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►SKETCH 4
A RT & A RCHITECTURE
IN M ATHEMATICS
IN
S TUDENTS ’ P ROJECTS
A BSTRACT
Ancient Greek scholars believed that everything around them was related to mathematics and numbers were the ultimate
reality. The most famous ancient mathematician and philosopher, Pythagoras, used to say “all is number”. Plato, another
famous ancient philosopher and mathematician, used geometric objects to represent the structure of the world. These objects still are called Platonic solids. Nowadays scientists spend their life searching for mathematical representations of
laws of physics, chemistry, astronomy, biology, economics and many other disciplines. At the same time an average human
does not see mathematics in his world at all.
Students’ projects in mathematics are an ample opportunity to develop students ‘awareness of mathematics in the world
around them. Such projects can bring a lot of benefits. They can improve the students’ problem solving skills, foster the
ability to look critically at the things around them, formulating problems or hypotheses, and finding solutions.
In this document we will explore a few everyday examples where an average person will not expect even a trace of mathematics. We will show what kind of mathematical ideas we can find there, what kind of mathematical questions we can state,
and what kind of solutions we may produce. Each of these examples can be used as an idea for a student’s project in mathematics.
I NTRODUCTION
Students’ projects in any school or university discipline are not a new
thing. We had them for centuries. They existed as long as universities and
schools existed. The most advanced forms of students’ projects are M.Sc.
and Ph.D. projects ended with an appropriate written thesis and a verbal
presentation. However, as the sciences became more abstract such projects tend to have more theoretical nature and quite often they are less
and less connected with the world around us. This last issue is especially
important for mathematics education. We know that architecture, selected
branches of art, biology and other disciplines have a very serious mathematical background. Unfortunately most of our students are not able to
see mathematics around them. They see nice buildings, nice pictures or
flowers but they do not see mathematics that is a foundation of these objects. This opens a gate for students’ projects where students will explore
some natural or human made objects, learn major principles of them and
investigate mathematics needed to understand them.
In recent years some countries started looking seriously at broken links
between mathematics, everyday life and human environment. In some
countries students’ projects and modeling were added to the mathematics
Art & Architecture in Students’ Projects in Mathematics |1
curriculum. The emphasis is usually put on modeling aspect as a final act
of understanding connection between mathematics, our environment and,
in some cases, predicting future events (see ‎[2], ‎[9], and ‎[10]). Blomhoj
and Hoff emphasize the importance of the whole modeling cycle and different phases of the modeling process. Many authors suggest strongly that
“a considerable part of the teaching should be organized as projects where
students in smaller teams or groups work independently with mathematical
as well as interdisciplinary problems and prepare a report” (see ‎[2]). This
requirement connects the concept of projects, modeling and team learning
– the three major learning outcomes of modern education. All this requires
a number of news skills from mathematics educators – finding appropriate
topics for students’ projects, learning modeling techniques, organizing and
supervising the team work.
In this document we will concentrate mainly on the first and the second
outcomes leaving the third one to the specialists in team learning concepts
(TBL). A very good introduction to TBL can be found in Michaelsen’ and
Sweet’ publications, e.g. ‎[8].
Some examples for this document were taken from author’s earlier publications, e.g. books: Sketches on Geometry and Art and Islamic Geometric
Patterns in Istanbul as well as from his new book The Art of Geometric Constructions that will be published later this year. All modeling tasks presented in this document were performed with dynamic geometry software
Geometer’s Sketchpad® by KCP Technologies Inc. Computer program Geometry Expressions® by Saltire Software was used to demonstrate selected
properties in example 1.
P AVEMENTS AND WALLS
For most of us pavements are simply patterns where we do not expect any
mathematics and we do not see it. We walk through them and we rarely
notice them as long as there is no hole where we can fall down or break
our leg. This is all. However sometimes pavements can hide a lot of interesting things. Let us see some examples.
E XAMPLE 1: A
PAVEMENT FROM
A BU D HABI
The original pavement presented in this example is a part of a large parking lot near one of the fruit and vegetable markets. The pavement is in a
very bad condition. Therefore, instead of using its photo we use here its
computer simulation. Very similar patterns often occur in a form of an iron
grid protecting windows and sometimes shop entrances.
