be denoted W1,2, it results that there are two determinations of the ex-centric circular supermathematics functions (EC-SMF): a principal one of index 1 cex1 , and a secondary one cex2 , of index 2, denoted cex1,2 . E and S were named ex-centre because they were excluded from the center O(0,0). This exclusion leads to the apparition of EM and implicitly of SM. By this, the number of mathematical objects grew from one to infinity: to a unique function from CM, for example cos , corresponds an infinity of functions cex , due to the possibilities of placing the ex-center S and/or E in the plane.

S(e, ) can take an infinite number of positions in the plane containing the unity or trigonometric circle. For each position of S and E we obtain a function cex . If S is a fixed point, then we obtain the ex-centric circular SM functions (EC-SMF), with fixed ex-center, or with constant s and . But S or E can take different positions, in the plane, by various rules or laws, while the

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straight line which generates the functions by its intersection with the circle, rotates with the angle around S and E.

Fig.1 Definition of Ex-Centric Circular Supermathematics Functions (EC-SMF)

In the last case, we have an EC-SMF of ex-center variable point S/E, which means s = s ()

and/or = (). If the variable position of S/E is represented also by EC-SMF of the same ex-center S(s, ) or by another ex-center S1[s1 = s1(), 1 = 1 ()], then we obtain functions of double ex-centricity. By extrapolation, well obtain functions of triple, and multiple ex-centricity. Therefore, EC-SMF are functions of as many variables as we want or as many as we need.

If the distances from O to the points W1,2 on the circle C(1,O) are constant and equal to the radius R = 1 of the trigonometric circle C, distances that will be named ex-centric radiuses, the distances from S to W1,2 denoted by r1,2 are variable and are named ex-centric radiuses of the unity circle C(1,O) and represent, in the same time, new ex-centric circular supermathematics functions (EC-SMF), which were named ex-centric radial functions, denoted rex1,2 , if are expressed in function of the variable named ex-centric and motor, which is the angle from the ex-center E. Or, denoted Rex1,2 , if it is expressed in function of the angle or the centric variable, the angle at O(0,0). The W1,2 are seen under the angles 1,2 from O(0,0) and under the angles and + from S(e, ) and E. The straight line d is divided by S d in the two semi-straight lines, one positive d + and the other negative d . For this reason, we can consider r1 = rex1 a positive oriented segment on d ( r1 > 0) and r2 = rex2 a negative oriented segment on d ( r2 < 0) in the negative sense of the semi-straight line d .

M1

W1

x

y

O

S E

cex1

sex1

cex2

sex2

OS = s OE = e OW1 = OW2 = 1 OM1 = OM2 = R SW1 = r1 = rex1 SW2 = r2 = rex2 EM1 = R.r1 = R.rex1 EM2 = R.r2 = R.rex2

W1OA = 1 W2OA = 2 SOA = S(s,) E(e,) M1,2 (R, 1,2) W1,2 (1, 1,2)

A

aex1,2 = 1,2 () = 1,2() = bex1,2 = = arcsin[s.sin(-)] cex1,2 = cos 1,2 sex1,2 = sin 1,2

W2

dex1,2 = d

d 2,1 =

=1 -

)(sin1)cos(.

22

s

s

Dex 1,2 = 2,1

dd

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Using simple trigonometric relations, in certain triangles OEW1,2, or, more precisely, writing the sine theorem (as function of ) and Pitagoras generalized theorem (for the variables 1,2) in these triangles, it immediately results the invariant expressions of the ex-centric radial functions:

r 1,2 () = rex1,2 = s.cos( ) )(sin1 22 s and

r 1,2 (1,2) = Rex 1,2 = )cos(..21 2 + ss . All EC-SMF have invariant expressions, and because of that they dont need to be tabulated,

tabulated being only the centric functions from CM, which are used to express them. In all of their expressions, we will always find one of the square roots of the previous expressions, of ex-centric radial functions.

Finding these two determinations is simple: for + (plus) in front of the square roots we always obtain the first determination (r1 > 0) and for the (minus) sign we obtain the second determination (r2 < 0). The rule remains true for all EC-SMF. By convention, the first determination, of index 1, can be used or written without index.

Some remarks about these REX (King) functions: The ex-centric radial functions are the expression of the distance between two points,

in the plane, in polar coordinates: S(s, ) and W1,2 (R =1, 1,2), on the direction of the straight line d, skewed at an angle in relation to Ox axis;

Therefore, using exclusively these functions, we can express the equations of all known plane curves, as well as of other new ones, which surfaced with the introduction of EM. An example is represented by Booths lemniscates (see Fig. 2, a, b, c), expressed, in polar coordinates, by the equation: () = R(rex1+rex 2) = 2 s.Rcos( - ) for R=1, = 0 and s [0, 3].

-1 -0.5 0.5 1

-0.4

-0.2

0.2

0.4

-1 -0.5 0.5 1-0.2-0.1

0.10.2

Fig.2,a Booths Lemniscates for R = 1 and numerical ex-centricity e [1.1, 2]

Fig. 2,b Booths Lemniscates for R = 1 and numerical ex-centricity e [2.1, 3]

Another consequence is the generalization of the definition of a circle: The Circle is the plane curve whose points M are at the distances r() = R.rex = R.rex [, E(e, )] in relation to a certain point from the circles plane E(e, ).

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If S O(0,0), then s = 0 and rex = 1 = constant, and r() = R = constant, we obtain the circles classical definition: the points situated at the same distance R from a point, the center of the circle.

Booth Lemniscate Functions

Polar coordinate equation with supermathematics circle functions rex 1,2 : = R (rex1 + rex2 ) for circle radius R = 1 and the numerical ex-centricity s [ 0, 1 ]

Fig. 2,c

The functions rex and Rex expresses the transfer functions of zero degree, or of

the position of transfer, from the mechanism theory, and it is the ratio between the

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parameter R(1,2), which positions the conducted element OM1,2 and parameter R.r1,2(), which positions the leader element EM1,2. Between these two parameters, there are the following relations, which can be deduced similarly easy from Fig. 1 that defines EC-SMF. Between the position angles of the two elements, leaded and leader, there are the following relations:

1,2 = arcsin[e.sin( )] = 1,2() = aex1,2

and

= 1,2 1,2(1,2 ) = 1,2 arcsin[ )cos(..21

)sin(.

2,12

2,1

+

ss

s ] = Aex (1,2 ).

The functions aex 1,2 and Aex 1,2 are EC-SMF, called ex-centric amplitude, because of their usage in defining the ex-centric cosine and sine from EC-SMF, in the same manner as the amplitude function or amplitudinus am(k,u) is used for defining the elliptical Jacobi functions:

sn(k,u) = sn[am(k,u)], cn(k,u) = cos[am(k,u)], or:

cex1,2 = cos(aex1,2 ) , Cex 1,2 = cos(Aex 1,2)

and sex 1,2 = sin (aex1,2 ), Sex 1,2 = cos (Aex 1,2 )

The radial ex-centric functions can be considered as modules of the position vectors

2,1r for the W1,2 on the unity circle C (1,O). These vectors are expressed by the following relations:

radrexr .2,12,1 =

,

where rad is the unity vector of variable direction, or the versor/phasor of the straight line direction d+, whose derivative is the phasor der = d(rad )/d and represents normal vectors on the straight lines OW1,2, directions, tangent to the circle in the W1,2. They are named the centric derivative phasors. In the same time, the modulus rad function is the corresponding, in CM, of the function rex for s = 0 = when rex = 1 and der 1,2 are the tangent versors to the unity circle in W1,2.

The derivative of the 2,1r vectors are the velocity vectors:

2,12,12,1

2,1 . derdexdrd

v ==

of the W1,2 C points in their rotating motion on the circle, with velocities of variable modulus v1,2 = dex1,2 , when the generating straight line d rotates around the ex-center S with a constant angular speed and equal to the unity, namely = 1. The velocity vectors have the expressions presented above, where der 1,2 are the phasors of centric radiuses R1,2 of module 1 and of 1,2 directions. The expressions of the functions EC-

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SM dex1,2 , ex-centric derivative of , are, in the same time, also the 1,2 () angles derivatives, as function of the motor or independent variable , namely

dex1,2 = d1,2 ()/d = 1 )(sin.1

)cos(.22

s

s

as function of , and

Dex 1,2 = d()/d1,2 = 2,1

22,1

2,12

2,1

Re)cos(.1

)cos(..21)cos(.1

xs

sss =+

,

as functions of 1,2 . It has been demonstrated that the ex-centric derivative functions EC-SM express the transfer functions of the first order, or of the angular velocity, from the Mechanisms Theory, for all (!) known plane mechanisms.

The radial ex-centric function rex expresses exactly the movement of push-pull mechanism S = R. rex , whose motor connecting rod has the length r, equal with e the real ex-centricity, and the length of the crank is equal to R, the radius of the circle, a very well-known mechanism, because it is a component of all automobiles, except those with Wankel engine.

The applications of radial ex-centric functions could continue, but we will concentrate now on the more general applications of EC-SMF.

Concretely, to the unique forms as those of the circle, square, parabola, ellipse, hyperbola, different spirals, etc. from CM, which are now grouped under the name of centrics, correspond an infinity of ex-centrics of the same type: circular, square (quadrilobe), parabolic, elliptic, hyperbolic, various spirals ex-centrics, etc. Any ex-centric function, with null ex-centricity (e = 0), degenerates into a centric function, which represents, at the same time its generating curve. Therefore, the CM itself belongs to EM, for the unique case (s = e = 0), which is one case from an infinity of possible cases, in which a point named eccenter E(e, ) can be placed in plane. In this case, E is overleaping on one or two points named center: the origin O(0,0) of a frame, considered the origin O(0,0) of the referential system, and/or the center C(0,0) of the unity circle for circular functions, respectively, the symmetry center of the two arms of the equilateral hyperbola, for hyperbolic functions.

