M02164
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
INNOMA (Innovation on Mathematics) with Curves and
Surfaces for Enrichment Mathematics Curriculum
H A Parhusip*
Mathematics Department , Science and Mathematics Faculty
Universitas Kristen Satya Wacana, Salatiga ,Jawa Tengah Indonesia
*
hanna.arini@staff.uksw.edu
Abstract. Innovation on mathematics with curves and surfaces are shown here to give
enrichment in mathematics curriculum. The innovation relates to present mathematics into
objects instead of presenting formulas directly. The enrichment here means that students may
learn more than the standard curriculum for particular topics. The simple circle equation sphere
are used to deliver ideas of innovation and enrichment yielding some new mathematical
expressions leading to enrichment of studying circle and sphere in schools. In a topic from
undergraduate mathematics, the parametric representation of circle and sphere are then
implemented for modifying a hypocycloid curve into modified curves and surfaces. The
derivative of each curve or surface is also illustrated. The visualizations are done with
MATLAB. Other innovations with algebraic surfaces are introduced using SURFER such that
the visualizations are done without writing a code. The resulting curves and surfaces are then
used to be curve stitching designs (can be used in schools), batik motifs (can be a mathematical
project), mathematical ornaments and motifs for several kinds of souvenirs (bags, t-shirt, glass)
for promoting mathematics.
Key words : circle, sphere, hypocycloid, curve stitching algebraic surfaces.
1. Introduction
Mathematics is mainly considered to be collection of formulas. Therefore mathematical symbols are
frequently introduced firstly in learning and teaching mathematics traditionally leading to negative
attitude of public images about the essence of mathematics (only numbers and formulas). On the other
hand mathematics is thought meaningful for wide areas. This is not known in basic level of education.
As a result, students gradually understand mathematics with lots of formulas rather than its
applications.
One way for breaking the gap between mathematics as formulas and its applications is presenting
mathematics by innovation and enrichment. An example in enrichment is done by involves offering
learners opportunities to pose as well as solve challenging (non-routine) problems that allow for
different methods, require fluency in the problem solving processes and encourage the identification of
elegant or efficient solutions (for problem-solving) (Piggott, 2007). An innovation in teaching some
topics of mathematics at the college Level is done by using Geogebra (Diković, 2009). In this paper,
an innovation here is meant by visualizing mathematics into some objects, popularizing mathematics
by exhibitions, mathematical games and art in different media such as films in you tubes and internet,
integrated topics in mathematics such as calculus, complex functions and geometry into a new topic
beyond calculus. The visualization can be started from various curves, surfaces using the known
1
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
software since mathematical operators and expression have geometrical meanings. In this paper,
mathematical operators and expression will be introduced through curves and surfaces.
There have been several software are developed such as Geogebra, Maple, MATLAB where
geometrical mathematics expression are easily visualized. Other software such as Cinderella and
Surfer are more challenging to present more dynamic physical operator and algebraic surfaces
respectively. The available software influences the development on teaching, learning and exploring
mathematics and applied science.
This paper is expected to inspire educators and researchers in schools and undergraduate mathematics
particularly to have their own enrichment in routine materials of teaching (curriculum based) into a
growing study such that the essence of research can be adjusted automatically by students. Some
examples of these explorations are shown to be innovation of some mathematical lessons in schools
and undergraduate mathematics.
.
2. Background for Creating Innovation
Several ideas for popularizing mathematics have been addressed by many groups in mathematics and
educations (Ahuja, 1996). The research of innovations here have been developed with curves and
surfaces meaning the known equations are explored, e.g. mapped with complex functions, extended
into 3 dimensional equations, using well known sequences and ratio, i.e. Fibonacci and Golden Ratio
(GR) for creating Fibonacci curves and surfaces Fibonacci (Parhusip,2016). Some motifs were
obtained and tried to find the similarity in natures such as flowers and animals where sacred geometry
was studied (Suryaningsih et.al.,2013). Hypocycloid curve and its derivative were explored into 3
dimensional with many different values of parameters where new curves and surfaces were collected
to be possible mathematical ornaments, accessories and designs for souvenirs
(Purwoto,2014)(Parhusip,2015) and exhibited for popularizing mathematics Some designs have been
used as designs in batik (called BATIMA) and for new curves stitching designs. The background to
start an innovation in mathematics is shown here as examples for a further individual learning.
2.1 Innovation on Circle
Before we do an innovation, one has already standard equations e.g. an equation for a circle, a ball in 3
dimensional. For instance, x 2 y 2 r 2 is a circle with a radius r and a center O on origin of
coordinate system. This equation is known in junior/senior schools. Similar expression is obtained in
undergraduate mathematics such as an astroid is formulated by x 2 / 3 y 2 / 3 r 2 / 3 . In number
theory, the expression x p y p r p is studied for positive integers x,y,r where p > 2, p integers.
1/ 2
Consider x p y p r p and p=4, we take r z such that the equation becomes
x4 y4 z 2
(1)
It was proven that x, y, z all nonzero and relatively prime integers (Boston,2003) , the Eq. (1)
has no solution. Note that in number theory we seek positive triple (x,y,z) satisfying Eq.(1). It is
known as Fermat Last Theorem, that this expression has no positive integer solutions stated since 1637
and proven formally in 1995 by Wiley. We see that the number theory focuses on a point containing
positive integer (x,y,r) for x p y p r p . In geometry, one has more pay attention on its geometrical
interpretation. An innovation is started.
2.2 Geometrical innovation in a modified circle equation
In this paper, we create an innovation in circle by observing values of p for a positive fixed
value of r where several innovations are encountered. For a given r, we may write :
xp yp rp
(2.a)
into
2
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
x r cos 2 / p , y r sin 2 / p .
(2.b)
Substituting the expression Eq.(2.b) into y r x , i.e.
p
p
p
y p r p r cos 2 / p
r p 1 cos 2 .
p
2/ p
2/ p
For 12. One may predict that for
p=k/3 with k 5 , Eq.(1) creates a family of curves, the same attitudes for arbitrary integer values of
p 5 . By extending the domain of the axes, we have still a close curve. By increasing the value of p,
i.e. p=13/3, 14/3, 15/3, 16/3, a family of weak kites appears since all curves behave the same as in Fig.
4. Again, the values of p are varied in the form k/3 similarities are observed as depicted in Fig.5. The
observations lead to patterns of weak kites for p started from 13/3 up to 33/3. Processing up to
p
p=100/3, similar patterns are obtained. Note that the solutions of z1 z 2 r can be complex
p
p
numbers. Therefore the real part solutions of z1 z 2 r are concluded to be family of weak kites
for p=k/3 with k 13,14,.... One may prove formally which is not shown here. As mentioned
above that the interest in this paper is the geometrical phenomenon of (Re z1,Re z2) satisfying.
p
p
p
z1 z 2 r p .
p
p
4
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
p=1,1/2 /, 1/3 and ¼
a weak circle for p=4
or 12/3.
p=1/3,2/3/, 3/3 and 4/3
p=13/3,14/3 , 15/3,16/3
p=5/3, p=6/3,
and p=8/3.
p=9/3, p=10/3, p=11/3
p=7/3 and 4=12/3.
p=17/3,18/3 ,19/3,20/3.
Figure 5. Real solutions of x y r
p
3.
p
p=21/3,22/3 ,23/3,24/3.
p
Innovation on Surfaces
3.1 Modified ball equation
Some well known curve and surface such as a circle and a ball are presented in Cartesian
coordinates, polar coordinates and parametric coordinates. The equation x y r ( x, y) R
describes a circle in Cartesian coordinate for p=2. In particular, an astroid can be obtained for p=2/3.
p
Rectangular representation of astroid is x
2/3
p
p
2
y 2 / 3 r 2 / 3 and its parametric is in the form
x r cos 3 and y r sin 3 . The second studied geometrical surface is a generalization of ball,
i.e.
xp yp zp rp .