2 | Author: Mirek Majewski, source http://symmetrica.wordpress.com
Fig. 1 Computer simulation of the pavement from Abu Dhabi
The pavement was created using two types
of square tiles. We have a left and right
version of the same tile. Each tile has deep
indentations making a pattern with a
square in the middle and four triangles.
The pavement shown in figure 1 seems to be completely unimportant but
its mathematical properties can make it very interesting. Let us start by
constructing a single tile and see what we can get out of it.
D
G
Construction of the left tile
C
K
H
Start by constructing a square with side equal to AB.
2.
From one of the corners draw a segment to a point on one of the opposite sides of the square (here this is segment DE).
3.
Construct segments BF=CG=DH, each one equal to AE.
4.
Construct lines or segments AF, BG, and CH.
L
F
J
Here we constructed the left tile. The right tile can be constructed exactly
the same way.
I
A
1.
E
B
What kind of mathematical questions we can state here? Here are some of
them.
1. Suppose that AE = BF = CG = DH, prove that triangles ABJ, BCK, CDL and DAI are congruent and each of them is a right triangle. Prove that quadrilateral IJKL is a square.
2. Suppose that AB = a and AE = a/2. Prove that:
a. AI = IJ = BJ.
b. JF = JB/2 = AI/2 = AF/5
c. AF = (a√5)/2
d. AI = (a√5)/5 = 2AF/5
e. Area of the square IJKL is equal to a2/5.
Art & Architecture in Students’ Projects in Mathematics |3
After this quite simple case we can be more inventive and ask more questions related to different proportions AE : AB. For example:
3. Suppose that AB = a and AE = a/3.
a. Show that AI = IJ/2 and JF = AJ/9.
b. Formulate and prove properties similar to 2.c, 2.d and 2.e.
We can also investigate a general case when a proportion of AE to AB is
any natural number.
4. Suppose that AB = a and AE = a/k. Prove that:
a. 𝐴𝐼 =
𝐼𝐽
𝑘−1
𝑎
and 𝐽𝐹 =
𝐴𝐽
𝑘2
.
b. 𝐴𝐹 = √𝑘 2 + 1
𝑘
c. 𝐴𝐼 =
d. 𝐸𝐼 =
𝑎√𝑘 2 +1
𝑘 2 +1
𝑎√𝑘 2 +1
𝑘(𝑘 2 +1)
e. Area of the triangle ∆𝐴𝐵𝐹 =
f. Area of the square IJKL is
𝑎2
2𝑘
𝑎2 (𝑘−1)2
𝑘 2 +1
One can use a computer program like Geometer’s Sketchpad or Geometry
Expressions to check first if the above properties are true and then prove
them by hand. We could also use Geometry Expressions to experiment
with the objects and investigate some new properties.
Geometry Expressions model for the
Abu Dhabi tile
Here we used the program to verify properties 4.a, … 4.f.
In exactly the same way we could use the computer
software to discover some new properties. For example
we could calculate the length of the diagonal of the central square, find a formula for the area of small triangles, etc.
4 | Author: Mirek Majewski, source http://symmetrica.wordpress.com
Finally one can look at our pavement and find here and prove the famous
Pythagoras theorem.
D
G
C
Pythagoras theorem in the Abu Dhabi pavement
K
H
L
x
J
a
F
b
Let 𝑎 = 𝐴𝐽, 𝑏 = 𝐽𝐵, 𝑐 = 𝐴𝐵 and 𝑥 = 𝑎 − 𝑏.
Then by observing the picture we can write down the statement:
𝑎𝑏
𝑐2 = 4
+ 𝑥2
2
This means that
𝑎𝑏
𝑐 2 = 4 + (𝑎 − 𝑏)2 = 2𝑎𝑏 + 𝑎2 − 2𝑎𝑏 + 𝑏 2 = 𝑎2 + 𝑏 2
2
and consequently 𝑐 2 = 𝑎2 + 𝑏 2
I
A
E
E XAMPLE 2: A
c
B
FLOOR FROM THE
NTHU
GUESTHOUSE IN
T AI-
WAN
There are some other pavements where we can see the Pythagoras theorem. For example a floor in the NTHU guesthouse in Taiwan looks like the
one shown on the next figure.