It was enough that a point E be eliminated from the center (O and/or C) to generate from the old CM a new world of EM. The reunion of these two worlds gave birth to the SM world.

This discovery occurred in the city of the Romanian Revolution from 1989, Timioara, which is the same city where on November 3rd, 1823 Janos Bolyai wrote: From nothing Ive created a new world. With these words, he announced the discovery of the fundamental formula of the first non-Euclidean geometry.

He from nothing, I in a joint effort, proliferated the periodical functions which are so helpful to engineers to describe some periodical phenomena. In this way, I have enriched the mathematics with new objects.

When Euler defined the trigonometric functions, as direct circular functions, if he wouldnt have chosen three superposed points: the origin O, the center of the circle C and S as a pole of a semi straight line, with which he intersected the trigonometric/unity circle, the EC-SMF would have been discovered much earlier, eventually under another name.

Depending on the way of the split (we isolate one point at the time from the superposed ones, or all of them at once), we obtain the following types of SMF:

O C S Centric functions belonging to CM;

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and those which belong to EM are:

O C S Ex-centric Circular Supermathematics Functions (EC-SMF); O C S Elevated Circular Supermathematics Functions (ELC-SMF); O C S Exotic Circular Supermathematics Functions (EXC-SMF). These new mathematics complements, joined under the temporary name of SM, are

extremely useful tools or instruments, long awaited for. The proof is in the large number and the diversity of periodical functions introduced in mathematics, and, sometimes, the complex way of reaching them, by trying the substitution of the circle with other curves, most of them closed.

To obtain new special, periodical functions, it has been attempted the replacement of the trigonometric circle with the square or the diamond. This was the proceeding of Prof. Dr. Math. Valeriu Alaci, the former head of the Mathematics Department of Mechanics College from Timioara, who discovered the square and diamond trigonometric functions. Hereafter, the mathematics teacher Eugen Visa introduced the pseudo-hyperbolic functions, and the mathematics teacher M. O. Enculescu defined the polygonal functions, replacing the circle with an n-sides polygon; for n = 4 he obtained the square Alaci trigonometric functions. Recently, the mathematician, Prof. Malvina Baica, (of Romanian origin) from the University of Wisconsin together with Prof. Mircea Crdu, have completed the gap between the Euler circular functions and Alaci square functions, with the so-called Periodic Transtrigonometric functios.

The Russian mathematician Marcusevici describes, in his work Remarcable sine functions the generalized trigonometric functions and the trigonometric functions lemniscates.

Even since 1877, the German mathematician Dr. Biehringer, substituting the right triangle with an oblique triangle, has defined the inclined trigonometric functions. The British scientist of Romanian origin Engineer George (Gogu) Constantinescu replaced the circle with the evolvent and defined the Romanian trigonometric functions: Romanian cosine and Romanian sine, expressed by Cor and Sir functions, which helped him to resolve some non-linear differential equations of the Sonicity Theory, which he created. And how little known are all these functions even in Romania!

Also the elliptical functions are defined on an ellipse. A rotated one, with its main axis along Oy axis. How simple the complicated things can become, and as a matter of fact they are! This paradox(ism) suggests that by a simple displacement/expulsion of a point from a center and by the apparition of the notion of the ex-center, a new world appeared, the world of EM and, at the same time, a new Universe, the SM Universe.

Notions like Supermathematics Functions and Circular Ex-centric Functions appeared on most search engines like Google, Yahoo, AltaVista etc., from the beginning of the Internet. The new notions, like quadrilobe quadrilobas, how the ex-centrics are named, and which continuously fill the space between a square circumscribed to a circle and the circle itself were included in the Mathematics Dictionary. The intersection of the quadriloba with the straight line d generates the new functions called cosine quadrilobe-ic and sine quadrilobe-ic.

The benefits of SM in science and technology are too numerous to list them all here. But we are pleased to remark that SM removes the boundaries between linear and non-linear; the linear belongs to CM, and the non-linear is the appanage of EM, as between ideal and real, or as between perfection and imperfection.

It is known that the Topology does not differentiate between a pretzel and a cup of tea. Well, SM does not differentiate between a circle (e = 0) and a perfect square (s = 1), between a circle

13

and a perfect triangle, between an ellipse and a perfect rectangle, between a sphere and a perfect cube, etc. With the same parametric equations we can obtain, besides the ideal forms of CM (circle, ellipse, sphere etc.), also the real ones (square, oblong, cube, etc.). For s [-1,1], in the case of ex-centric functions of variable , as in the case of centric functions of variable , for s[-,+], it can be obtained an infinity of intermediate forms, for example, square, oblong or cube with rounded corners and slightly curved sides or, respectively, faces. All of these facilitate the utilization of the new SM functions for drawing and representing of some technical parts, with rounded or splayed edges, in the CAD/ CAM-SM programs, which dont use the computer as drawing boards any more, but create the technical object instantly, by using the parametric equations, that speed up the processing, because only the equations are memorized, not the vast number of pixels which define the technical piece. The numerous functions presented here, are introduced in mathematics for the first time, therefore, for a better understanding, the author considered that it was necessary to have a short presentation of their equations, such that the readers, who wish to use them in their applications development, be able to do it. SM is not a finished work; its merely an introduction in this vast domain, a first step, the authors small step, and a giant leap for mathematics. The elevated circular SM functions (ELC-SMF), named this way because by the modification of the numerical ex-centricity s the points of the curves of elevated sine functions sel as of the elevated circular function elevated cosine cel is elevating in other words it rises on the vertical, getting out from the space {-1, +1] of the other sine and cosine functions, centric or ex-centric. The functions cex and sex plots are shown in Fig. 3, where it can be seen that the points of these graphs get modified on the horizontal direction, but all remaining in the space [-1,+1], named the existence domain of these functions. The functions cel and sel plots can be simply represented by the products: cel 1,2 = rex1,2 . cos and Cel 1,2 = Rex 1,2. cos sel 1,2 = rex 1,2 . sin and Sel 1,2 = Rex 1,2. sin and are shown Fig. 4.

The exotic circular functions are the most general SM, and are defined on the unity circle which is not centered in the origin of the xOy axis system, neither in the eccenter S, but in a certain point C (c,) from the plane of the unity circle, of polar coordinates (c, ) in the xOy coordinate system. Many of the drawings from this album are done with EC-SMF of ex-center variable and with arcs that are multiples of (n.). The used relations for each particular case are explicitly shown, in most cases using the centric mathematical functions, with which, as we saw, we could express all SM functions, especially when the image programs cannot use SMF. This doesnt mean that, in the future, the new math complements will not be implemented in computers, to facilitate their vast utilization.

14

Fig. 3,a The ex-centric circular supermathematics function (EC-SMF) ex-centric cosine of cex for = 0, [0, 2]

Fig. 3,b The ex-centric circular supermathematics function (EC-SMF) ecentric sine of sex for = 0, [0, 2]

Numerical ex-centricity s = e/R [ -1, 1] The computer specialists working in programming the computer assisted design software

CAD/CAM/CAE, are on their way to develop these new programs fundamentally different, because the technical objects are created with parametric circular or hyperbolic SMFs, as it has been exemplified already with some achievements such as airplanes, buildings, etc. in http://www.eng.upt.ro/~mselariu and how a washer can be represented as a toroid ex-centricity (or as an ex-centric torus), square or oblong in an axial section, and, respectively, a square plate with a central square hole can be a square torus of square section. And all of these, because SM doesnt make distinction between a circle and a square or between an ellipse and a rectangle, as we mentioned before.

But the most important achievements in science can be obtained by solving some non-linear

problems, because SM reunites these two domains, so different in the past, in a single entity. Among these differences we mention that the non-linear domain asks for ingenious approaches for each problem. For example, in the domain of vibrations, static elastic characteristics (SEC) soft non-linear (regressive) or hard non-linear (progressive) can be obtained simply by writing y = m. x, where m is

15

not anymore m = tan as in the linear case (s = 0 ), but m = tex1,2 and depending on the numerical ex-centricity s sign, positive or negative, or for S placed on the negative x axis ( = ) or on the positive x axis ( = 0), we obtain the two nonlinear elastic characteristics, and obviously for s=0 well obtain the linear SEC.

1 2 3 4 5 6

-1-0.5

0.51

1.52

1 2 3 4 5 6

-2-1.5-1

-0.5

0.51

1 2 3 4 5 6

-1-0.5

0.51

1.5

1 2 3 4 5 6

-1.5-1

-0.5

0.51

Fig. 4,a ELC-SMF elevated cosine of - cel , for s [-1, +1], = 0, [0, 2].

Fig. 4,b ELC-SMF elevated sine of - sel , for s [-1, +1], = 1, [0, 2].

Due to the fact that the functions cex and sex , as well Cex and Sex and their

combinations, are solutions of some differential equations of second degree with variable coefficients, it has been stated that the linear systems (Tchebychev) are obtained also for s = 1, and not only for s = 0. In these equations, the mass ( the point M) rotates on the circle with a double angular speed = 2. (reported to the linear system where s = 0 and = = constant) in a half of a period, and in the other half of period stops in the point A(R,0) for e = sR = R or = 0 and in A(R, 0) for e = s.R = 1, or = . Therefore, the oscillation period T of the three linear systems is the same and equal with T = / 2. The nonlinear SEC systems are obtained for the others values, intermediates, of s and e. The projection, on any direction, of the rotating motion of M on the circle with radius R, equal to the oscillation amplitude, of a variable angular speed = .dex ( after dex function) is an non-linear oscillating motion.