(3)
This equation gives us a ball with a center O and its radius r for p=2. Furthermore, one may extend the
Eq. (3) to be a 3 dimensional astroid by using spherical coordinates, i.e.
x sin 3 cos 3 ; y sin 3 sin 3 ; z cos 3 , 0 2 ; 0 .
(4)
and the geometrical illustration is shown in Fig. 6.
Figure 6. Ilustration of 3 dimensional astroid using spherical coordinates.
As in 2 dimensional case, one may express Eq.(3) into parametric equations, i.e.
x sin 2 / p cos 2 / p ; y sin 2 / p sin 2 / p ; z cos 2 / p ;0 2 ; 0 .
(5).
Since 3 dimensional astroid can be defined, one generalizes Eq.(3) into Eq.(5) and try to find
the appeared geometrical phenomena for different values of p . Thus, the surfaces given by Eq.(5) will
be the real parts of solutions
z1 z2 z3 r p ; z1 , z 2 , z3 C 3 r , p R .
p
p
p
5
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
3.2 Innovation from modified hypocycloid curve
The third type innovation done in this paper is a generalization of 2 dimensional hypocycloid
curve into 3 dimensional surfaces, i.e. (Purwoto, et.al, 2014)
a b
x sin a b cos b cos
x sin ;
b
a b
y sin a b sin b sin
y sin ;
b
z cos
.
(6)
r
; r x 2 y 2 and a, b are real numbers. Originally, a and b denote radius of 2
where
sin
different circles, a > b. In this case, a and b are freely chosen. Another innovation is done by
replacing the minus signs in the Eq.(6) to be positive. The innovation can still be created by presenting
of each derivative to create the new curves (from x , y ) and surfaces. This paper provides some
similar patterns in curves and surfaces for the generalization defined above.
3.3 Innovation from Algebraic Surfaces
According to the equations used to create surfaces, algebraic surfaces are surfaces created from
polynomials of three variables, e.g. ellipsoid, sphere classified as quadric surfaces. Other
classifications are quartic surfaces, Caley cubic, Barth Sextic, and many more. One may refer to
Wikipedia for other variation of algebraic surfaces. There is an available software called SURFER to
visualize those algebraic surfaces such that a user simply writes an algebraic equation to create a
surface without writing a code. The used of SURFER is not only for presenting algebraic surfaces for
education but also for architecture and art (). The examples for some algebraic surfaces using
SURFER will be shown.
Among all surfaces, the mathematical studies usually involve several types of curvatures on
curves and surfaces. In this paper, the curvatures are not discussed since the main focus is about
innovating curves and surfaces. Finally, the next chapter shows innovation results for 3 classifications
above, i.e. innovation on curves and surfaces from circle- sphere equations, modified hypocycloid and
algebraic surfaces.
4
Innovation Results
4.1 Innovation Surfaces of Generalized Ball Equations
Note that the derivatives of the equations may create other curve and surfaces. In 3
dimensional case (see Eq.5), we may have some possible triple to visualize the surfaces due to the
derivatives, i.e. x ; x ; x ; y ; y ; y ; z ; z and z . Since z =0 ,the number of possible
8
surfaces are C 3 =56 . Furthermore, the second derivatives may be created.
Since p=2/3 has led to a known geometry (astroid), we try for p=k/3, k=1, 2,…,20. One
observes for p=12,…,20 the surfaces have the same families. The collections of surfaces are called
parking surfaces. Some surfaces are shown in Fig.7.
6
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
Figure 7. A family of parking surfaces for p=13/3,14//3,15/3/16/3 (from left to right)
Note that the 2 dimensional projections have the same images as shown in Figure 5.
Finally some surfaces are found by varying the values of p. Started with astroid in 3
dimensional (p=2/3) and vary for p=2/k, k=4, 5, 6, 7, 8, 9, one observes geometrical patterns as
depicted in Fig. 8. Weak astroids are observed for p=2/3, 2/5, 2/7 and 2/9 with diminishing surfaces
area. Generally, one may have weak astroids for p=2/(2n+1), for n is natural number.
3
Figure 8. Visualization of Re z1 , Re z 2 , Re z3 C ; r R , p=2/3, 2/4, 2/5, 2/6, 2/7, 2/8, 2/9 (from
left to right).
As we notice, The real solutions z1 z2 z3 r ; have closed astroid surfaces with
decreasing area of surfaces for p=2/3, p/5, p/7. On the other hand the open surfaces are found for
p=2/4, 2/6, 2/8. One may increase the value of denominator and we have the area of surfaces tends to
zero for all p. Some variations of parameters have been observed in the results of surfaces where the
p
p
p
p
value of p / k , k=1,2,3,…,15.
For irrational numbers of p some interesting patterns of surfaces are found. One observes that
families of surfaces are started from p / 5 . Closed surfaces called weak astroid are obtained for
p / 5 , p / 8 , p / 11 and p / 14 with each area is diminishing. Otherwise, for
p / 6 , p / 7 , ..., p / 13 , the open surfaces are shown with similar pattern in Fig. 9-10. By
studying the surfaces created by the derivatives, open surfaces are obtained for p=2/k, k is an odd
number and closed surfaces for p=2/t where t is an even number (see Fig. 11). As above, the area of
surfaces are decreasing for increasing values of k and t. These surfaces are possible to be some
mathematical ornaments or for other purposes such as motifs for batik.
Figure 9. Surfaces for p / k , k=1,2,3,4,5
Figure 10. Surfaces for p / k , k=6,7,…,15
Figure 11. Surfaces
dx dy dz
; ; ,p=2/k,
d d d
k=3,4,…,10.
4.2 Result of innovation for curve stitching and mathematical art
7
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
As mention previous sections, various curves and surfaces have been obtained. The obtained
designs can be 2 dimensional and 3 dimensional. This paper focuses on presenting the 2 dimensional
designs for curve stitching that can be used for students to be material for innovation on mathematics.
Originally curve stitching has been designed with creating line or parabola into different kinds of
motifs. This research has obtained many curves from a modification of hypocycloid curve into kind of
polygons called Connected Pseudo Polygon (CPP). Several designs of CPP for curve stitching are
displayed in Fig.11-12.
1.5
500
3
400
1
2
300
0.5
200
1
100
0
0
0
-0.5
-100
-1
-1
-200
-300
-1.5
-2
-400
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
a=1, p=24, q=12, dan b=2 , n=50 (the
number of points).
-500
-500
-400
-300
-200
-100
0
100
200
300
400
-3
-3
500
-2.5
a=457, p=35, q=33, b=-(p/q), n=50
(the number of points).
-2
-1.5
-1
-0.5
0
0.5
1
1.5
a=1, p=24, q=12, b=2, n=50
(the number of points).
Figure 11. Modified Hypocycloid curve
a b ;
x( ) ( a b ) cos b cos
y( ) ( a b ) sin b sin a b .
b
b
50
80
2500
150
40
2000
60
30
40
100
1500
20
1000
50
10
20
500
0
0
0
0
-10
-500
-20
-20
-1000
-50
-30
-40
-1500
-40
-60
-100
-50
-50
-80
-80
-60
-40
-20
0
20
40
60
-40
-30
-20
-10
0
10
20
30
40
-2000
50
-2500
-2500 -2000 -1500 -1000
80
Derivative of modified
hypocycloid curve ; a=20, p=800,
q=-500, b=1.6 , n=25 (the
number of points)
a=20, p=250, q=-20, b=12.5 ,
n=25 (the number of points).