Fig. 2 Pythagoras on the floor at the NTHU in Taiwan
This pavement was created using two kinds of square tiles. In
this particular example the side of large tile is twice the side
of the smaller tile. This particular floor can be also used to
illustrate a proof of the Pythagoras theorem. If we connect
appropriate vertices of small or large tiles like it was shown
on the picture we will get the same square as the one we had
in the previous example.
Art & Architecture in Students’ Projects in Mathematics|5
Another proof of the Pythagoras theorem
A
B
C
The picture enclosed here shows another proof of the Pythagoras theorem that one can see on this floor. Here the AB2 =
the area of the light-shaded shapes plus the area of the darkshaded shapes. We can easily see that the light shapes form a
square with the side CB, and the dark shaded shapes form a
square with side equal to AC.
There are numerous other ways to draw lines on this floor
that lead to other proofs of the Pythagoras theorem.
Now, imagine a similar pavement created using two square tiles but the
proportion of sides of tiles is not necessary 1:2. How to construct such
pavement? Will it also illustrate the Pythagoras theorem?
The two examples shown above represent three different points of view:
1. Detailed calculations
2. General formulae proving
3. Constructions of geometric figures.
To these three points of view we can add one more. For example we can
look at both examples and try to determine what kind of symmetries they
have? Do they have mirror symmetry lines, or points where the whole pattern can be rotated, etc.? Going further we can also try to decide to which
symmetry group the pattern belongs. Then we will find that the pattern
from example 1 has a signature 4*2 and the other one is 442.
Floors and pavements can give us almost infinite number of interesting
geometric properties. Many theorems related to areas of geometric objects
can be illustrated and proved by looking at floors or pavements. Let us
look at one more example. In this example we use two rectangular tiles
and we will show how they can be used to illustrate one of the most famous facts in calculus – the Cauchy-Schwartz inequality.
6 | Author: Mirek Majewski, source http://symmetrica.wordpress.com
E XAMPLE 3: T HE C AUCHY -S CHWARTZ
INEQUALITY
Fig. 3. Cauchy-Schwartz inequality
Let us denote AB by X and BC by Y. Then from the enclosed
picture we see that
𝑋 = √𝑎2 + 𝑑 2 and 𝑌 = √𝑐 2 + 𝑏 2
C
Thus
Y
|𝑎𝑏 + 𝑐𝑑| ≤ |𝑎||𝑏| + |𝑐||𝑑| ≤
≤ 𝑋𝑌 = √𝑎2 + 𝑑 2 × √𝑐 2 + 𝑏 2
B
A
X
d
b
c
a
I suggest my readers to look around and see what kind of pavements and
floors are in their place and see if they can discover any interesting mathematical properties or proofs of some already known theorems. Here is
one of such problems.
Fig. 4. The three tiles problem
The enclosed picture shows a floor with three different tiles.
There are two squares – one large and one small, and one rectangular tile. What mathematical properties we can discover
on this floor.
I SLAMIC ART AND ARCHITECTURE
Islamic art brings us into the world of geometric constructions based on
segments. Almost each pattern in mosques, Islamic books, and sometimes
every day tools, can present an interesting and challenging problem worth
to be considered as a student’s project or at least part of it. Such projects
can bring quite different challenge to our students. Here we may not have
much calculations, or proving a formulae, but the challenge is to construct
such patterns using Euclidean tools and principles. However, still some of
the old Persian patterns may hide some mathematical theorems or their
proofs. Let us see an interesting example from Islamic architecture.
Art & Architecture in Students’ Projects in Mathematics |7
E XAMPLE 4 I SLAMIC
ARCHITECTURE
Fig. 5. Islamic architecture
The photograph shown to the left presents a marble
window grid in one of the old tombs in Turkey.
The fragment of the grid enclosed inside the rectangle
ABEF will be called a repeat unit and it can be used to
recreate construction of the whole grid.
E
A
F
B
By analyzing the pattern shown in fig. 5 we can notice that we have here a
very clear geometric structure. The pattern in each rectangle is identical as
the one inside the rectangle ABEF or it is a mirror copy of it. Therefore, if
we wish to model this grid we need to construct the pattern enclosed by
the rectangle ABEF, so called repeat unit, and then using translations and
reflections create the whole grid. Here is one of many ways of modeling
this grid.