The discovery of king function rex , with its properties, facilitated the apparition of a hybrid method (analytic-numerical), by which a simple relation was obtained, with only two terms, to calculate the first degree elliptic complete integral K(k), with an unbelievable precision, with a minimum of 15 accurate decimals, after only 5 steps. Continuing with the next steps, can lead us to a new relation to compute K (k), with a considerable higher precision and with possibilities to expand the method to other elliptic integrals, and not only to those. After 6 steps, the relation of E (k) has the same precision of computation.

s [0, 1]

s [ -1, 0]

16

The discovery of SMF facilitated the apparition of a new integration method, named integration through the differential dividing. We will stop here, letting to the readers the pleasure to delight themselves by viewing the drawings from this album. [Translated from Romanian by Marian Niu and Florentin Smarandache]

Mircea Eugen elariu

17

Selariu SuperMathematics Functions &

Other SuperMathematics Functions

18

D O U B L E C L E P S Y D R A

-0.50

0.5

-0.5

0

0.5

-1

-0.5

0

0.5

1

-0.5

0

0.5

M

===

sexzucexyucexx

sin.cos.

, Eccenter S(1,0) s = 1, = 0, t [0, 3]; u [0, 2]

19

S U P E R M A T H E M A T I C S F L O W E R S

0.5 1 1.5 2

0.5

1

1.5

2

0.5 1 1.5 2

0.5

1

1.5

2

X = dex ( /`2), s = s0 cos 4 , = 0 Y = dex , s0 [0, 1], [ 0, 2 ]

X = dex ( + /`2), s = s0 cos 6 , = 0 Y = dex , s0 [0, 1], [ 0, 2 ]

0.5 1 1.5 2

0.5

1

1.5

2

0.5 1 1.5 2

0.5

1

1.5

2

X = dex ( + /`2), s = s0 cos 8 , = 0 Y = dex , s0 [0, 1], [ 0, 2 ]

X = dex ( + /`2), s = s0 cos 15 , = 0 Y = dex , s0 [0, 1], [ 0, 2 ]

20

The Ballet of the Functions

0 1 2 3 4 5 6

-1

0

1

2

0 1 2 3 4 5 6-0.4

-0.2

0

0.2

0.4

0.6

F(x) = sign(cos x).cos x / (1+ tann x)n, , n [0, 10] F (x) = sign(sin x).sin x / (1+ tann x)n, , n [0, 10]

0 1 2 3 4 5 6-0.2

-0.1

0

0.1

0 1 2 3 4 5 6

-0.2

-0.1

0

0.1

0.2

F (x) = sign(cos x) tan x (1 + Abs(tan n x))n, n [0, 10] F (x) = sign(sin x) tan x (1 + Abs(tan n x))n, n [0, 10]

0 1 2 3 4 5 6

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 1 2 3 4 5 6

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

F (x) = sign(sin x) tan x / (1 + Abs(tan x))n, n [0, 10] F (x) = sign(cos x) tan x / (1 + Abs(tan x))n, n [0, 10]

21

J a c u z z i

-2-1

0

1

2 -2

-1

0

1

2

-1

-0.5

0

0.5

1

-2-1

0

1

2

-2

-1

0

1

2 -2

-1

0

1

2

-1

-0.5

0

0.5

1

-2

-1

0

1

2

M

=====

]3,0[....1,.]2,0[,..sin).1(

2sin,..cos).1(

ssexzuucexy

usucexx M

======

]3,0[.....................1,.]2,0[,cos,.sin).1(

sin........,.........cos).1(

ssexzuusucexy

usucexx

I m p l o d e d J a c u z z i -2

-10

12

-0.5

0

0.5-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

M

======

]3,0[.....................1,.]2,0[,sin,.sin).1(

sin........,.........cos).1(

ssexzuusucexy

usucexx

22

D a m a g e d p a r t o f T I T A N I C

-0.50

0.5

-2

-1

0

1

2

-1

-0.5

0

0.5

1

-0.50

0.5

-2

-1

0

1

2

M } ]2,0[],3,0[,1)cos.1(cos).1(

==+==

ussexz

ucexyucexx

23

K A Z A T C I O K (Russian Popular Dance)

-2 -1 1 2

-2

-1

1

2

24

F l y i n g B i r d 1

-2 -1 1

-2

-1

1

2

M +===

==3cos)8.0',0(.5sin1

3cos])1,0[,0,(.5sin12

2

sSsexyscexx

, S=s.cos5, ]2,0[ ,s ]1,0[

-1 -0.5 0.5 1 1.5

-1

-0.5

0.5

1

1.5

M ]2,0[],1,0[,5cos.)5cos(.)((sin1)8.0,(

)5cos(.)3(sin1),(2

2

=

+==+= ssS

SsSssexySsSscexx

25

F l y i n g B i r d 2

-1 -0.5 0.5 1 1.5

-2

-1

1

2

M ]2,0[,0],1,0[,3cos9sin1)8.0,(

3cos.5cos.3sin1),(2

2 =

++==+= s

ssexysscexx

-1 -0.5 0.5 1 1.5

-1.5

-1

-0.5

0.5

1

1.5

M ]2,0[,0],1,0[,5cos.3cos.9sin1)8.0,(

3cos.5cos.3sin1),(2

2 =

++==

+= ssssexy

sscexx

26

M U L T I C O L O R E D S U N

-2 -1 1 2

-2

-1

1

2

M ]2,0[],1,1[,sin..20cos..20

==

sdexydexx

27

R e d S u n

-2 -1 1 2

-2

-1

1

2

M ]2,0[],1,1[,sin..20cos..20

==

sdexydexx

28

T h e d o u b l e N o z z l e f o r N A S A -2 -1 0 1 2

-2-1

01

2

-2

0

2

-2-1

01

2

-2 -1 0 1 2-1

01

-2

0

2

-10

1

-2 -1 0 1 2

-2-1

01

2

-2

0

2

-2-1

01

2

-2 -1 0 1 2

-2-1

01

2

-2

0

2

-2-1

01

2

M

===

szxnyxnx

3sin..Recos..Re

, s [-1, 1], [0, 2], n = 2 ,3 ,4, 5.

29

T h e s u p e r m a t h e m a t i c s C o m e t

-4 -3 -2 -1 1

-2

-1

1

2

M ]2,0[),0],1,0[(,sin.cos.

=

==

sSdexydexx

30

The Lake of Swans The Dance of Swords

0 1 2 3 4 5 6

-1

0

1

2

0 1 2 3 4 5 6

-1.5

-1

-0.5

0

0.5

1

1.5

nn xxxsign

)tan1(cos)(cos

+ , n [0, 10]; x [0, 2] xxxsign

ntan1cos)(

+ , n [-10, 10]; x [0, 2]

The Nut Cracker The Decease of Swan

0 1 2 3 4 5 6

-1

-0.5

0

0.5

1

0 1 2 3 4 5 6

-0.4

-0.2

0

0.2

0.4

0.6

xxxsign

n 2tan15cos)(sin

+ , n [-10, 10]; x [0, 2] nn xxxsign

)tan1(sin)(sin

+ , n [0, 10]; x [0, 2]

31

T h e F l o w e r i n g

0.6 0.8 1.2 1.4

0.6

0.8

1.2

1.4

0.4 0.6 0.8 1.2 1.4 1.6

0.6

0.8

1.2

1.4

M

====

2,0[,,4cos.,

,4sin.,4cos.),

1

21

ssdexysssdexx

s [0, 1], = - /2, [0, 2,]

M

====

2,0[,,8cos.,

,8sin.,8cos.),

1

21

ssdexysssdexx

s [0, 1], = - /2, [0, 2,]

0.5 1 1.5 2

0.6

0.8

1.2

1.4

0.4 0.6 0.8 1.2 1.4 1.6

0.6

0.8

1.2

1.4

M

========

0,8cos.,2cos.,2/,2sin.,2cos.,

2/32

31

22

1

ssssdexyssssdexx

s [0, 1], [0, 2]

========

0,2cos.,2cos.,2/,2sin.,2cos.,

2/32

31

22

1

ssssdexyssssdexx

s [0, 1], [0, 2]

32

The supermathematics ring surface

-4-2

02

4 -4

-2

0

24

-1-0.500.51

-4-2

02

4

-4-2

02

4 -4

-2

0

2

4

-1-0.50

0.51

-4-2

02

4

M

+=+=

qquqyquqx

sinsin).cos3(cos).cos3(

s =1, = 0, [0, 2], u [ , ]

-20

2-2

0

2

-1-0.5

00.51

-20

2

-20

2 -2

0

2-1

-0.50

0.51

-20

2

M

+=+=

sexucexyucexx

sin).2(cos).2(

s =1, = 0, [0, 2], u [0, 2]

33

The ex-centric sphere

-2-1

01

2-1

0

1-1-0.50

0.5

1

-2-1

01

2

-1-0.5

00.5

1

-1-0.5

00.5

1

-1

-0.5

0

0.5

1

-1-0.5

00.5

1

-1-0.5

00.5

1

M

===

sin.sin.cos.cos.cos.

dexzudexyudexx

S(s [0,1], =0) M

===

sin.cossin.cos.sincos.cos.cos

qzuqyuqx

S(s [0,1], =0)

-1-0.5

00.5

1

-1-0.5

00.5

1

-1

-0.5

0

0.5

1

-1-0.5

00.5

1

-1-0.5

00.5

1

-1-0.5

00.5

1

-1-0.5

00.5

1

-1

-0.5

0

0.5

1

-1-0.5

00.5

1

-1-0.5

00.5

1

M

===

uzusexyucexx

sinsin.cos.