-150
-150
-100
-50
0
50
100
-500
0
500
1000
1500
2000
2500
150
p=450;q=-10;a=20;b=(p/q); n=13 (the number
of points).
p=450;q=10;a=20;b=(p/q); n=13 (the number of
points). Repeated by
Fibonacci sequence.
Figure 12. Modified hypocycloid curve :
a b y (a b) sin b sin a b .
x (a b) cos b cos
b
b
The designs of curve stitching strongly depend on the order of arranging the material. Thus the designs
require numbering on each vertex. Finally by following the numbers, the curve stitching can be made.
The examples are shown in Fig.13. Furthermore, if one requires a particular size, the code can provide
the minimum one dimensional material.
Figure 13. Curve stitching based on modified hypocycloid curve.
4.3 Results of surfaces innovations
Several mathematical ornaments and art are created based on the modified hypocycloid and
weak balls obtained in Section 3. Figure 14 and Fig.15 illustrate 2 pairs of designs and the related
products made by at least 11 home industries for all products. In this paper, the results from 4 home
industries are displayed. Figure 16 illustrates designs and the related products called BATIMA (as one
of Indonesian innovations in 2016) and puzzles with the same designs. Thus the same designs can be
used for several purposes. There are also designs can be used for an architecture for a building in
future.
8
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
dx dy dz p=1;q=4; a=1;
; ;
d d d
b=-(p/q)
Second derivative of hypocycloid for
p=1;q=4;a=1;b=-(p/q)
dx dy p=1;q=4;a=1;b=-(p/q)
, , z
d d
Figure 14. Design and its surface of Derivative of hypocycloid
’
x p y p z p r p for p / 7
Derivative of hypocycloid for
p=1;q=6; a=1;b=-(p/q) (second
pair, right)
The second derivative of
hypocycloid for
a=1;p=1;q=4;b=p/q.
Figure 15. Design and its surface of modified hypocycloid and weak balls
Figure 16. Designs and the related products (puzzle and batik (BATIMA))
4.4 Innovation on Algebraic Surfaces
A sphere is one of algebraic surfaces in quadric surfaces. The other quadric surfaces are, Cone,
ellipsoid, paraboloid. The algebraic surfaces are governed by polynomial surfaces. A new
development for creating algebraic surfaces is a software called surfer (found by Imaginary research
group from Germany). This software can be easily used to visualize without doing programming as
above. Several surfaces are displayed here as examples. Figure18 can be models for Christmas
ornaments called Santa Xmas Ornaments, and Xmas-hats are also created. Additionally, the designs
can also be used to be batik motifs.
Figure 18.Santa Xmas-Ornaments by an algebraic surface, i.e:
2
2
2
x y z 2 2 xyz 1 x 1 y 1 z 1 2x 1 y 1z 1 1 0
2
2
9
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
Figure 19. Xmas-hats from algebraic surfaces
as combination of several 3 and 5 spheres with 1 ellipsoid.
Conclusion
Creations curves and surfaces as innovations on mathematics are discussed to provide
enrichment of mathematics curriculum. The exponent in the circle equation is changed to be any real
number leading to find the complex sets of solutions and the related geometrical solutions. This
approach is extended to a sphere where the geometrical surfaces as real parts of the solutions. The
spherical coordinates are used to express the modified hypocycloid curve into new surfaces including
the derivatives to give visualisation between the equation and its derivative.
The results are classified into 3 types of innovations. The first is an innovation on extending
circle and sphere into higher exponent in the equations yielding to enrichment of curves and surfaces
into weak kites and weak balls respectively. The modified hypocycloid curves are the second
innovation finally collected to be designs for curve stitching and the surfaces employed to be batik
motifs, puzzles and mathematical ornaments. The designs are also used for several designs on
souvenirs. The batik creation called BATIMA has been selected as one of 108 Indonesian innovations
in 2016 by BIC (Business Innovation Center).
The algebraic surfaces as the third innovation have also been used to be surfaces innovation
through SURFER where the expression can be typed directly to create the surfaces. The Xmas
Ornaments and Xmas-hats are shown here to be examples of the surfaces.
Remark
The approach in this paper very rare met with reputable and modern journals since the
innovation means that the creation beyond the standard related topics. Therefore, author has very
limited references.
Acknowledgment
Special thank to Dr. Bianca Violet from Imaginary Mathematisches Forschungsinstitut
Oberwolfach,Germany who has inspired me to use SURFER for creating algebraic surfaces.
References
[1] Ahuja O P 1996. Let’s Popularize Mathematics, The Mathematics Educator , Vol. l , No. 1. 8298.
[2] Piggott J 2007 The nature of mathematical enrichment: a case study of Implementation, Educate,
Vol.7, No.2, pp. 30-45,University of Cambridge, Faculty of Education.
[3] Diković L 2009 Applications GeoGebra into teaching some topics of Mathematics at the College
Level, Computer Science and Information Systems , 6(2), 191-203.
[4] Parhusip H A 2014 Arts revealed in calculus and its extension. International Journal of
Statistics and Mathematics, 1(3): 002-009, Premier-Publisher.
[5] Parhusip H A 2015 Disain ODEMA (Ornament Decorative Mathematics) untuk Populerisasi
Matematika, Proceeding, Seminar Nasional Matematika,Sains dan Informatika, dalam rangka
Dies Natalis ke 39 UNS, FMIPA, UNS, ISBN-978-602-18580-3-5 ,pp. 8-15 (in Indonesian).
10
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
[6]
Parhusip H A 2010 Learning Complex Function And Its Visualization With Matlab, South East
Asian Conference On Mathematics And Its Applications Proceedings Institut Teknologi
Sepuluh Nopember, Surabaya , ISBN 978 – 979 – 96152 – 5 – 1,pp.373-384.
[7]
1
Parhusip H A and Sulistyono 2009 Pemetaan w dan hasil pemetaannya, Prosiding,
z
Seminar Nasional Matematika dan Pendidikan Matematika,FMIPA UNY 5 Des 2009, ISBN :
978-979-16353-3-2,T-16, hal. 1127-1138 (Indonesian).
[8] Purwoto Parhusip H A, Mahatma T 2014 Perluasan Kurva Parametrik Hypocycloid Dimensi
Menjadi 3 Dimensi Dengan Sistem Koordinat Bola, prosiding Seminar Nasional UNNES,8 Nov
2014, ISBN 978-602-1034-06-4; hal.326-336 (Indonesian).
[9] Hartkopf A and Matt A D 2013 SURFER in Math Art, Education and Science Communication
Proceedings of Bridges: Mathematics, Music, Art, Architecture, Culture.
[10] Suryaningsih V Parhusip H A, Mahatma T 2013 Kurva Parametrik dan Transformasinya untuk
Pembentukan Motif Dekoratif, Prosiding, Seminar Nasional Matematika dan Pendidikan
Matematika UNY,9 Nov, ISBN:978-979-16353-9-4,hal. MT – 249-258.
[11] Parhusip H A 2016 Sains dan Matematika untuk Art untuk Technoeducation, Science Festival ,
organized by LK-FSM, UKSW, 26 Feb 2016, di gedung GKJ Salatiga. DOI:
10.13140/RG.2.1.4388.8402
[12] Parhusip H A 2014 Arts revealed in Calculus and Its Extension International Journal of
Statistics and
Mathematics, 1(3): 002-009, Premier-Publisher, available online on :
http://premierpublishers.org/ijsm/articles
11
SCHEDULE
International Conference on Mathematics :
Education, Theory, and Application (ICMETA) 2016
6th & 7th December 2016, Hotel ALILA
Solo, Central Java, Indonesia
Department of Mathematics
Faculty of Mathematics and Natural Sciences
Universitas Sebelas Maret
1st Day, Tuesday, December 6, 2016
Time Schedule
Time
’
Registration
08.00 - 08.10
’
1st Day Opening
Ballroom Hotel Alila
Opening of Master of Ceremony
08.10 - 08.20
’
Samsuri Sutarna, M.Sn
Lecturer of Institute Seni Indonesia-Surakarta
Central Java
08.20 - 08.40
’
Vox Magistra FKIP UNS
07.00 am08.00am
Venue
Activities
Duration
Lobby Ballroom
Traditional Dance
Performance :
Tari Klana Topeng
a. Singing Traditional
b. Singing National
Anthem (audiences)
WELCOME SPEECH
08.40 - 08.50
Dr. Dewi Retno Sari Saputro, M.Kom.