Construction of the window grid
D
E
F
A
C
A
C
We start by drawing a segment AB and dividing it into
three equal parts. Point C is one the dividing points.
2.
Construct lines perpendicular to AB and passing through
its ends. Construct point D by intersecting two circles ○(C,
CB) and ○(B,CB).
3.
Construct a line passing through D and parallel to AB.
Points of intersection of this line with the other two lines
label as E and F. This way we created the rectangle ABEF.
4.
By drawing segments EC, CD and DB we divide the rectangle ABCD into two identical equilateral triangles and
two halves of such a triangle. Now one can easily calculate that FB = (AB√3)/3. The proportion 1:√3/3 is one
of most popular proportions used in Islamic art and architecture.
B
D
E
1.
F
B
8 | Author: Mirek Majewski, source http://symmetrica.wordpress.com
D
E
F
M
H
20
40
20
A
40
G
5.
Divide angles ACE, ECD and BCD into parts equal to 20°
and 40°. Here comes the angle trisection problem.
6.
Divide segment CD into three equal parts and mark the
points of division as G and H.
7.
Construct a line passing through the point H and perpendicular to the segment CD.
8.
Label as M the point of intersection of the line created in
7 and one of the rays going from the point C.
9.
By drawing the thick segments shown on the figure next
to this text we create part of the final pattern.
20
40
B
C
D
E
F
M
H
G
N
A
B
C
D
E
F
M
11. Draw the remaining part of the pattern. We can easily
notice that the pattern needs some improvements – there
are large empty areas in the left-bottom corner of the
rectangle and in the top-right corner of the rectangle. In
order to fill this area we can use segments EC and DB as
mirrors and reflect abouth them fragments of already
created pattern.
H
G
N
P
A
E
B
C
D
F
M
12. By drawing circle ○(T,TR)we obtain point U that allows
us to make adjustments of the left side of the pattern. In
exactly the same way we make adjustments in the topright side of the pattern.
H
R
U
10. Repeat steps 5, 6, 7, 8 and 9 for the points D and G. This
way we create a framework for the second part of the
pattern.
S
T
G
P
A
C
B
Art & Architecture in Students’ Projects in Mathematics |9
D
E
F
M
13. The final pattern bounded by the rectangle ABEF may
look like the one shown to the left of this text.
SUMMARY of geometry skills needed to create this pattern:
division of a segment into three parts, constructing parallel
and perpendicular lines, angle trisection, using mirror reflection.
H
G
P
A
B
C
From here there is a simple task of creating the window grid shown on the
photograph. We create a rectangular grid shown below and fill each cell of
the grid by the pattern that we created a while ago. Letters on the grid
show how the repeat unit was placed.
A'
B'
A''
E'
F'
E''
A'
B'
A''
E
F
E'
A
B
A'
Creation of the window
grid
Far left – rectangular grid
that makes a framework
for the window grid
Left – final geometry of the
window grid
Far Left – a 3D model of
the window grid
Left – stained glass using
the same geometry
10 | A u t h o r : M i r e k M a j e w s k i , s o u r c e h t t p : / / s y m m e t r i c a . w o r d p r e s s . c o m
E XAMPLE 5: P ERSIAN
WINDOWS
While travelling through the Middle East, or at least looking at photographs from this region, we can notice an incredible beauty of shapes of
windows in Iran or further to the East in Central Asia. We can wonder
what kind of techniques were used by ancient masons to create such
shapes. What geometric constructions were used to create such beauty?
Let us try to rediscover their construction techniques.
Fig. 6 A Persian window
Sher-Dor Madrasah near the Timur Mausoleum, so called Gur-e-Amir Mausoleum in
Samarkand
In the next sequence of steps we will show one of the methods used to create the window from the above photograph. In fact, ancient Persian masters developed a number of such methods. From old books we can easily
find out that some of the Persian craftsman were great geometers.
A r t & A r c h i t e c t u r e i n S t u d e n t s ’ P r o j e c t s i n M a t h e m a t i c s | 11
H
Construction of a Persian window shape
I
A
D
E
C
F
G
B
1.
Let us start by drawing a segment AB, finding its center C
and constructing a bisector of AB.
2.
Divide segments AC and CB into three equal parts. Points of
division label as D, E (left side) and F, G (the right side).
3.
Draw two circles ○(F,FA) and ○(F,FD)
4.