S(s [0,1], =0) M

===

uqzuqyuqx

sin.cossin.sincos.cos

S(s [0,1], =0)

[0, 2], u [, ], S[ s [ 0, 1], = 0]

34

T h e s u p e r m a t h e m a t i c s Screw Surface -20 -10

0 1020

-20-10

010

-20

0

20

40

60-20

-100

10

M

++===++=++=

)13

2sin(13

8.4)0,1,4

(.8

)]13

2cos(5sin[132

)]13

2cos(5cos[132

ussexz

uy

ux

, u [0, 2], [0, 26]

35

T h e T r o j a n H o r s e 0

2

4

6

-1-0.5

00.5

1

-4

-2

0

2

4

-1-0.5

00.5

1

-4

-2

0

2

4

02

46

-1-0.500.5

1

-4

-2

0

2

4

-1-0.500.5

1

-4

-2

0

2

4

X = , y = s, Z = cosq [, S(s, )], S[s [ - 1, 1], = 0], [0, 2]

X = , y = s, Z = sinq [, S(s, )], S[s [ - 1, 1], = 0], [0, 2]

36

T h e A m p h o r a s

-2 -1 01 2

-10

1

-10

-7.5

-5

-2.5

0

-10

1

-10

1

-1

0

1

0

2

4

6

-1

0

1

M

==+==+=

]2,0[],5.0,6.3[,...9.0

sin.)2

cos,(.21

cos.)98.0,(.212

2

z

ssexssy

scexssx

M ]2,0[,

]2.2,0[sin.Recos.Re

===

szxyxx

37

B U D D H A

-4-2

02

4-2-1

01

2

-10

-7.5

-5

-2.5

0

-2-1

01

2

M

==+==+=

]2,0[],5.0,6.3[,9.0sin.)cos,(..214.1

cos.)98.0,(.212

22

12

sszsscexssy

sssexssx

38

J E T P L A N E

0

2

4

6

8 -2

0

2

0

1

2

0

2

4

6

8

39

T R O U B L E D L A N D o r

D O U B L E A N A L Y N I C A L E X C E N T R I C F U N C T I O N

0

5

10 1

2

3

4

0

0.5

1

0

5

10

cex2a(x,S(s, = 0), ) = cos{ )]2

arcsin(2

[2

xxx }

40

S E L F - P I E R C E B O D Y

-1-0.5

00.5

1

-1-0.5

00.5

1

-1

-0.5

0

0.5

1

-1-0.5

00.5

1

-1-0.5

00.5

1

-1-0.5 0 0.5 1

-1

-0.50

0.51-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.50

0.51-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1-0.5

00.5

1

-1-0.5

00.5

1

-1

-0.5

0

0.5

1

-1-0.5

00.5

1

-1-0.5

00.5

1

M

====

])2,0[(.2sin])2,0[(.2sin

)9.0,3(

scexzssexysbexx

41

H I L L S a n d V A L L E Y S

-2

0

2-2

0

2

-8-6-4-20

-2

0

2

z = 10/(1 + 1.3 x2 + y2)[sex[(x2 - y2), S(s = 0.8, = 0)] Rex[(x2 y2), S(s = 0.64, = 0)], x, y [ - , + ]

-2

0

20

1

2

3

-6-4-20

-2

0

2

z = 10/(1 + 1.3 x2 + y2)[sex[(x2 - y2 - 8), S(s = 0.8, = 0)] Rex[(x2 y2 - 8), S(s = 0.64, = 0)], x [ - , + ], y [ 0, + ],

42

B e r n o u l l i s L e m n i s c a t e, C a s s i n i s O v a l s a n d o t h e r s

0.4 0.6 0.8 1.2 1.4 1.6

0.5

1

1.5

2

M ]2,0[),2

],1,0[(2cos.,2()cos,(

00

0 =

====

sSssdexyssdexx

43

Continuous transforming of a circle into a haystack

0.4 0.6 0.8 1.2 1.4 1.6

0.2

0.4

0.6

0.8

1

1.2 1.4 1.6 1.8 2

0.4

0.6

0.8

1.2

1.4

1.6

M ]2,0[],1,0[))0,cos(,(

))2

,cos)(,(0

0

0

====== s

ssSdexy

ssSdexx

44

C y c l i c a l S y m m e t r y

0.4 0.6 0.8 1.2 1.4 1.6

0.25

0.5

0.75

1

1.25

1.5

1.75

0.4 0.6 0.8 1.2 1.4 1.6

0.25

0.5

0.75

1

1.25

1.5

1.75

0.5 1 1.5 2

0.5

1

1.5

2

M M ]2,0[],1,0[))0,4cos(,(

))2

,4cos)(,(0

0

0

====== s

ssSdexy

ssSdexx

45

S m a r a n d a c h e S t e p p e d F u n c t i o n s

2 4 6 8 10 12

2.5

5

7.5

10

12.5

15

6 8 10 12

-10

-5

5

10

F(n, s, ) = Ssf () = [ bex(, s = 1) dex ].s.dex n, n = 10, s = [0.2, 0.4, 0.6, 0.9, 1]

46

S C R I B B L I N G S W I T H . . . H E A D A N D T A L E

-4 -2 2 4

-4

-2

2

4

-4 -2 2 4

-4

-2

2

4

M ]2,0[],1,0[,32)1(32)1(

=+=

ssexsexycexcexx

47

Q U A D R I P O D

-2

0

2

-2

0

2

-1

-0.5

0

0.5

1

-2

0

2

z = cexq [x.y, S(s = 0.8, = 0)], x, y [ , + ]

48

E X C E N T R I C S Y M M E T R Y

-4 -2 2 4

-4

-2

2

4

-4 -2 2 4

-4

-2

2

4

M ]2,0[),0],1,0[]0,1[()53sin(2.3

72.3 =+

=+=

andssSbexsexy

cexcexx

-4 -2 2 4

-4

-2

2

4

M ]2,0[),0],1,1[()53sin(2.3

72.3 =+

=+=

sSbexsexy

cexcexx

49

D R A C U L A S C A S T L E

-2

0

2-2

0

2

-1-0.5

00.51

-2

0

2

-10

1-2

0

20

2.55

7.510

-10

1

Z = cosq (x.y), s = 0.8, x, y [ , + ]

Z = 1 / cosq (x.y). cos (x.y), s = 0.8, x, y [ , + ]

-2

0

2-2

0

2-4-2024

-2

0

2

Z = 1 / cosq (x.y), s = 0.8, x,y [ , + ]

50

T U N I N G F O R K

-4-2

02

4

-10

0

10

-1-0.500.51

-4-2

02

4

51

d e x - O I D S

0 12 3

-1-0.500.5

10

0.5

1

1.5

2

0 12 3

0 1 2 3

-1-0.500.51

0

0.5

1

1.5

2

0 1 2 3

z = dex(x, y), x [0, ], S( s y [-1,1], = 0) r e x O I D S

01 2

3

-1-0.500.51

0

0.5

1

1.5

2

01 2

3

z = Rex(x, y), x [0, ], S( s y [-1,1], = 0)

52

E x c e n t r i c g e o m e t r y , p r i s m a t i c s o l i d s

00.5

11.5

2

0

0.5

11.5

2

0

0.5

1

1.5

00.5

11.5

2

0

0.5

11.5

2

00.5

11.5

2

00.5

11.5

2

-1.5

-1

-0.5

0

00.5

11.5

2

00.5

11.5

2

M ]0,1[..]..1,0[],2,0[,5.1

)2/,4()0,4(

=====

sorssz

dexydexx

-1 -0.5 0 0.5 1

-1-0.5

00.5

1

-3

-2

-1

0

1-1

-0.50

0.51

-1 -0.5 0 0.5 1

-1-0.5

00.5

1

-3

-2

-1

0

1-1

-0.50

0.51

M1

===

2sincos.

1

1

1

szqyqsx

, M2

===

szqyqx

sincos

2

2

, [0, 2], S( = 0, s [-1,1])

53

V a s e

-1 0 1

-10

1

-10

-7.5

-5

-2.5

0

-10

1

-2 -1 0 1 2-1

01

-10

-7.5

-5

-2.5

0

-10

1

M ]2,0[,]0,6.3[,

sin).(Recos).(Re

===

szsxysxx

M

==+===+==

]2,0[],2/,6.3[,.9.0)5cos,(.21sin),(Resin

]98.0,[.21cos),(Recos

222

12

1

sszsssexsssxy

sscexsssxx

54

H E X A G O N A L T O R U S

-4-2

02

4 -4

-2

0

2

4

-1-0.5

00.51

-4-2

02

4

-4-2

02

4 -4

-2

0

2

4

-1-0.5

00.51

-4-2

02

4

M ]2,0[],2,0[,]0,1(,[

sin]).0,1(,[3(cos]).0,1(,[3(

111

111

111

=====+===+=

ssSssexz

sSscexysSscexx

55

O p e n s q u a r e t o r u s M ]2.2,0[],2,0[,)sin)cos3(cos).cos3(

=+=+=

ssz

qsyqsx

-4-2

02

4

-4

-2

0

24

0

2

4

6

-4-2

02

4

-4

-2

0

24

S q u a r e t o r u s M ]2.2,0[],2,0[,sin

)1,(sin)cos3()1,(cos).cos3(

1

1

==+==+=

ssz

sqsysqsx

-4-2

02

4 -4

-2

0

2

4

-1-0.5

00.51

-4-2

02

4

56

D o u b l e s q u a r e t o r u s ]2,0[,,)1,(sin

sin).cos1/)1,(cos3(cos).sin1/)1,(cos3(

2

2

=+=+=

sqz

sqysqx

-4-2

02

4 -4

-2

0

2

4

-1-0.5

00.51

-4-2

02

4

-2

0

2-2

0

2

-1-0.5

00.51

-2

0

2

S q u a r e t o r u s ]2,0[,)1,(sin

sin)].1,(cos3[cos)].1,(cos2[

=+=+=

ssqz

sqysqx

57

S i n u o u s C o r r u g a t e W a s h e r s

o r N a n o - P e r i s t a l t i c E n g i n e

-4

-2

0

2

4-4

-2

0

2

4

0

1

2

3

-4

-2

0

2

4

M

+

++=

+=+=

30

)3/2

08sin(2.0)1,(sin3.0

]sin).1,(cos3[]cos)].1,(cos3[

sqz

sqysqx

58

I G L O O M ]2,0[,,

1.0

),(sin4

cos),(cos4

sin.22

),(sin4

sin),(cos4

cos.22

4

===

s

sz

sqsqsy

sqsqsx

-1

0

1-1

0

1

0

0.5

1

1.5

-1

0

1

T h e m a g i c c a r p e t ]1,1[],2,0[)),0,(,( 2 === syxysSxbexz

0

2

4

6 -1-0.5

00.5

1-0.500.5

0

2

4

6

-0.500.5

59

Multiple Ex Centric Circular SuperMathematics Functions

1 2 3 4 5 6

-1

-0.5

0.5

1

sex with dauble ex centre: y = s2ex = sin[-arcsin[s.sin]-arcsin[s.sin[-arcsin[s.sin ]]]]

ex centre S( s [-1, 1], = 0 ), [0, 2]