Chair of ICMETA
’
08.50 - 09.00
’
Prof. Ari Handono Ramelan,
M.Sc.(Hons),Ph.D.
Dean of FMIPA
09.00 - 09.10
’
Prof. Dr. Ravik Karsidi, M.S.
Rector of Universitas Sebelas
Maret
09.10 - 09.25
5’
Coffee Break
Lobby Ballroom Hotel Alila
Keynote Session - Ballroom Hotel Alila
Prof. Drs. Tri Atmojo K, M,Sc. Ph.D.
Department of Mathematics, Faculty of
Mathematics and Natural Sciences
Universitas Sebelas Maret, Indonesia
Moderator
09.25 - 10.10
’
Prof. Dr. Edy Tri Baskoro, M.Sc.
Department of Mathematics, Faculty of
Mathematics and Natural Sciences, Bandung
Institute of Technology , Indonesia
Keynote Speaker I
10.10 - 10.55
’
Keynote Speaker II
10.55 - 11.10
5’
Dr. Darfiana Nur, M.Sc.
Statistical Science in the School of Computer
Science, Engineering and Mathematics,
Flinders University, Australia
Panel Discussion and Q & A Session
Putranto Hadi Utomo, M.Si.
Department of Mathematics, Faculty of
Mathematics and Natural Sciences
Universitas Sebelas Maret, Indonesia
Moderator
SCHEDULE-ICMETA, DECEMBER 6-7, 2016
1
’
11.10 - 11.55
11.55-12.40
12.40 -12.55
’
12.55 -13.40
45’
Dr. G.R. (Ruud) Pellikaan
Department of Mathematics and Computer
Science, Technische Universiteit Eindhoven,
Netherland
Keynote Speaker III
Prof. Dr. Mohd Bin Omar
Institute of Mathematical Sciences, Faculty of
Science, University of Malaya, Malaysia
Panel Discussion and Q & A Session
Keynote Speaker IV
Lunch
Lobby Ballroom Hotel Alila
Paper Presentation Parallel Session 1
’
13.40 - 15.00
15.00 - 15.30
’
Mathematics Education
a. Room 1
5 papers
b. Room 2
5 papers
c. Room 3
5 papers
Combinatorics
a. Room 4
5 papers
b. Room 5
5 papers
Statistics
a. Room 6
5 papers
b. Room 7
5 papers
Applied Mathematics
a. Room 8
5 papers
b. Room 9
5 papers
c. Room 10
5 papers
Coffee Break
Room Meeting Hotel Alila
Room Meeting Hotel Alila
Paper Presentation Parallel Session 2
15.30 - 16.45
’
16.45 -17.00
5’
Mathematics Education
a. Room 1
4 papers
b. Room 2
5 papers
c. Room 3
5 papers
Combinatorics
a. Room 4
6 papers
b. Room 5
5 papers
Statistics
a. Room 6
5 papers
b. Room 7
5 papers
Applied Mathematics
a. Room 8
4 papers
b. Room 9
4 papers
c. Room 10
4 papers
End of 1st Day-Closing Remarks
Certificate Distribution
Room Meeting Hotel Alila
SCHEDULE-ICMETA, DECEMBER 6-7, 2016
2
2nd Day, Wednesday, December 7, 2016
Time Schedule
Time
07.00 - 08.00
Activities
Duration
’
Registration
Venue
Lobby Ball Room
Hotel Alila
2nd Day Opening
Ballroom Hotel Alila
08.00 - 08.15
’
08.15 - 08.30
20’
Opening of Master of Ceremony
Coffee Break
Lobby Ball Room
Hotel Alila
Keynote Session
Ballroom Hotel Alila
Dra. Mania Roswitha, M.Si.
Department of Mathematics, Faculty of
Mathematics and Natural Sciences
Universitas Sebelas Maret, Indonesia
Moderator
08.30 - 09.25
’
Dr. Hanna Arini Parhusip
Department of Mathematics, Faculty of Science
and Mathematics, UKSW, Salatiga Indonesia
Keynote Speaker V
09.25-10.10
’
Prof. Dr. Kenjiro T. Miura
Realistic Modeling Laboratory, Department of
Mechanical Engineering, Shizuoka University,
Japan
Keynote Speaker VI
10.10-10.25
’
Panel Discussion and Q & A Session
Keynote Session-Ballroom Hotel Alila
Drs. Isnandar Slamet, M.Sc., Ph.D.
Department of Statisticss, Faculty of
Mathematics and Natural Sciences
Universitas Sebelas Maret, Indonesia
Moderator
10.25 - 11.10
’
Dr. Sutanto, DEA
Department of Mathematics, Faculty of
Mathematics and Natural Sciences
Universitas Sebelas Maret, Indonesia
Keynote Speaker VII
11.10 - 11.55
’
Dr. Benoît Liquet
Laboratory of Mathematics and Their
Applications, Université de Pau et des Pays de
l'Adou, France
Keynote Speaker VIII
11.55 - 12.10
’
Panel Discussion and Q & A Session
SCHEDULE-ICMETA, DECEMBER 6-7, 2016
3
12.10 - 13.15
’
Lobby Ballroom
Hotel Alila
Lunch
Paper Presentation Parallel Session 3
13.15 - 15.00
’
a.
b.
c.
a.
b.
a.
b.
c.
15.00 - 15.30
’
15.30 - 16.30
’
16.30 - 16.45
5’
Mathematics Education
Room 1
5 papers
Room 2
5 papers
Room 3
5 papers
Combinatorics
Room 4
5 papers
Statistics
Room 5
5 papers
Room 6
5 papers
Room Algebra & Analysis
Room 7
5 papers
Applied Mathematics
Room 8
5 papers
Room 9
5 papers
Room 10
5 papers
Room Meeting Hotel
Alila
Coffee Break
Paper Presentation Parallel Session 4
Mathematics Education
a. Room 1
4 papers
b. Room 2
5 papers
c. Room 3
4 papers
Combinatorics
Room 4
5 papers
Statistics
a. Room 5
4 papers
b. Room 6
4 papers
Room Algebra & Analysis
Room 7
3 papers
Applied Mathematics
a. Room 8
4 papers
b. Room 9
5 papers
c. Room 10
4 papers
nd
End of 2 Day -Closing Remarks
Certificate Distribution
Room Meeting Hotel
Alila
SCHEDULE-ICMETA, DECEMBER 6-7, 2016
4
International Conference on Mathematics :
Education, Theory and Application (ICMETA)
December 6th&7th, 2016
http://ICMETA.uns.ac.id
No
: 021/ICMETA/2016
Subject :Invitation as Keynote Speaker for the ICMETA
Dear Dr Hanna Arini Parhusip
We are pleased to invite you to the International Conference on Mathematics : Education,
Theory & Application (ICMETA) scheduled to be held between (Desember 6
Secretariat ICMETA 2016 :
2nd floor, A Building
Department of Mathematics, Faculty of Mathematics and Natural Sciences
SebelasMaret University
Ir. Sutami 36 A Solo, Central Java, Indonesia
e-mail : icmeta@mail.uns.ac.id
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
INNOMA (Innovation on Mathematics) with Curves and
Surfaces for Enrichment Mathematics Curriculum
H A Parhusip*
Mathematics Department , Science and Mathematics Faculty
Universitas Kristen Satya Wacana, Salatiga ,Jawa Tengah Indonesia
*
hanna.arini@staff.uksw.edu
Abstract. Innovation on mathematics with curves and surfaces are shown here to give
enrichment in mathematics curriculum. The innovation relates to present mathematics into
objects instead of presenting formulas directly. The enrichment here means that students may
learn more than the standard curriculum for particular topics. The simple circle equation sphere
are used to deliver ideas of innovation and enrichment yielding some new mathematical
expressions leading to enrichment of studying circle and sphere in schools. In a topic from
undergraduate mathematics, the parametric representation of circle and sphere are then
implemented for modifying a hypocycloid curve into modified curves and surfaces. The
derivative of each curve or surface is also illustrated. The visualizations are done with
MATLAB. Other innovations with algebraic surfaces are introduced using SURFER such that
the visualizations are done without writing a code. The resulting curves and surfaces are then
used to be curve stitching designs (can be used in schools), batik motifs (can be a mathematical
project), mathematical ornaments and motifs for several kinds of souvenirs (bags, t-shirt, glass)
for promoting mathematics.
