City
Geometry
Tumbling
Triangles
and
Rampant
Rhombuses
Name:
20
Net of Architectural Fragment
2
19
Federation Square
The tiling design of Federation Square is based on a single right-angled triangle. The right angle
sides of the triangle are in the ratio 1 : 2. The architects who designed Federation Square decided to
use the tiling described in 1994 by Charles Radin of the University of Texas.
The tiling has been named the pinwheel tiling because there are infinitely many rotations of the
tiles. The sizes of the angles of the triangles are irrational numbers, so rotating the tile will never
bring it back exactly to its initial position.
Notice how five triangles fit together to form a larger triangle which has the same shape as the
smaller ones. This keeps on repeating, but each time the triangles are rotated around and end up in a
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different position. The design is called a nonperiodic tiling because it can theoretically go on
forever with the triangles continually ending up in very slightly different rotations.
Net of Melbourne Central cone
Look at all the different shapes which are formed when the triangles fit together.
Cut out 40 of the triangle shapes from page 13. Paste your triangles onto a large sheet of
paper to make each of the following shapes. Write the name of each shape beside it.
Using two of the triangle tiles, make each of the following shapes:
A rectangle
A kite
An isosceles triangle
A different isosceles triangle
Using four of the triangle tiles, make each of the following shapes:
A square
A rectangle
A parallelogram
What quadrilateral shape is formed if four of the triangles are placed with their right angles
together?
Using eight of the triangles, make each of the following shapes:
An isosceles triangle
A kite
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17
Storey Hall, RMIT
A bit of history: why is Storey Hall purple and green?
Early in the 20th century Storey Hall was used for meetings of an Irish society and green is the Irish
national colour. The Hall was also used by the Suffragettes (an organisation of women who fought
for women’s rights) and their colours were purple and green.
Finding the sum of the interior angles of any polygon
By dividing a polygon into triangles, can you see how we can calculate the sum of the
interior angles of the polygon?
…………………………………………………………………………………………….……
………………………………………………………………………………………………….
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5
Finding the size of the interior angles of any regular polygon
Storey Hall: Fat and thin rhombuses
…………………………………………………………………………………………….
Why do regular hexagons tessellate? In the regular hexagon tessellation, look at where the
three hexagons meet in the middle. What is it about the angles at this point which allows the
hexagons to tessellate?
…………………………………………………………………………………………….
…………………………………………………………………………………………….
Which regular polygons will tessellate?
…………………………………………………………………………………………….
…………………………………………………………………………………………….
Why are there only three regular polygons which will tessellate with themselves?
…………………………………………………………………………………………….
…………………………………………………………………………………………….
Why do regular octagons and squares tessellate?
…………………………………………………………………………………………….
…………………………………………………………………………………………….
Periodic and non-periodic tessellations
When tiles fit together exactly, we say that they tessellate. Regular hexagons fit together exactly as
you can see in the drawing below. A beehive is made up of hexagon-shaped cells which fit together
like this. When regular hexagons fit together, the pattern in one place is the same as anywhere else
in the tessellation. The hexagon tiling could go on forever with the same pattern. Similarly with this
tessellation formed by octagons and squares. These are repeating or periodic tessellations.
A non-periodic tiling is a tiling that does not have repeating pattern.
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15
Rhombuses and squares
Use your ruler and pencil to draw a rhombus and a square.
What features do a rhombus and a square have in common?
…………………………………………………………………………………………….
…………………………………………………………………………………………….
Sir Roger Penrose, a Professor of Mathematics at Oxford University, wondered if it was possible to
find some tile shapes which would tessellate with each other forever, but with a pattern which kept
changing so that the exact same pattern would never be repeated. He came up with these two very
simple rhombuses, and it is these rhombuses which form the design of the outside and inside of
Storey Hall at RMIT. The fat and thin rhombuses are known as Penrose tiles.
Fat rhombus
Thin rhombus
Sir Roger Penrose
Use a protractor to measure the smallest angle in each rhombus.
Can you work out the other angles in each rhombus without measuring them?
Make a sketch of how the rhombuses could be put together to form a periodic tiling.
However, we can also put the rhombuses together so that they form a non-periodic tiling. To do this
we need some markings on them a bit like the bumps and cut-outs on the pieces of a jigsaw puzzle.
The pattern on these fat and thin rhombuses tells us how we fit them together – when we put the
tiles together, the patterns must match.