1 2 3 4 5 6

-1

-0.5

0.5

1

y = sin[ arctan[s.sin2 /Rex 2 ]]]

1 2 3 4 5 6

-1

-0.5

0.5

1

y = cos[ arctan(s.sin2 )] / 2cos.sa

60

E X C E N T R I C T O R U S R I N G

M

===+==+=

],0[]2,0[

],2,0[,)8.0,(

sin)]}sin.9.0,2(cos[2{cos)]}8.0,2(cos[2{

ussexz

uusbexyusbexx

-20

2-2

0

2-1

-0.50

0.51

-20

2

-20

2 0123-1-0.500.51

-20

2

-1-0.500.51

M

===+==+=

],0[]2,0[

],2,0[,)8.0,(

sin)]}sin.9.0,3(cos[2{cos)]}8.0,3(cos[2{

ussexz

uusbexyusbexx

-20

2-2

0

2

-1-0.5

00.51

-20

2

-3-2

-10

-2

0

2

-1-0.5

00.51

-3-2

-10

-2

0

2

61

H Y P E R S O N I C J E T A I R P L A N E

M ]2,0[],0],1,0[(,

Re)2/sin(.1(

Resin.1

=

+=

==

sS

xsz

xsy

sx

-10

-7.5

-5

-2.5

0 01

230

1

2

3

-10

-7.5

-5

-2.5

0

0

1

2

3

0

2

4

6

8-0.500.5

-0.500.5

0

2

4

6

8

-0.500.5

S ( s [ 0, 0.8], = 0)

62

P L U M P V A S E

M ]2,0[),0,3.2,0[(,.sin..cos.25.Recos.cos.25.Re

2

2

=

=+=+=

sSsz

ssxsyssxsx

-2 -1 0 1 2

-2

0

2

0

2

4

6

-2

0

2

-2-1

01

2

-2

02

0

2

4

-2-1

01

2

-2

02

- 20

2

- 2

0

2

0

2

4

- 20

2

- 2

0

2

63

A R R O W S

-0.4-0.2 00.20.4

-1

0

1

2

-0.5-0.25

00.250.5

-0.4-0.2 00.20.4

-1

0

1

2

-0.4-0.200.2

0.4

-1

0

1

2

-0.5-0.2500.250.5

-0.4-0.200.2

0.4

-1

0

1

2

-0.5-0.2500.250.5

-0.4-0.200.2

0.4

-1

0

1

2

-0.5-0.2500.250.5

-0.4-0.200.2

0.4

-1

0

1

2

-0.4

-0.200.20.4

-1

0

1

2

-0.5-0.25

00.250.5

-0.4-0.20

0.20.4

-1

0

1

2

M ]2,0[),0),2

,2

[(,sin.5.0

cos)1,0,().1,0,(2cos.cos1).1,0,( 2

=

==

=sS

szsdelcexyssexx

64

H Y P E R B O L I C Q U A D R A T I C C Y L I N D E R 1

R I G H T : n = 4, m = 1 R O T A T E D n = 1, m = 0.5

0 0.5 1 1.5 2

00.5

11.5

2

-1

0

1

00.5

11.5

2

-1-0.5 0

0.51

-1-0.5

00.5

1

-1

0

1

-1-0.5

00.5

1

M ]2,0[),0],1,1[(,5.1

)0,,(sin)0,,(cos

=

=====

sSsz

snqysnqx

m

m

65

H Y P E R B O L I C Q U A D R A T I C C Y L I N D E R 2

01

2

00 . 5

11 . 5

2

- 1

0

1

00 . 5

11 . 5

2

00 . 5

11 . 5

2

0

1

2

- 1

0

1

00 . 5

11 . 5

2

0

1

2

66

E X - C E N T R I C F U L L S P R I N G -1

0

1

-1

0

1

-1

0

1

2

3

-1

0

1

M

======

==

==

))0,1(,4

(

cos3.0

)cos3.0))0,1(,

4(2.0

)0,1(,4

(2.0

sSaexz

y

xsSaex

sSaex

67

E X - C E N T R I C E M P T Y S P R I N G

-20

-100

10

-10

0

10

0

20

40

-10

0

10

68

E X C E N T R I C P E N T A G O N H E L I X

-500

50

-50

0

50

0

50

100

150

-50

0

50

69

C Y L I N D E R S w i t h C O L L A R S

-20

2

-2

02

-5

0

5

-2

02

-20

2

-2

0

2

-5

0

5

-2

0

2

70

U n i c u r s a l S u p e r m a t h e m a t i c s F u n c t i o n s 1

-2 2 4 6 8

-1

-0.5

0.5

1

]2

3,2

[,8.48.35.3

,45.2

1,,

)cos.2cos

sin.2cos.arctansin(cos.2cos

sin.2cos.arctan

==

++=++=

cba

cbaycb

ax

O l d w o m a n f r o m C a r p a t i M o u n t a i n ( R o m a n i a )

2 4 6 8 10 12

-1

-0.5

0.5

1

71

U n i c u r s a l S u p e r m a t h e m a t i c s F u n c t i o n s 2

2 4 6 8 10 12

-1

-0.5

0.5

1

2 4 6 8 10 12

-1

-0.5

0.5

1

72

U n i c u r s a l Su p e r m a t h e m a t i c s F u n c t i o n s 3

1 2 3 4 5 6

-1

-0.5

0.5

1

W a l k i n g P i n g u i n s

-2.5 2.5 5 7.5 10 12.5

-1

-0.5

0.5

1

73

T o D o u b l e a n d S i m p l e C a n o e

- 0 . 5- 0 . 2 5

00 . 2 5

0 . 5

- 1

0

1

2

- 0 . 5

- 0 . 2 5

0

0 . 2 5

0 . 5

- 0 . 5- 0 . 2 5

00 . 2 5

0 . 5

- 1

0

1

2

- 1- 0 . 5

00 . 5

1

- 1

0

1

2

- 0 . 5- 0 . 2 5

00 . 2 50 . 5

- 1- 0 . 5

00 . 5

1

74

R o m a n i a n f o l k d a n c e

-4

-20

24

-4

-2

0

2

4

-5

-2.5

0

2.5

5

-4

-2

0

2

4

75

P i l l o w

22 . 5

33 . 5

4

2

2 . 5

33 . 5

4

- 1

- 0 . 5

0

0 . 5

1

22 . 5

33 . 5

4

2

2 . 5

33 . 5

4

22.5

33.5

4

2

2.5

3

3.54

-1

-0.5

0

0.5

1

22.5

33.5

4

2

2.5

3

3.54

76

T e r r a w i t h M a r k e d M e r i d i a n s

2

2.5

3

3.5

4

2

2.5

3

3.5

4

-1

-0.5

0

0.5

1

2

2.5

3

3.5

4

2

2.5

3

3.5

4

M

==+=

sexzucexy

cexxsin.

cos.3, S( s u [0, 3 /2], [0, 2]

77

S u p e r m a t h e m a t i c s C o l u m n s

?? x = cosq t y = sinq t , t [0, 2] z u [-1,1]

-1 -0.5 00.5 1

-1-0.5

00.5

1

-2

0

2

4

-1-0.5

00.5

1

-0.5 -0.25 0 0.25 0.5

-0.5-0.25

00.25

0.5

0

2

4

78

A e r o d y n a m i c S o l i d - 1

0

1

- 1- 0 . 5

00 . 5

1- 1- 0 . 5

0

0 . 5

1

- 1- 0 . 5

0

0 . 5

1

-10 1

-1-0.5

00.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

79

H a l v i n g C u r v e

x = cosq , y = sinq , with numerical ex-center : S( s, = 0 ), [0, 8 ]

S (s = 1, = 0) S (s = 0.89, = 0)

5 10 15 20 25

-1

-0.5

0.5

1

5 10 15 20 25

-1

-0.5

0.5

1

S (s = 0, = 0) S (s = 0.5, = 0)

5 10 15 20 25

-1

-0.5

0.5

1

5 10 15 20 25

-1

-0.5

0.5

1

S (s = 0.5, = 0) , x x2 S (s = 0.5, = 0), y y 2

5 10 15 20 25

-1

-0.5

0.5

1

5 10 15 20 25

-1

-0.5

0.5

1

80

The crook lines (s 0) - a generalization of straight lines ( s = 0 )

y = m. aex (x, S )=m{x-arcsin [s. sin(x - )]}, Numerical ex centre S (s, ) fixed m = 1, S (s [- 1, 0], = 0 ) m = 1, S (s [0, +1], = 0 )