Key words : circle, sphere, hypocycloid, curve stitching algebraic surfaces.
1. Introduction
Mathematics is mainly considered to be collection of formulas. Therefore mathematical symbols are
frequently introduced firstly in learning and teaching mathematics traditionally leading to negative
attitude of public images about the essence of mathematics (only numbers and formulas). On the other
hand mathematics is thought meaningful for wide areas. This is not known in basic level of education.
As a result, students gradually understand mathematics with lots of formulas rather than its
applications.
One way for breaking the gap between mathematics as formulas and its applications is presenting
mathematics by innovation and enrichment. An example in enrichment is done by involves offering
learners opportunities to pose as well as solve challenging (non-routine) problems that allow for
different methods, require fluency in the problem solving processes and encourage the identification of
elegant or efficient solutions (for problem-solving) (Piggott, 2007). An innovation in teaching some
topics of mathematics at the college Level is done by using Geogebra (Diković, 2009). In this paper,
an innovation here is meant by visualizing mathematics into some objects, popularizing mathematics
by exhibitions, mathematical games and art in different media such as films in you tubes and internet,
integrated topics in mathematics such as calculus, complex functions and geometry into a new topic
beyond calculus. The visualization can be started from various curves, surfaces using the known
1
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
software since mathematical operators and expression have geometrical meanings. In this paper,
mathematical operators and expression will be introduced through curves and surfaces.
There have been several software are developed such as Geogebra, Maple, MATLAB where
geometrical mathematics expression are easily visualized. Other software such as Cinderella and
Surfer are more challenging to present more dynamic physical operator and algebraic surfaces
respectively. The available software influences the development on teaching, learning and exploring
mathematics and applied science.
This paper is expected to inspire educators and researchers in schools and undergraduate mathematics
particularly to have their own enrichment in routine materials of teaching (curriculum based) into a
growing study such that the essence of research can be adjusted automatically by students. Some
examples of these explorations are shown to be innovation of some mathematical lessons in schools
and undergraduate mathematics.
.
2. Background for Creating Innovation
Several ideas for popularizing mathematics have been addressed by many groups in mathematics and
educations (Ahuja, 1996). The research of innovations here have been developed with curves and
surfaces meaning the known equations are explored, e.g. mapped with complex functions, extended
into 3 dimensional equations, using well known sequences and ratio, i.e. Fibonacci and Golden Ratio
(GR) for creating Fibonacci curves and surfaces Fibonacci (Parhusip,2016). Some motifs were
obtained and tried to find the similarity in natures such as flowers and animals where sacred geometry
was studied (Suryaningsih et.al.,2013). Hypocycloid curve and its derivative were explored into 3
dimensional with many different values of parameters where new curves and surfaces were collected
to be possible mathematical ornaments, accessories and designs for souvenirs
(Purwoto,2014)(Parhusip,2015) and exhibited for popularizing mathematics Some designs have been
used as designs in batik (called BATIMA) and for new curves stitching designs. The background to
start an innovation in mathematics is shown here as examples for a further individual learning.
2.1 Innovation on Circle
Before we do an innovation, one has already standard equations e.g. an equation for a circle, a ball in 3
dimensional. For instance, x 2 y 2 r 2 is a circle with a radius r and a center O on origin of
coordinate system. This equation is known in junior/senior schools. Similar expression is obtained in
undergraduate mathematics such as an astroid is formulated by x 2 / 3 y 2 / 3 r 2 / 3 . In number
theory, the expression x p y p r p is studied for positive integers x,y,r where p > 2, p integers.
1/ 2
Consider x p y p r p and p=4, we take r z such that the equation becomes
x4 y4 z 2
(1)
It was proven that x, y, z all nonzero and relatively prime integers (Boston,2003) , the Eq. (1)
has no solution. Note that in number theory we seek positive triple (x,y,z) satisfying Eq.(1). It is
known as Fermat Last Theorem, that this expression has no positive integer solutions stated since 1637
and proven formally in 1995 by Wiley. We see that the number theory focuses on a point containing
positive integer (x,y,r) for x p y p r p . In geometry, one has more pay attention on its geometrical
interpretation. An innovation is started.
2.2 Geometrical innovation in a modified circle equation
In this paper, we create an innovation in circle by observing values of p for a positive fixed
value of r where several innovations are encountered. For a given r, we may write :
xp yp rp
(2.a)
into
2
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
x r cos 2 / p , y r sin 2 / p .
(2.b)
Substituting the expression Eq.(2.b) into y r x , i.e.
p
p
p
y p r p r cos 2 / p
r p 1 cos 2 .
p
2/ p
2/ p
For 12. One may predict that for
p=k/3 with k 5 , Eq.(1) creates a family of curves, the same attitudes for arbitrary integer values of
p 5 . By extending the domain of the axes, we have still a close curve. By increasing the value of p,
i.e. p=13/3, 14/3, 15/3, 16/3, a family of weak kites appears since all curves behave the same as in Fig.
4. Again, the values of p are varied in the form k/3 similarities are observed as depicted in Fig.5. The
observations lead to patterns of weak kites for p started from 13/3 up to 33/3. Processing up to
p
p=100/3, similar patterns are obtained. Note that the solutions of z1 z 2 r can be complex
p
p
numbers. Therefore the real part solutions of z1 z 2 r are concluded to be family of weak kites
for p=k/3 with k 13,14,.... One may prove formally which is not shown here. As mentioned
above that the interest in this paper is the geometrical phenomenon of (Re z1,Re z2) satisfying.
p
p
p
z1 z 2 r p .
p
p
4
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
p=1,1/2 /, 1/3 and ¼
a weak circle for p=4
or 12/3.
p=1/3,2/3/, 3/3 and 4/3
p=13/3,14/3 , 15/3,16/3
p=5/3, p=6/3,
and p=8/3.
p=9/3, p=10/3, p=11/3
p=7/3 and 4=12/3.
p=17/3,18/3 ,19/3,20/3.
Figure 5. Real solutions of x y r
p
3.
p
p=21/3,22/3 ,23/3,24/3.
p
Innovation on Surfaces
3.1 Modified ball equation
Some well known curve and surface such as a circle and a ball are presented in Cartesian
coordinates, polar coordinates and parametric coordinates. The equation x y r ( x, y) R
describes a circle in Cartesian coordinate for p=2. In particular, an astroid can be obtained for p=2/3.
p
Rectangular representation of astroid is x
2/3
p
p
2
y 2 / 3 r 2 / 3 and its parametric is in the form
x r cos 3 and y r sin 3 . The second studied geometrical surface is a generalization of ball,
i.e.
xp yp zp rp .