Draw a number of rays through the point F and intersecting
with the large circle on the other side from the point F. There
is no need to have equal angles between them.
5.
Select one of the rays from F (here it is FJ). Through the intersection points of the ray with both circles produce line
perpendicular to AB and another one parallel to AB. The
perpendicular line should go through the intersection with
larger circle and the parallel one through the point of intersection with smaller circle.
6.
Mark the point of intersection of these lines (here it is the
point L).
7.
Repeat the same procedure for the remaining rays. We
should obtain a number of points located between the two
circles.
H
I
J
L
A
K
D
E
C
F
G
B
8.
Connect these points with segments to obtain a polygonal
line that can be used to model the shape of the left side of the
window.
9.
The picture shows the obtained curve. In a very similar way
we may construct the right side of the window.
H
I
J
L
A
SUMMARY of geometry skills needed to create this pattern: constructing perpendicular and parallel lines, creating a locus of a
point
K
D
E
C
F
G
B
12 | A u t h o r : M i r e k M a j e w s k i , s o u r c e h t t p : / / s y m m e t r i c a . w o r d p r e s s . c o m
H
J
L
A
We can easily notice that the line we created in this example is a
locus of a point L generated by moving point J along the large
circle. Students can go further with their investigation and find
out that the curve created is an ellipse and develop its accurate
graph using a geometry software. From here they can start investigating what are the other methods of creating ellipse and
other conic sections.
I
K
D
E
C
F
G
B
One can easily notice that the window shape we constructed is not exactly
the same as the one on the photograph. Therefore, a student while doing
his project may ask which points were used to construct this particular
window, and how it works with other window shapes from Central Asia or
Middle East? This opens a gate to many interesting investigations.
G OTHIC TRACERY
In Islamic art we deal with geometry of segments and polygons. Gothic
tracery leads us to the world of geometry of circles, arcs and tangency
problems. Gothic designs may look incredibly complicated but many simple examples of tracery can be a very good topics for students projects. Let
us look at one more example in this document.
E XAMPLE 6: G OTHIC
TRACERY
Fig. 7 A complex Gothic
vault
This vault comes from
one of English cathedrals
and it shows a complicated construction of
circles, arcs and segments.
A r t & A r c h i t e c t u r e i n S t u d e n t s ’ P r o j e c t s i n M a t h e m a t i c s | 13
In the above photograph we can see one of the typical geometric problems
occurring in Gothic tracery – construct a circle that is tangent to two other
circles and a line. Let us see how to solve this problem.
Construction of a circle tangent to a line in a given point and another circle
B
A
E
B
A
Our starting point is a circle and a line with a given point A. Our
objective is to find a circle that will be tangent to the line in the
point A and to the given circle.
1.
Construct a line perpendicular to the given line in the point A.
2.
We mark on it a segment length equal to the radius of a given
circle and one end in point A. The other end of the segment is
labeled as C.
3.
Draw a segment connecting C and B and construct its bisector.
The point of intersection of the bisector with perpendicular
line (point E) is the center of the circle we need.
4.
If we mark the point C on the other side of the line then we get
another circle that fulfills our request. This time the original
circle is inside of the tangent circle.
5.
Both constructions are frequently used in gothic tracery designs.
D
C
E
C
D
A
B
With this knowledge we can show how one can construct a simple tracery
similar to the one shown in the figure 7.
14 | A u t h o r : M i r e k M a j e w s k i , s o u r c e h t t p : / / s y m m e t r i c a . w o r d p r e s s . c o m
G
D
C
Construction of a simple tracery
1.
S
R
H
F
P
A
Q
C
S
H
R
U
2. Draw diagonals of the square. By connecting intersection points of diagonals and circles create a small
square in the space between circles.
3. Our goal will be to construct four small circles filling
spaces between small square and large circles, e.g.
HSP, PQE, etc.
B
E
G
D
Construct a square ABCD, and draw four circles each one
with center in a vertex of the square and radius equal
AB/2.
T
P
X
F
4.
Use the construction of the circle tangent to a line in a
given point and a circle. Here TU is equal AB/2, UB is was
bisected in V. Bisector of UB intersecting with line HF gives
us point X that is a center of the circle that should be inscribed in the space QRF.
5.