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

y = m. aex (x, S), Numerical ex centre S (s, ) variable: s = xx

ssin.cos

0

- 3 - 2 - 1 1 2 3

- 4

- 2

2

4

S ( s = s0 / cos x .sin x s0 [- 1, 0], = 0 )

81

The crook lines (s 0) - a generalization of straight lines ( s = 0 )

y = m. aex (x, S )=m{x-arcsin [s. sin(x - )]}, Numerical ex centre S (s, ) variable m = 1, S (s = s0.cos2x, s0 [- 1, 0], = 0 ) m = 1, S (s= s0.cos2x, s0 [0, +1], = 0 )

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

m = 1, S (s= s0.cos2x, s0 [-1, +1], = 0 ), x [ - 3 / 2, 3 / 2 ]

- 4 - 2 2 4

- 6

- 4

- 2

2

4

6

82

The crook lines (s 0, s = 1) - a generalization of straight lines ( s = 0 )

y = m. aex (x, S )= m{x-arcsin [s. sin(x - )]}, Numerical ex centre S (s, ) fixed

-3 -2 -1 1 2 3

-4

-2

2

4

Numerical ex centre S (s, ) fixed : S (s [ - 1 , 1 ] , = 1 )

83

A R A B E S Q U E S 1

-1 -0.5 0.5 1

-1

-0.5

0.5

1

- 1 - 0 . 5 0 . 5 1

- 1

- 0 . 5

0 . 5

1

84

S T A R S x = Dex 10 . Rex 10 .cos y = Dex 10 . Rex 10 .sin

S( s [-1,1 ] , = 0), [0, 2]

-1 -0.5 0.5 1

-1.5

-1

-0.5

0.5

1

1.5

-1 -0.5 0.5 1

-1.5

-1

-0.5

0.5

1

1.5

-2 -1.5 -1 -0.5 0.5 1 1.5

-1.5

-1

-0.5

0.5

1

1.5

x = (1-cos 5 ) cos / Rex 10 . y = (1- cos 5 ) sin / Rex 10 .

85

H y s t e r e t i c C u r v e s 1

-4 -2 2 4

-1

-0.5

0.5

1

x = 1

M

}00.1,95,0,9.0,8.0,6.0,4.0,2.0{

]2

,2

[),0,5.0

1],1,0[(,

cos1

)sin())1,(,(sin

sin1

)cos())1,(,(cos

22

22

==

=

===

===

s

sS

ssSqy

ssSqx

yx

y

x

-4 -2 2 4

-1

-0.5

0.5

1

x = 0.5

86


-4 -2 2 4

-4

-2

2

4

x = 0.5; y= 0.5 =

-4 -2 2 4

-4

-2

2

4

x = 0.5; y= 1

87


-4 -2 2 4

-4

-2

2

4

x = 0.5; y= 0.5

-4 -2 2 4

-4

-2

2

4

x = 0.2; y= 0.5

88

E x C e n t r i c C i r c u l a r C u r v e s w i t h E x C e n t r i c V a r i a b l e

- 1 - 0 . 5 0 . 5 1

- 1

- 0 . 5

0 . 5

1

0.1 S t e p

-1 -0.5 0.5 1

-1

-0.5

0.5

1

0 . 2 S t e p

89

F i l i g r e e 1

-1 -0.5 0.5 1

-1

-0.5

0.5

1

s 0.1 u [-1, 1] with 0.1 S t e p

- 1 - 0 . 5 0 . 5 1

- 1

- 0 . 5

0 . 5

1

s 0.1 u [-1, 1] with 0.2 S t e p

90

F I L I G R E E 2

-1 -0.5 0.5 1

-1

-0.5

0.5

1

s 0.1 u [0, 1] with 0.1 S t e p s 0.1 u [-1, 0] with 0.1 S t e p

-1 -0.5 0.5 1

-1

-0.5

0.5

1

-1 -0.5 0.5 1

-1

-0.5

0.5

1

91

U s h a p e d C u r v e s i n 2 D a n d 3 D

y = arctan{1/[sex . Abs[cex ]]}, S(s [-1,1] , = 0), [ 0, 2]

1 2 3 4 5 6

-60

-40

-20

20

40

60

0

2

4

6

-0.500.5

-1

0

1

-1

0

1

92

E x p l o s i o n s

-1 -0.5 0.5 1

-1

-0.5

0.5

1

-1 -0.5 0.5 1

-1

-0.5

0.5

1

]2,0[)0],1,0[(

,

10Resin.8cos.

Recos.10cos.

=

== sS

xsy

xsx

]2,0[

)0],1,0[(,

8Resin.7cos.

Recos.10cos.

=

== sS

xsy

xsx

93

S p a t i a l F i g u r e

M ]2,0[),0],1,0[(,)]cos3(,2[3

5Re5

20

=

===

=sS

ssSCexzy

xbexx

-10

1

-2

0

2

-2

0

2

-10

1

-2

0

2

94

P l a n e t s a n d s t a r s

-1 -0.5 0.5 1

-1

-0.5

0.5

1

-2 -1 1 2

-2

-1

1

2

-2 -1 1 2

-2

-1

1

2

95

E x C e n t r i c C i r c l e ( n = 1 ) a n d A s t e r o i d ( n = 2, 4 , 6 )

M ]2,0[),0],1,1[(, =

==

sSsexycexx

n

n

-1 -0.5 0.5 1

-1

-0.5

0.5

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

n = 1 n = 2

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

n = 4 n = 6

96

E x C e n t r i c A s t e r o i d ( n = 3, 5 ,7 ,9)

]2,0[),0],1,1[(, =

==

sSsexycexx

n

n

-1 -0.5 0.5 1

-1

-0.5

0.5

1

-1 -0.5 0.5 1

-1

-0.5

0.5

1

n = 3 n = 5

-0.4 -0.2 0.2 0.4

-0.4

-0.2

0.2

0.4

-0.2 -0.1 0.1 0.2

-0.2

-0.1

0.1

0.2

n = 7 n = 9

97

E x C e n t r i c L e m n i s c a t e s

M { x = dex[, S(s = s0. cos, = - / 2]; y = dex[, S( s = s0.cos 2, = 0)] }, s0 [0, 1], [0, 2]

0.4 0.6 0.8 1.2 1.4 1.6

0.5

1

1.5

2

98

B u t t e r f l y w i t h S y m m e t r i c a l C e n t e r 1

M { x = dex[, S(s = s0. cos, = - / 2]; y = dex[, S( s = s0.sin 2, = 0)] }, s0 [0, 1], [0, 2]

0.4 0.6 0.8 1.2 1.4 1.6

0.5

1

1.5

2

99


M { x = dex[, S(s = s0. cos, = - / 2]; y = dex[, S( s = s0.sin 2, = 0)] }, s0 [0, 1], [0, 2]

0 . 6 0 . 8 1 . 2 1 . 4

0 . 5

1

1 . 5

2

100


0 . 6 0 . 8 1 . 2 1 . 4

0 . 2 5

0 . 5

0 . 7 5

1

1 . 2 5

1 . 5

1 . 7 5

101

B u t t e r f l y R a p i d l y F l a p p i n g t h e W i n g s

0.5 1 1.5 2

0.5

1

1.5

2

102

F l o w e r w i t h F o u r P e t a l e s

0.25 0.5 0.75 1 1.25 1.5 1.75

0.25

0.5

0.75

1

1.25

1.5

1.75

103

E c C e n t r i c P y r a m i d

0.5

1

1.5

0

0.5

1

1.5

2

-1

-0.5

0

0.5

1

0.5

1

1.5

0

0.5

1

1.5

2

104

A e r o d y n a m i c P r o f i l e w i t h S u p e r m a t h e m a t i c s F u n c t i o n s 1

M { x = 2 cex( + / 6, S( s = 0.6, = 0)), y = 2 sex(( + / 6, S( s [ -1, 0], = 0) )}, [0, 2]

-2 -1 1 2

-1

-0.5

0.5

1

M { x = 2 cex( + / 6, S( s = 0.6, = 0)), y = sex(( + / 6, S( s [ 0, 1], = 0) )}, [0, 2]

-2 -1 1 2

-1

-0.5

0.5

1

M { x = 2 cex( + / 6, S( s = 0.2, = 0)), y = sex(( + / 2 , S( s [ 0, 0.9], = 0) )}, [0, 2]

-1 -0.5 0.5 1

-1

-0.5

0.5

1

105

A e r o d y n a m i c P r o f i l e w i t h S u p e r m a t h e m a t i c s F u n c t i o n s 2

-1 -0.5 0.5 1

-1

-0.5

0.5

1

M { x = 2 cex( + / 6, S( s = 0.2, = 0)), y = sex(( + / , S( s [ 0, 0.9], = 0) )}, [0, 2]

-2-1.5

-1-0.5

0-1

-0.5

0

-2

-1.5

-1

-0.5

0

-1

-0.5

0

106

S A L T C e l l a r

-2

-1

0

1

2

-2

-1

0

1

2

-2

-1

0

1

2

-2

-1

0

1

2

-2

-1

0

1

2

M ]2,0[),0],1,1[(,.2

)2/(

1,2

2,1

=+

==

=sS

szrexy

rexx

107

S U P E R M A T H E M A T I C S T O W E R

-5

-2.5

0

2.5

5

-5

0

5

-4

-2

0

2

4

-5

-2.5

0

2.5

5

-5

0

5

M ]2,0[),0],2,0[(,sin..2

)).0,1(,().cos3())0,1(,().3(

=

==+===+=

sSsdexs

dexsSsexsysScexdexsx

108

THE CONTINUOUS TRANSFORMATION OF A RIGHT TRIANGLE INTO ITS HYPOTENUSE

-1 -0.5 0.5 1

-1

-0.5

0.5

1

-1 -0.5 0.5 1

-1

-0.5

0.5

1

x=cex(, S(s, = 0)), y=cex(, S(-s, =0)), S(s [0,1], = 0), [ 0, ]