(3)
This equation gives us a ball with a center O and its radius r for p=2. Furthermore, one may extend the
Eq. (3) to be a 3 dimensional astroid by using spherical coordinates, i.e.
x sin 3 cos 3 ; y sin 3 sin 3 ; z cos 3 , 0 2 ; 0 .
(4)
and the geometrical illustration is shown in Fig. 6.
Figure 6. Ilustration of 3 dimensional astroid using spherical coordinates.
As in 2 dimensional case, one may express Eq.(3) into parametric equations, i.e.
x sin 2 / p cos 2 / p ; y sin 2 / p sin 2 / p ; z cos 2 / p ;0 2 ; 0 .
(5).
Since 3 dimensional astroid can be defined, one generalizes Eq.(3) into Eq.(5) and try to find
the appeared geometrical phenomena for different values of p . Thus, the surfaces given by Eq.(5) will
be the real parts of solutions
z1 z2 z3 r p ; z1 , z 2 , z3 C 3 r , p R .
p
p
p
5
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
3.2 Innovation from modified hypocycloid curve
The third type innovation done in this paper is a generalization of 2 dimensional hypocycloid
curve into 3 dimensional surfaces, i.e. (Purwoto, et.al, 2014)
a b
x sin a b cos b cos
x sin ;
b
a b
y sin a b sin b sin
y sin ;
b
z cos
.
(6)
r
; r x 2 y 2 and a, b are real numbers. Originally, a and b denote radius of 2
where
sin
different circles, a > b. In this case, a and b are freely chosen. Another innovation is done by
replacing the minus signs in the Eq.(6) to be positive. The innovation can still be created by presenting
of each derivative to create the new curves (from x , y ) and surfaces. This paper provides some
similar patterns in curves and surfaces for the generalization defined above.
3.3 Innovation from Algebraic Surfaces
According to the equations used to create surfaces, algebraic surfaces are surfaces created from
polynomials of three variables, e.g. ellipsoid, sphere classified as quadric surfaces. Other
classifications are quartic surfaces, Caley cubic, Barth Sextic, and many more. One may refer to
Wikipedia for other variation of algebraic surfaces. There is an available software called SURFER to
visualize those algebraic surfaces such that a user simply writes an algebraic equation to create a
surface without writing a code. The used of SURFER is not only for presenting algebraic surfaces for
education but also for architecture and art (). The examples for some algebraic surfaces using
SURFER will be shown.
Among all surfaces, the mathematical studies usually involve several types of curvatures on
curves and surfaces. In this paper, the curvatures are not discussed since the main focus is about
innovating curves and surfaces. Finally, the next chapter shows innovation results for 3 classifications
above, i.e. innovation on curves and surfaces from circle- sphere equations, modified hypocycloid and
algebraic surfaces.
4
Innovation Results
4.1 Innovation Surfaces of Generalized Ball Equations
Note that the derivatives of the equations may create other curve and surfaces. In 3
dimensional case (see Eq.5), we may have some possible triple to visualize the surfaces due to the
derivatives, i.e. x ; x ; x ; y ; y ; y ; z ; z and z . Since z =0 ,the number of possible
8
surfaces are C 3 =56 . Furthermore, the second derivatives may be created.
Since p=2/3 has led to a known geometry (astroid), we try for p=k/3, k=1, 2,…,20. One
observes for p=12,…,20 the surfaces have the same families. The collections of surfaces are called
parking surfaces. Some surfaces are shown in Fig.7.
6
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
Figure 7. A family of parking surfaces for p=13/3,14//3,15/3/16/3 (from left to right)
Note that the 2 dimensional projections have the same images as shown in Figure 5.
Finally some surfaces are found by varying the values of p. Started with astroid in 3
dimensional (p=2/3) and vary for p=2/k, k=4, 5, 6, 7, 8, 9, one observes geometrical patterns as
depicted in Fig. 8. Weak astroids are observed for p=2/3, 2/5, 2/7 and 2/9 with diminishing surfaces
area. Generally, one may have weak astroids for p=2/(2n+1), for n is natural number.
3
Figure 8. Visualization of Re z1 , Re z 2 , Re z3 C ; r R , p=2/3, 2/4, 2/5, 2/6, 2/7, 2/8, 2/9 (from
left to right).
As we notice, The real solutions z1 z2 z3 r ; have closed astroid surfaces with
decreasing area of surfaces for p=2/3, p/5, p/7. On the other hand the open surfaces are found for
p=2/4, 2/6, 2/8. One may increase the value of denominator and we have the area of surfaces tends to
zero for all p. Some variations of parameters have been observed in the results of surfaces where the
p
p
p
p
value of p / k , k=1,2,3,…,15.
For irrational numbers of p some interesting patterns of surfaces are found. One observes that
families of surfaces are started from p / 5 . Closed surfaces called weak astroid are obtained for
p / 5 , p / 8 , p / 11 and p / 14 with each area is diminishing. Otherwise, for
p / 6 , p / 7 , ..., p / 13 , the open surfaces are shown with similar pattern in Fig. 9-10. By
studying the surfaces created by the derivatives, open surfaces are obtained for p=2/k, k is an odd
number and closed surfaces for p=2/t where t is an even number (see Fig. 11). As above, the area of
surfaces are decreasing for increasing values of k and t. These surfaces are possible to be some
mathematical ornaments or for other purposes such as motifs for batik.
Figure 9. Surfaces for p / k , k=1,2,3,4,5
Figure 10. Surfaces for p / k , k=6,7,…,15
Figure 11. Surfaces
dx dy dz
; ; ,p=2/k,
d d d
k=3,4,…,10.
4.2 Result of innovation for curve stitching and mathematical art
7
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
As mention previous sections, various curves and surfaces have been obtained. The obtained
designs can be 2 dimensional and 3 dimensional. This paper focuses on presenting the 2 dimensional
designs for curve stitching that can be used for students to be material for innovation on mathematics.
Originally curve stitching has been designed with creating line or parabola into different kinds of
motifs. This research has obtained many curves from a modification of hypocycloid curve into kind of
polygons called Connected Pseudo Polygon (CPP). Several designs of CPP for curve stitching are
displayed in Fig.11-12.
1.5
500
3
400
1
2
300
0.5
200
1
100
0
0
0
-0.5
-100
-1
-1
-200
-300
-1.5
-2
-400
-2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
a=1, p=24, q=12, dan b=2 , n=50 (the
number of points).
-500
-500
-400
-300
-200
-100
0
100
200
300
400
-3
-3
500
-2.5
a=457, p=35, q=33, b=-(p/q), n=50
(the number of points).
-2
-1.5
-1
-0.5
0
0.5
1
1.5
a=1, p=24, q=12, b=2, n=50
(the number of points).
Figure 11. Modified Hypocycloid curve
a b ;
x( ) ( a b ) cos b cos
y( ) ( a b ) sin b sin a b .
b
b
50
80
2500
150
40
2000
60
30
40
100
1500
20
1000
50
10
20
500
0
0
0
0
-10
-500
-20
-20
-1000
-50
-30
-40
-1500
-40
-60
-100
-50
-50
-80
-80
-60
-40
-20
0
20
40
60
-40
-30
-20
-10
0
10
20
30
40
-2000
50
-2500
-2500 -2000 -1500 -1000
80
Derivative of modified
hypocycloid curve ; a=20, p=800,
q=-500, b=1.6 , n=25 (the
number of points)
a=20, p=250, q=-20, b=12.5 ,
n=25 (the number of points).
-150
-150
-100
-50
0
50
100
-500
0
500
1000
1500
2000
2500
150
p=450;q=-10;a=20;b=(p/q); n=13 (the number
of points).
p=450;q=10;a=20;b=(p/q); n=13 (the number of
points). Repeated by
Fibonacci sequence.