In exactly the same way we construct the remaining three
circles.
Q
V
A
E
B
A simple tracery with circles tangent to a circle and a line
After constructing geometry of the tracery in a computer program student can copy and paste it in any vector graphics program, e.g. Illustrator, Xara Designer, etc. Here student can
create multiple copies of the model, each one with different size
of the strokes, add fills or apply multiple graphics filters.
Design shown here was created in Geometer’s Sketchpad and
then copied to Inkscape (a freeware graphics program). Three
copies of it were created. Each copy had different stroke size:
8pt (black), 6pt (yellow), and 1 point (black on the top).
A r t & A r c h i t e c t u r e i n S t u d e n t s ’ P r o j e c t s i n M a t h e m a t i c s | 15
S UMMARY
Each of the examples presented in this document shows that mathematics
is present in almost any architectural object. We have seen mathematics
on floors, pavements, walls, windows and ceilings. Most of these examples
involve very simple geometry and reasonably easy numerical or symbolic
calculations. By analyzing such examples students can improve their problem solving skills, develop modeling and visualization skills, discover a link
between the world around them and mathematics, and see importance of
mathematics for an average person.
In this document we were concerned about mathematics and projects in
mathematics. However, projects described here can also be treated as students’ projects in an art course. There are numerous benefits for the art
education. Students will learn about design of architectural elements, architectural decorations in particular periods of time, styles, etc. Finally one
can consider described here projects as interdisciplinary works that fit
into area of many courses in school education. This is something that educators call an integrated science.
C REDITS
This document was originally presented during ATCM 2013 Korea conference and published in the ATCM 2013 Korea proceedings.
All illustrations and text were done by the author. All sketches, with one
exception, were completed using Geometer’s Sketchpad®, a computer program by KCP Technologies, now part of the McGraw-Hill Education. More
about Geometer’s Sketchpad can be found at Geometer’s Sketchpad Resource Center at http://www.dynamicgeometry.com/.
All rights reserved. No part of this document can be copied or reproduced
without permission of the author and appropriate credits note.
B IBLIOGRAPHY
[1]
[2]
[3]
[4]
[5]
Archibald R. C., Euclid’s Book on Divisions of Figures, Cambridge, University Press, 1915.
Blomhoj M., Hoff Kjeldsen T., Teaching mathematical modeling through project work,
ZDM – The International Journal on Mathematics Education, Springer Verlag, Volume 38,
Issue 2 , pp 163-177, 2006.
Bussey W.H., Geometric Constructions without the Classical Restriction to Ruler and Compasses, The American Mathematical Monthly, Vol. 43, No. 5 (May, 1936), pp. 265-280.
Cundy M., Rollett A.P., Mathematical Models, Oxford University Press, 1954.
Majewski M., Sketches on geometry and art: between East and West, publisher Axiomat
Torun, Poland, 2012.
16 | A u t h o r : M i r e k M a j e w s k i , s o u r c e h t t p : / / s y m m e t r i c a . w o r d p r e s s . c o m
Majewski M., Islamic Geometric Ornament in Istanbul, Nicholas Copernicus University
Publishing House, Poland, 2011.
[7] Majewski M., Sketches on geometry and art: the art of geometric constructions, publisher
Axiomat Torun, Poland, (to be published in 2013).
[8] Michaelsen L. K., Sweet M., The Essential Elements of Team-Based Learning, published in
New Directions for Teaching and Learning, no. 116, Winter 2008 © Wiley Periodicals,
Inc. Published online in Wiley InterScience (www.interscience.wiley.com (also at
http://medsci.indiana.edu/c602web/tbl/reading/michaelsen.pdf)
[9] Niss M., Mathematical competencies and the learning of mathematics: the danish KOM
project, Technical Report, IMFUFA, Roskilde University, 2002.
[10] Noss R., Pachler N., The Challenge of New Technologies: Doing Old Things in a New Way,
or Doing New Things? In P. Mortimore, editor, Understanding Pedagogy and its impact
on learning, pages 195-211. Sage, London, 1999.
[6]
MIROSLAW MAJEWSKI, PROF. DR.
PROFESSOR EMERITUS AT NEW YORK INSTITUTE OF TECHNOLOGY
ABU DHABI CAMPUS,
UNITED ARAB EMIRATES
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