THE CONTINUOUS TRANSFORMATION OF QUADRATE (Rx = Ry)

OR RECTANGLE (Rx Ry) INTO ITS DIAGONAL

-1 -0.5 0.5 1

-1

-0.5

0.5

1

x = Rx cex(, S(s, = 0)), y = Ry cex(, S(-s, =0)), S(s [ -1, 1 ], = 0), [ 0, ]

109

Mo d i f i e d c e x a n d s e x

]2,0[),0],1,1[(

),2Re2sin.arctansin( 22

=+==

sSx

sMsexy

]2,0[),0],1,1[(

),2Re2sin.arctancos( 22

=+==

sSx

sMcexy

1 2 3 4 5 6

-1

-0.5

0.5

1

1 2 3 4 5 6

-1

-0.5

0.5

1

]2,0[),0],1,1[(

),2Re2sin.arctancos( 21

===

sSx

sMcexy

]2,0[),0],1,1[(

),2Re2sin.arctansin( 21

===

sSx

sMsexy

1 2 3 4 5 6

-1

-0.5

0.5

1

1 2 3 4 5 6

-1

-0.5

0.5

1

1 2 3 4 5 6

-1

-0.5

0.5

1

1 2 3 4 5 6

-1

-0.5

0.5

1

110

S i n u o u s S u r f a c e w i t h A n a l i t i c a l S u p e r m a t h e m a t i c s F u n c t i o n s

]4,20[],2.1,4[],3sin5cos21.0arcsin[),( =

xxxxf

-4

-2

0

2 5

10

15

20

-0.2-0.1

0

0.1

0.2

-4

-2

0

2

S e c t i o n f o r = 8 and x [ - 5, 10 ]

- 1 0 1 0 2 0 3 0

- 0 . 2

- 0 . 1

0 . 1

0 . 2

111

S u p e r m a t h e m a t i c s S p i r a l

x = 0.3 cos Exp[0.2 (0.25 arcsin(sin0.25 ) y = 0.3 cos Exp[0.2 (0.25 arcsin(sin0.25 )

-10 -5 5 10

-10

-5

5

10

-1 -0.5 0.5 1

-1

-0.5

0.5

1

[ 0, 80] [ 0, 40]

S u p e r m a t h e m a t i c s P a r a b l e s

2 4 6 8 10

-3

-2

-1

1

2

3

]1,1[),0],1,1[(, +=

==

sSaexyaexx

112

A R A B E S Q U E S 2

-6 -4 -2 2 4 6

-1

1

2

3

4

-6 -4 -2 2 4 6

-1

1

2

3

4

113

S u p e r m a t h e m a t i c s f u n c t i o n s cex xy and sex xy

cex xy = cos[xy-arcsin[s.sin (xy - ]] for s = 0.4 and s = 0.9 ;x [-3,3], y [-3,3

1 2 3 4 5 6 71

2

3

4

5

6

7

1 2 3 4 5 6 71

2

3

4

5

6

7

S (s = 0.4, = 0) S (s = 0.9, = 0)

sex xy = sin[xy-arcsin[s.sin (xy - ]] for s = 0.4 and s = 0.9 ;x [-3,3], y [-3,3

1 2 3 4 5 6 71

2

3

4

5

6

7

1 2 3 4 5 6 71

2

3

4

5

6

7

S (s = 0.4, = 0) S (s = 0.9, = 0)

114

S u p e r m a t h e m a t i c s f u n c t i o n s rex xy and dex xy

dex1 xy = 1 s. cos xy / Sqrt[ 1 s2 sin2 xy]

1 2 3 4 5 6 71

2

3

4

5

6

7

1 2 3 4 5 6 71

2

3

4

5

6

7

S (s = 0.4, = 0) S (s = 0.9, = 0)

rex1 xy = s. cos xy - Sqrt[ 1 s2 sin2 xy]

1 2 3 4 5 6 71

2

3

4

5

6

7

1 2 3 4 5 6 71

2

3

4

5

6

7

S (s = 0.4, = 0) S (s = 0.9, = 0)

115

E x C e n t r i c F o l k l o r e C a r p e t 1

116


117


118

W A T E R F A L L I N G

5 10 15 20

0.2

0.4

0.6

0.8

1

5 10 15 20

0.2

0.4

0.6

0.8

1

F( , s) = 0.05 .cos .dex = 0.05 . cos (s)(1-s.cos / 22 sin.1 s )

119

S I N G L E and D O U B L E K C Y L I N D E R

-1 -0.5 0 0.5 1

-1-0.5

00.5

1

120

121

122

123

124-1-0.5

00.5

1

01

2

-1

0

1

120

121

122

123

124

-1

0

1

M

===

szsexycexx

.400

,

s = 0.3, = 0, [, ]

M

=+=+=

szssexyscexx

.400.sin.cos

,

s = 0.3, = 0, [, ]

120

S U P E R M A T H E M A T I C A L K N O T S H A P E D B R E A D a n d

O N E C R A C K N E L ( P R E T Z E L )

-2

0

2

-2

0

2

-1-0.5

00.5

1

-2

0

2

-4-2

0

2

4 -4

-2

0

2

4

-1-0.5

00.51

-4-2

0

2

4

M

=+=

=+=

szsy

sbexsx

sin)cos3.(sin

)]1,(cos[5.13.[cos

, [0, 2] , s [0, 2]

121

S i x C o n o p y r a m i d s

-1-0.5

00.5

1

-1

-0.5

0

0.51

-1

-0.5

0

0.5

1

-1-0.5

00.5

1

-1

-0.5

0

0.51

M for one conopyramid

===

szqsyqsx

sin.cos.

, s 0, 1], [0, 2]

122

F O U R C O N O P Y R A M I D S V I E W E D F R O M A B O V E

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

-3

-2

-1

1

2

3

M

==

qsyqsx

sin.cos.

, [0, 2], s [ 0, 2]]

123

D O U B L E C O N O P Y R A M I D

or the transformation of circle into a square with circular ex-centric supermathematics function dex or quadrilobic functions cosq and sinq v

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

M

===

szqyqx

sincos

, S[s [-1 , 1], = 0], [0, 2]

124

E X - C E N T R I C S a n d V A L E R I U A L A C I C U A D R O L O B S

-1 -0.5 0.5 1

-1

-0.5

0.5

1

-1-0.5

00.5

1

-1-0.5

00.5

1

-1

-0.5

0

0.5

1

-1-0.5

00.5

1

-1-0.5

00.5

1

-1 -0.5 0 0.5 1

-1-0.5

00.5

1

0

2

4

-1-0.5

00.5

1

-10

1

-10

1

0

1

2

3

4-1

01

-2 -1 1 2

-2

-1

1

2

-1 -0.5 0.5 1

-1

-0.5

0.5

1

125

P E R F E C T C U B E

-1-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1-0.5

0

0.5

1

-1

-0.5

0

0.5

1

X = cosq( , e = 1) . cosq ( e, = 1)

Y = cosq ( + /2, e = 1) = - sinq(, e = 1) Z = cosq (e + /2, = 1 ) = - sinq (u, e = 1)

126

E x c e n t r i c c i r c u l a r c u r v e s 1

M

==

)),(,()),(,(

xy

xx

sSsexysScexx

, x = y = 0

-1 -0.5 0.5 1

-1

-0.5

0.5

1

-1 -0.5 0.5 1

-1

-0.5

0.5

1

sx [ 0, 1], sy = 1 sx = 1, s y [ 0, 1]

-1 -0.5 0.5 1

-1

-0.5

0.5

1

-1 -0.5 0.5 1

-1

-0.5

0.5

1

sx [ 0, 1], sy = 0.5 sx = 0.5, s y [ 0, 1]

127

E x c e n t r i c c i r c u l a r c u r v e s 2

M

==

)),(,()),(,(

xy

xx

sSnsexysSmcexx

, x = y = 0

o r E x c e n t r i c L i s s a j o u s c u r v e s

-1 -0.5 0.5 1

-1

-0.5

0.5

1

-1 -0.5 0.5 1

-1

-0.5

0.5

1

-1 -0.5 0.5 1

-1

-0.5

0.5

1

-1 -0.5 0.5 1

-1

-0.5

0.5

1

m = 2, n = 3, s x [0, 1], sy = 0.5 m = 2, n = 3, s y [0, 1], sx = 0.5

128

References in SuperMathematics:

1 Selariu Mircea FUNCTII CIRCULARE EXCENTRICE Com. I Conferinta Nationala de Vibratii in Con-

structia de Masini , Timisoara , 1978,

pag.101...108.

2 Selariu Mircea FUNCTII CIRCULARE

EXCENTRICE si EXTENSIA LOR.

Bul .St.si Tehn. al I.P. TV Timisoara, Seria Mecanica,

Tomul 25(39), Fasc. 1-1980, pag. 189...196

3 Selariu Mircea STUDIUL VIBRATIILOR LIBERE ale UNUI SISTEM

NELINIAR, CONSERVATIV cu AJUTORUL

FUNCTIILOR CIRCULARE EXCENTRICE

Com. I Conf. Nat. Vibr.in C.M.

Timisoara,1978, pag. 95...100

4 Selariu Mircea APLICATII TEHNICE ale FUNCTIILOR

CIRCULARE EXCENTRICE

Com.a IV-a Conf. PUPR, Timisoara,

1981, Vol.1. pag. 142...150

5 Selariu Mircea THE DEFINITION of the ELLIPTIC EX-CENTRIC

with FIXED EX-CENTER

A V-a Conf. Nat. de Vibr. in Constr. de

Masini,Timisoara, 1985, pag. 175...182

6 Selariu Mircea ELLIPTIC EX-CENTRICS with MOBILE EX-

CENTER

IDEM pag. 183...188

7 Selariu Mircea CIRCULAR EX-CENTRICS and HYPERBOLICS

EX-CENTRICS

Com. a V-a Conf. Nat. V. C. M.