Figure 12. Modified hypocycloid curve :
a b y (a b) sin b sin a b .
x (a b) cos b cos
b
b
The designs of curve stitching strongly depend on the order of arranging the material. Thus the designs
require numbering on each vertex. Finally by following the numbers, the curve stitching can be made.
The examples are shown in Fig.13. Furthermore, if one requires a particular size, the code can provide
the minimum one dimensional material.
Figure 13. Curve stitching based on modified hypocycloid curve.
4.3 Results of surfaces innovations
Several mathematical ornaments and art are created based on the modified hypocycloid and
weak balls obtained in Section 3. Figure 14 and Fig.15 illustrate 2 pairs of designs and the related
products made by at least 11 home industries for all products. In this paper, the results from 4 home
industries are displayed. Figure 16 illustrates designs and the related products called BATIMA (as one
of Indonesian innovations in 2016) and puzzles with the same designs. Thus the same designs can be
used for several purposes. There are also designs can be used for an architecture for a building in
future.
8
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
dx dy dz p=1;q=4; a=1;
; ;
d d d
b=-(p/q)
Second derivative of hypocycloid for
p=1;q=4;a=1;b=-(p/q)
dx dy p=1;q=4;a=1;b=-(p/q)
, , z
d d
Figure 14. Design and its surface of Derivative of hypocycloid
’
x p y p z p r p for p / 7
Derivative of hypocycloid for
p=1;q=6; a=1;b=-(p/q) (second
pair, right)
The second derivative of
hypocycloid for
a=1;p=1;q=4;b=p/q.
Figure 15. Design and its surface of modified hypocycloid and weak balls
Figure 16. Designs and the related products (puzzle and batik (BATIMA))
4.4 Innovation on Algebraic Surfaces
A sphere is one of algebraic surfaces in quadric surfaces. The other quadric surfaces are, Cone,
ellipsoid, paraboloid. The algebraic surfaces are governed by polynomial surfaces. A new
development for creating algebraic surfaces is a software called surfer (found by Imaginary research
group from Germany). This software can be easily used to visualize without doing programming as
above. Several surfaces are displayed here as examples. Figure18 can be models for Christmas
ornaments called Santa Xmas Ornaments, and Xmas-hats are also created. Additionally, the designs
can also be used to be batik motifs.
Figure 18.Santa Xmas-Ornaments by an algebraic surface, i.e:
2
2
2
x y z 2 2 xyz 1 x 1 y 1 z 1 2x 1 y 1z 1 1 0
2
2
9
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
Figure 19. Xmas-hats from algebraic surfaces
as combination of several 3 and 5 spheres with 1 ellipsoid.
Conclusion
Creations curves and surfaces as innovations on mathematics are discussed to provide
enrichment of mathematics curriculum. The exponent in the circle equation is changed to be any real
number leading to find the complex sets of solutions and the related geometrical solutions. This
approach is extended to a sphere where the geometrical surfaces as real parts of the solutions. The
spherical coordinates are used to express the modified hypocycloid curve into new surfaces including
the derivatives to give visualisation between the equation and its derivative.
The results are classified into 3 types of innovations. The first is an innovation on extending
circle and sphere into higher exponent in the equations yielding to enrichment of curves and surfaces
into weak kites and weak balls respectively. The modified hypocycloid curves are the second
innovation finally collected to be designs for curve stitching and the surfaces employed to be batik
motifs, puzzles and mathematical ornaments. The designs are also used for several designs on
souvenirs. The batik creation called BATIMA has been selected as one of 108 Indonesian innovations
in 2016 by BIC (Business Innovation Center).
The algebraic surfaces as the third innovation have also been used to be surfaces innovation
through SURFER where the expression can be typed directly to create the surfaces. The Xmas
Ornaments and Xmas-hats are shown here to be examples of the surfaces.
Remark
The approach in this paper very rare met with reputable and modern journals since the
innovation means that the creation beyond the standard related topics. Therefore, author has very
limited references.
Acknowledgment
Special thank to Dr. Bianca Violet from Imaginary Mathematisches Forschungsinstitut
Oberwolfach,Germany who has inspired me to use SURFER for creating algebraic surfaces.
References
[1] Ahuja O P 1996. Let’s Popularize Mathematics, The Mathematics Educator , Vol. l , No. 1. 8298.
[2] Piggott J 2007 The nature of mathematical enrichment: a case study of Implementation, Educate,
Vol.7, No.2, pp. 30-45,University of Cambridge, Faculty of Education.
[3] Diković L 2009 Applications GeoGebra into teaching some topics of Mathematics at the College
Level, Computer Science and Information Systems , 6(2), 191-203.
[4] Parhusip H A 2014 Arts revealed in calculus and its extension. International Journal of
Statistics and Mathematics, 1(3): 002-009, Premier-Publisher.
[5] Parhusip H A 2015 Disain ODEMA (Ornament Decorative Mathematics) untuk Populerisasi
Matematika, Proceeding, Seminar Nasional Matematika,Sains dan Informatika, dalam rangka
Dies Natalis ke 39 UNS, FMIPA, UNS, ISBN-978-602-18580-3-5 ,pp. 8-15 (in Indonesian).
10
Material Keynote speaker for “The 1st International Conference on Mathematics : Education, Theory and
Application (ICMETA) (in Science and Art), at hotel Alila Solo, 6-7th Des 2016 organized by MATH Dept FMIPA
UNS, Solo-Indonesia
[6]
Parhusip H A 2010 Learning Complex Function And Its Visualization With Matlab, South East
Asian Conference On Mathematics And Its Applications Proceedings Institut Teknologi
Sepuluh Nopember, Surabaya , ISBN 978 – 979 – 96152 – 5 – 1,pp.373-384.
[7]
1
Parhusip H A and Sulistyono 2009 Pemetaan w dan hasil pemetaannya, Prosiding,
z
Seminar Nasional Matematika dan Pendidikan Matematika,FMIPA UNY 5 Des 2009, ISBN :
978-979-16353-3-2,T-16, hal. 1127-1138 (Indonesian).
[8] Purwoto Parhusip H A, Mahatma T 2014 Perluasan Kurva Parametrik Hypocycloid Dimensi
Menjadi 3 Dimensi Dengan Sistem Koordinat Bola, prosiding Seminar Nasional UNNES,8 Nov
2014, ISBN 978-602-1034-06-4; hal.326-336 (Indonesian).
[9] Hartkopf A and Matt A D 2013 SURFER in Math Art, Education and Science Communication
Proceedings of Bridges: Mathematics, Music, Art, Architecture, Culture.
[10] Suryaningsih V Parhusip H A, Mahatma T 2013 Kurva Parametrik dan Transformasinya untuk
Pembentukan Motif Dekoratif, Prosiding, Seminar Nasional Matematika dan Pendidikan
Matematika UNY,9 Nov, ISBN:978-979-16353-9-4,hal. MT – 249-258.
[11] Parhusip H A 2016 Sains dan Matematika untuk Art untuk Technoeducation, Science Festival ,
organized by LK-FSM, UKSW, 26 Feb 2016, di gedung GKJ Salatiga. DOI:
10.13140/RG.2.1.4388.8402
[12] Parhusip H A 2014 Arts revealed in Calculus and Its Extension International Journal of
Statistics and
Mathematics, 1(3): 002-009, Premier-Publisher, available online on :
http://premierpublishers.org/ijsm/articles
11
SCHEDULE
International Conference on Mathematics :
Education, Theory, and Application (ICMETA) 2016
6th & 7th December 2016, Hotel ALILA
Solo, Central Java, Indonesia
Department of Mathematics
Faculty of Mathematics and Natural Sciences
Universitas Sebelas Maret
1st Day, Tuesday, December 6, 2016
Time Schedule
Time
’
Registration
08.00 - 08.10
’
1st Day Opening
Ballroom Hotel Alila
Opening of Master of Ceremony
08.10 - 08.20
’
Samsuri Sutarna, M.Sn
Lecturer of Institute Seni Indonesia-Surakarta
Central Java
08.20 - 08.40
’
Vox Magistra FKIP UNS
07.00 am08.00am
Venue
Activities
Duration
Lobby Ballroom
Traditional Dance
Performance :
Tari Klana Topeng
a. Singing Traditional
b. Singing National
Anthem (audiences)
WELCOME SPEECH
08.40 - 08.50
Dr. Dewi Retno Sari Saputro, M.Kom.