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7
A Storey Hall treasure hunt
Federation Square triangles
Tick each of these as you find them:
Parallelogram that is not a rhombus
(
)
Rhombus door-handle
(
)
Trapezium
(
)
Pentagon
(
)
Decagon
(
)
Look for this pattern on the tiles in Storey Hall.
(
)
The two tiles will fit together in many ways. Look for the fat and thin rhombuses at Storey
Hall and see how they fit together.
( )
The diagrams below show how the fat and thin rhombuses fit together to make 10-sided
polygons. What are these 10-sided polygons called?
( )
Are there equal numbers of fat and thin rhombuses in the tessellation?
Number of fat rhombuses ……………… Number of fat rhombuses ………………
The Golden Ratio is
1+ 5
:1
2
Turn this into a decimal correct to 3 decimal places.
Compare this with the ratio of fat to thin rhombuses.
If the numbers of fat and thin rhombuses in a huge tiled area, for example as large as the
MCG, the ratio of fat to thin rhombuses is very close to the golden ratio.
Cut out the fat and thin rhombuses on page 15 and use the patterns on the rhombuses to
make a tessellation.
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13
Melbourne Central
When it was decided to build Melbourne Central, the architects had to protect the historic red brick
Shot Tower. They came up with the idea of surrounding it with a huge steel and glass cone.
Some cone calculations
Use the architect’s drawing on the next page and a ruler to find these measurements:
Height of the cone on the drawing
Scale shown on the drawing:
Height of the actual cone
=
.......... cm
1 cm is equivalent to
=
.......... m
.......... m
Use the architect’s drawing and a ruler to find these measurements:
Diameter of the cone on the drawing =
Scale shown on the drawing:
Diameter of the actual cone
.......... cm
1 cm is equivalent to
=
.......... m
.......... m
Can you work out the circumference of the cone? Why would the developers have wanted to
know this?
What angle does the sloping edge of the cone make with the floor of the cone? (Use a
protractor to measure the angle on the architect’s drawing)
What sort of triangle would you get if you cut the cone in half in a vertical direction?
Cut out the cone on page 17 and use sellotape to stick it together.
What is the volume of the inside of the cone?
Suggest why the architects needed to know the surface area of the outside of the cone.
Find a way of working out the surface area of the outside of the cone.
The Shot Tower
Built in 1889, the Shot Tower was a factory for
making lead shot. Lead ingots (bars of lead)
were heated in a furnace to melt them, then the
molten lead was dropped from the top of the
tower into a 1-metre deep trough of water at the
bottom. As the molten lead fell, it formed into
spherical balls which solidified as they landed in
the water. The factory was operating until 1961.
Use the architect’s drawing to estimate the distance the lead shot fell.
How long would the lead shot take to fall? d = 1 gt 2 where g is the acceleration due to gravity,
2
d metres is the distance the lead falls and t seconds is the time taken. g ≈ 10 m/sec 2
Calculate the speed of the lead shot as it hit the water. v = gt
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Architectural Fragment
Outside the State Library is sculpture called Architectural Fragment which was designed by the
Victorian sculptor, Petrus Spronk. Petrus got the idea for the design from the shape of the State
Library building and from fragments of old buildings he saw when he visited the island of Samos in
Greece. Samos was the island where Pythagoras lived over 2000 years ago.
Who was Pythagoras?
…………………………………………………………………………………………….
…………………………………………………………………………………………….
Petrus decided to use right-angled triangles for faces of the sculpture. In fact, one of the faces is
based on a 3:4:5 triangle.
The framework of the sculpture is constructed from steel, with slabs of Port Fairy bluestone bolted
and cemented into place. The sculpture was built at the quarry and stone-cutting plant in Port Fairy
then it was transported to Swanston Street, where it was set into the pavement.
Count the little squares in each of the three larger squares. Do you notice anything about the
numbers of little squares.
…………………………………………………………………………………………….
What is the hypotenuse of a right-angled triangle?
What was Pythagoras’ theorem.
…………………………………………………………………………………………….
Each of these sets of measurements represents the lengths of the three sides of a triangle.
Which of the triangles could be right-angled triangles?
4cm, 5cm, 7cm
6cm, 9cm, 11cm
5cm, 12cm, 13cm
9cm, 12cm, 15cm
The set of numbers 3, 4, 5 is called a Pythagorean triplet. Find some more Pythagorean
triplets.
…………………………………………………………………………………………….
After you have been to Federation Square, apply Pythagoras’ theorem to the Federation
Square triangles.
…………………………………………………………………………………………….
Use the net on page 19 to make a model of Architectural Fragment.
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