Timisoara, 1985, pag. 189...194.

8 Selariu Mircea EX-CENTRIC LISSAJOUS FIGURES IDEM, pag. 195...202

9 Selariu Mircea FUNCTIILE SUPERMATEMATICE CEX si SEX-

SOLUTIILE UNOR SISTEME MECANICE

NELINIARE

Com. a VII-a Conf.Nat. V.C.M.,

Timisoara,1993, pag. 275...284.

10 Selariu Mircea SUPERMATEMATICA Com.VII Conf. Internat. de Ing. Manag. si

Tehn.,TEHNO95 Timisoara, 1995, Vol. 9:

Matematica Aplicata,. pag.41...64

11 Selariu Mircea FORMA TRIGONOMETRICA a SUMEI si a

DIFERENTEI NUMERELOR COMPLEXE

Com.VII Conf. Internat. de Ing. Manag. si

Tehn., TEHNO95 Timisoara, 1995, Vol.

9: Matematica Aplicata,.,pag. 65...72

12 Selariu Mircea MISCAREA CIRCULARA EXCENTRICA Com.VII Conf. Internat. de Ing. Manag. si

Tehn. TEHNO95., Timisoara, 1995 Vol.7:

Mecatronica, Dispozitive si Rob.Ind.,pag.

85...102

13 Selariu Mircea RIGIDITATEA DINAMICA EXPRIMATA

CU FUNCTII SUPERMATEMATICE

Com.VII Conf. Internat. de Ing. Manag. si

Tehn., TEHNO95 Timisoara, 1995 Vol.7:

Mecatronica, Dispoz. si Rob.Ind.,pag.

185...194

129

14 Selariu Mircea DETERMINAREA ORICAT DE EXACTA A

RELATIRI DE CALCUL A INTEGRALEI

ELIPTICE COMPLETE

D E SPETA INTAIA K(k)

Bul. VIII-a Conf. de Vibr. Mec.,

Timisoara,1996, Vol III,

pag.15 ... 24.

15 Selariu Mircea FUNCTII SUPERMATEMATICE CIRCULARE

EXCENTRICE DE VARIABILA CENTRICA

A VIII_a Conf. Internat. de Ing. Manag. si

Tehn. TEHNO98, Timisoara, 1998, pag.

531...548

16 Selariu Mircea FUNCTII DE TRANZITIE INFORMATIONALA A VIII_a Conf. Internat. de Ing. Manag. si

Tehn. TEHNO98, Timisoara, 1998,

pag.549..556

17 Selariu Mircea FUNCTII SUPERMATEMATICE EXCENTRICE DE

VARIABILA CENTRICA CA SOLUTII ALE UNOR

SISTEME OSCILANTE NELINIARE

A VIII_a Conf. Internat. de Ing. Manag. si

Tehn. TEHNO98, Timisoara, 1998, pag.

557..572

18 Selariu Mircea TRANSFORMAREA RIGUROASA IN CERC A

DIAGRAMEI POLARE A COMPLIANTEI

Bul. X Conf. VCM ,Bul St. Si Tehn. Al

Univ. Poli. Timisoara, Seria Mec. Tom. 47

(61) mai 2002, Vol II pag. 247260

19 Selariu Mircea INTRODUCEREA STRAMBEI IN MATEMATICA Luc.Simp. Nat. Al Univ. Gh. Anghel

Drobeta Tr. Severin, mai 2003, pag.

171178

20 Petrisor

Emilia

ON THE DYNAMICS OF THE DEFORMED

STANDARD MAP

Workshop Dynamicas Days94, Budapest,

si Analele Univ.din Timisoara, Vol.XXXIII,

Fasc.1-1995, Seria Mat.-Inf.,pag. 91105

21 Petrisor

Emilia

SISTEME DINAMICE HAOTICE Seria Monografii matematice, Tipografia

Univ. de Vest din Timisoara, 1992

22 Petrisor

Emilia

RECONNECTION SCENARIOS AND THE

THERESHOLD OF RECONNECTION IN THE

DYNAMICS OF NONTWIST MAPS

Chaos, Solitons and Fractals, 14 ( 2002)

117127

23 Cioara

Romeo

FORME CLASICE PENTRU FUNCTII CIRCULARE

EXCENTRICE

Proceedings of the Scientific

Communications Meetings of "Aurel

Vlaicu" University, Third Edition, Arad,

1996, pg.61 ...65

24 Preda Horea REPREZENTAREA ASISTATA A

TRAIECTORILOR IN PLANUL FAZELOR A

VIBRATIILOR NELINIARE

Com. VI-a Conf.Nat.Vibr. in C.M.

Timisoara, 1993, pag.

25 Selariu Mircea

Ajiduah Crist.

Bozantan

INTEGRALELE UNOR FUNCTII

SUPERMATEMATICE

Com. VII Conf.Intern.de Ing.Manag.si

Tehn. TEHNO95 Timisoara. 1995,Vol.IX:

Matem.Aplic. pag.73...82

130

Emil (USA)

Filipescu Avr.

26 Selariu Mircea CALITATEA CONTROLULUI CALITATII Buletin AGIR anul II nr.2 (4) -1997

27 Selariu Mircea

Fritz Georg

(G)

Meszaros A.

(G)

ANALIZA CALITATII MISCARILOR PROGRAMATE

cu FUNCTII SUPERMATEMATICE

IDEM, Vol.7: Mecatronica, Dispozitive si

Rob.Ind.,

pag. 163...184

28 Selariu Mircea

Szekely

Barna

( Ungaria )

ALTALANOS SIKMECHANIZMUSOK

FORDULATSZAMAINAK ATVITELI FUGGVENYEI

MAGASFOKU MATEMATIKAVAL

Bul.St al Lucr. Prem.,Universitatea din

Budapesta, nov. 1992

29 Selariu Mircea

Popovici

Maria

A FELSOFOKU MATEMATIKA ALKALMAZASAI Bul.St al Lucr. Prem., Universitatea din

Budapesta, nov. 1994

30 Konig

Mariana

Selariu Mircea

PROGRAMAREA MISCARII DE CONTURARE A

ROBOTILOR INDUSTRIALI cu AJUTORUL

FUNCTIILOR TRIGONOMETRICE CIRCULARE

EXCENTRICE

MEROTEHNICA, Al V-lea Simp. Nat.de

Rob.Ind.cu Part .Internat. Bucuresti, 1985

pag.419...425

31 Konig

Mariana

Selariu Mircea

PROGRAMAREA MISCARII de CONTURARE ale

R I cu AJUTORUL FUNCTIILOR

TRIGONOMETRICE CIRCULARE EXCENTRICE,

Merotehnica, V-lea Simp. Nat.de RI cu

participare internationala, Buc.,1985,

pag. 419 ... 425.

32 Konig

Mariana

Selariu Mircea

THE STUDY OF THE UNIVERSAL PLUNGER IN

CONSOLE USING THE ECCENTRIC CIRCULAR

FUNCTIONS

Com. V-a Conf. PUPR, Timisoara, 1986,

pag.37...42

33 Staicu

Florentiu

Selariu Mircea

CICLOIDELE EXPRIMATE CU AJUTORUL

FUNCTIEI SUPERMATEMATICE REX

Com. VII Conf. Internationala de

Ing.Manag. si Tehn ,Timisoara

TEHNO95pag.195-204

34 Gheorghiu

Em. Octav

Selariu Mircea

Bozantan

Emil

FUNCTII CIRCULARE EXCENTRICE DE SUMA si

DIFERENTA DE ARCE

Ses.de com.st.stud.,Sectia

Matematica,Timisoara, Premiul II pe 1983

35 Gheorghiu

Octav,

Selariu

Mircea,

FUNCTII CIRCULARE EXCENTRICE. DEFINItII,

PROPRIETATI, APLICATII TEHNICE.

Ses. de com.st.stud. Sectia Matematica,

premiul II pe 1985.

131

Cojerean

Ovidiu

36 Filipescu

Avram

APLICAREA FUNCTIILOR ( ExPH ) EXCENTRICE

PSEUDOHIPERBOLICE IN TEHNICA

Com.VII-a Conf. Internat.de Ing. Manag.

si Tehn. TEHNO'95, Timisoara, Vol. 9.

Matematica aplicata., pag. 181 ... 185

37 Dragomir

Lucian

(Toronto

- Canada )

UTILIZAREA FUNCTIILOR SUPERMATEMATICE

IN CAD / CAM : SM-CAD / CAM. Nota I-a:

REPREZENTARE IN 2D




38 Selariu

Serban

UTILIZAREA FUNCTIILOR SUPERMATEMATICE

IN CAD / CAM : SM-CAD / CAM. Nota I I -a:

REPREZENTARE IN 3D




39 Staicu

Florentiu

DISPOZITIVE UNIVERSALE de PRELUCRARE a

SUPRAFETELOR COMPLEXE de TIPUL

EXCENTRICELOR ELIPTICE

Com. Ses. anuale de com.st. Oradea ,

1994

40 George

LeMac

THE EX-CENTRIC TRIGONOMETRIC

FUNCTIONS: an extension of the classical

trigonometric functions.

The University of Western Ontario,

London, Ontario, Canada

Depertment of Applied Mathematics

May 18, 2001, pag.1...78

41 Selariu Mircea PROIECTAREA DISPOZITIVELOR DE

PRELUCRARE, Cap. 17 din PROIECTAREA

DISPOZITIVELOR

Editura Didactica si Pedagogica,

Bucuresti, 1982, pag. 474 ... 543

Documents

Techno-Art of Selariu SuperMathematics Functions