Chair of ICMETA
’
08.50 - 09.00
’
Prof. Ari Handono Ramelan,
M.Sc.(Hons),Ph.D.
Dean of FMIPA
09.00 - 09.10
’
Prof. Dr. Ravik Karsidi, M.S.
Rector of Universitas Sebelas
Maret
09.10 - 09.25
5’
Coffee Break
Lobby Ballroom Hotel Alila
Keynote Session - Ballroom Hotel Alila
Prof. Drs. Tri Atmojo K, M,Sc. Ph.D.
Department of Mathematics, Faculty of
Mathematics and Natural Sciences
Universitas Sebelas Maret, Indonesia
Moderator
09.25 - 10.10
’
Prof. Dr. Edy Tri Baskoro, M.Sc.
Department of Mathematics, Faculty of
Mathematics and Natural Sciences, Bandung
Institute of Technology , Indonesia
Keynote Speaker I
10.10 - 10.55
’
Keynote Speaker II
10.55 - 11.10
5’
Dr. Darfiana Nur, M.Sc.
Statistical Science in the School of Computer
Science, Engineering and Mathematics,
Flinders University, Australia
Panel Discussion and Q & A Session
Putranto Hadi Utomo, M.Si.
Department of Mathematics, Faculty of
Mathematics and Natural Sciences
Universitas Sebelas Maret, Indonesia
Moderator
SCHEDULE-ICMETA, DECEMBER 6-7, 2016
1
’
11.10 - 11.55
11.55-12.40
12.40 -12.55
’
12.55 -13.40
45’
Dr. G.R. (Ruud) Pellikaan
Department of Mathematics and Computer
Science, Technische Universiteit Eindhoven,
Netherland
Keynote Speaker III
Prof. Dr. Mohd Bin Omar
Institute of Mathematical Sciences, Faculty of
Science, University of Malaya, Malaysia
Panel Discussion and Q & A Session
Keynote Speaker IV
Lunch
Lobby Ballroom Hotel Alila
Paper Presentation Parallel Session 1
’
13.40 - 15.00
15.00 - 15.30
’
Mathematics Education
a. Room 1
5 papers
b. Room 2
5 papers
c. Room 3
5 papers
Combinatorics
a. Room 4
5 papers
b. Room 5
5 papers
Statistics
a. Room 6
5 papers
b. Room 7
5 papers
Applied Mathematics
a. Room 8
5 papers
b. Room 9
5 papers
c. Room 10
5 papers
Coffee Break
Room Meeting Hotel Alila
Room Meeting Hotel Alila
Paper Presentation Parallel Session 2
15.30 - 16.45
’
16.45 -17.00
5’
Mathematics Education
a. Room 1
4 papers
b. Room 2
5 papers
c. Room 3
5 papers
Combinatorics
a. Room 4
6 papers
b. Room 5
5 papers
Statistics
a. Room 6
5 papers
b. Room 7
5 papers
Applied Mathematics
a. Room 8
4 papers
b. Room 9
4 papers
c. Room 10
4 papers
End of 1st Day-Closing Remarks
Certificate Distribution
Room Meeting Hotel Alila
SCHEDULE-ICMETA, DECEMBER 6-7, 2016
2
2nd Day, Wednesday, December 7, 2016
Time Schedule
Time
07.00 - 08.00
Activities
Duration
’
Registration
Venue
Lobby Ball Room
Hotel Alila
2nd Day Opening
Ballroom Hotel Alila
08.00 - 08.15
’
08.15 - 08.30
20’
Opening of Master of Ceremony
Coffee Break
Lobby Ball Room
Hotel Alila
Keynote Session
Ballroom Hotel Alila
Dra. Mania Roswitha, M.Si.
Department of Mathematics, Faculty of
Mathematics and Natural Sciences
Universitas Sebelas Maret, Indonesia
Moderator
08.30 - 09.25
’
Dr. Hanna Arini Parhusip
Department of Mathematics, Faculty of Science
and Mathematics, UKSW, Salatiga Indonesia
Keynote Speaker V
09.25-10.10
’
Prof. Dr. Kenjiro T. Miura
Realistic Modeling Laboratory, Department of
Mechanical Engineering, Shizuoka University,
Japan
Keynote Speaker VI
10.10-10.25
’
Panel Discussion and Q & A Session
Keynote Session-Ballroom Hotel Alila
Drs. Isnandar Slamet, M.Sc., Ph.D.
Department of Statisticss, Faculty of
Mathematics and Natural Sciences
Universitas Sebelas Maret, Indonesia
Moderator
10.25 - 11.10
’
Dr. Sutanto, DEA
Department of Mathematics, Faculty of
Mathematics and Natural Sciences
Universitas Sebelas Maret, Indonesia
Keynote Speaker VII
11.10 - 11.55
’
Dr. Benoît Liquet
Laboratory of Mathematics and Their
Applications, Université de Pau et des Pays de
l'Adou, France
Keynote Speaker VIII
11.55 - 12.10
’
Panel Discussion and Q & A Session
SCHEDULE-ICMETA, DECEMBER 6-7, 2016
3
12.10 - 13.15
’
Lobby Ballroom
Hotel Alila
Lunch
Paper Presentation Parallel Session 3
13.15 - 15.00
’
a.
b.
c.
a.
b.
a.
b.
c.
15.00 - 15.30
’
15.30 - 16.30
’
16.30 - 16.45
5’
Mathematics Education
Room 1
5 papers
Room 2
5 papers
Room 3
5 papers
Combinatorics
Room 4
5 papers
Statistics
Room 5
5 papers
Room 6
5 papers
Room Algebra & Analysis
Room 7
5 papers
Applied Mathematics
Room 8
5 papers
Room 9
5 papers
Room 10
5 papers
Room Meeting Hotel
Alila
Coffee Break
Paper Presentation Parallel Session 4
Mathematics Education
a. Room 1
4 papers
b. Room 2
5 papers
c. Room 3
4 papers
Combinatorics
Room 4
5 papers
Statistics
a. Room 5
4 papers
b. Room 6
4 papers
Room Algebra & Analysis
Room 7
3 papers
Applied Mathematics
a. Room 8
4 papers
b. Room 9
5 papers
c. Room 10
4 papers
nd
End of 2 Day -Closing Remarks
Certificate Distribution
Room Meeting Hotel
Alila
SCHEDULE-ICMETA, DECEMBER 6-7, 2016
4
International Conference on Mathematics :
Education, Theory and Application (ICMETA)
December 6th&7th, 2016
http://ICMETA.uns.ac.id
No
: 021/ICMETA/2016
Subject :Invitation as Keynote Speaker for the ICMETA
Dear Dr Hanna Arini Parhusip
We are pleased to invite you to the International Conference on Mathematics : Education,
Theory & Application (ICMETA) scheduled to be held between (Desember 6
Secretariat ICMETA 2016 :
2nd floor, A Building
Department of Mathematics, Faculty of Mathematics and Natural Sciences
SebelasMaret University
Ir. Sutami 36 A Solo, Central Java, Indonesia
e-mail : icmeta@mail.uns.ac.id