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FOR STUDENTS Investigating the maths inside: Stargazing with the SKA Activity 4 Seeking spirals © Maths Inside 2016 except where otherwise indicated. This document may be used, reproduced, communicated and adapted free of charge for non-commercial educational purposes provided all acknowledgements associated with the material are retained. Maths Inside is a UTS project in collaboration with CSIRO and AAMT. Page 1 of 11
Transcript of Home | University of Technology Sydney - Introduction · Web viewThe Archimedean spiral has loops...
FOR STUDENTS
Investigating the maths inside:Stargazing with the SKA
Activity 4Seeking spirals
For more Maths Inside activities see https://mathsinside.uts.edu.au
© Maths Inside 2016 except where otherwise indicated. This document may be used, reproduced, communicated and adapted free of charge for non-commercial educational purposes provided all acknowledgements associated with the material are retained. Maths Inside is a UTS project in collaboration with CSIRO and AAMT.
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Investigating the maths inside: Stargazing with the SKA ACTIVITY 4: SEEKING SPIRALS
Introduction
The photograph below shows the shell of a chambered nautilus that has been cut in half to reveal
the chambers.
A spiral is a curve traced by a point that moves around a fixed point. The moving point gets
further and further away from the fixed point.
© Maths Inside 2016 except where otherwise indicated. This document may be used, reproduced, communicated and adapted free of charge for non-commercial educational purposes provided all acknowledgements associated with the material are retained. Maths Inside is a UTS project in collaboration with CSIRO and AAMT.
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Investigating the maths inside: Stargazing with the SKA ACTIVITY 4: SEEKING SPIRALS
© Maths Inside 2016 except where otherwise indicated. This document may be used, reproduced, communicated and adapted free of charge for non-commercial educational purposes provided all acknowledgements associated with the material are retained. Maths Inside is a UTS project in collaboration with CSIRO and AAMT.
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Investigating the maths inside: Stargazing with the SKA ACTIVITY 4: SEEKING SPIRALS
Archimedean spirals
The Archimedean spiral has loops spaced at equal intervals. It is named after Archimedes, who
lived in the third century BC and wrote a book on spirals.
Logarithmic spirals
The curve of the shell of the chambered nautilus is a logarithmic spiral.
© Maths Inside 2016 except where otherwise indicated. This document may be used, reproduced, communicated and adapted free of charge for non-commercial educational purposes provided all acknowledgements associated with the material are retained. Maths Inside is a UTS project in collaboration with CSIRO and AAMT.
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Investigating the maths inside: Stargazing with the SKA ACTIVITY 4: SEEKING SPIRALS
Finding spirals
Sunflowers
Pinecones Roses
© Maths Inside 2016 except where otherwise indicated. This document may be used, reproduced, communicated and adapted free of charge for non-commercial educational purposes provided all acknowledgements associated with the material are retained. Maths Inside is a UTS project in collaboration with CSIRO and AAMT.
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Investigating the maths inside: Stargazing with the SKA ACTIVITY 4: SEEKING SPIRALS
Fruits and Vegetables Shells
Spiral Galaxies Hurricanes
The Milky Way has several spiral arms, each of them a logarithmic spiral of about 12 degrees.
Faces
For more details go to the following sites
https://io9.gizmodo.com/5985588/15-uncanny-examples-of-the-golden-ratio-in-nature?IR=T
Vi Hart
Part 1 https://www.youtube.com/watch?v=ahXIMUkSXX0
Part 2: http://youtu.be/lOIP_Z_-0Hs
Part 3: http://youtu.be/14-NdQwKz9w
© Maths Inside 2016 except where otherwise indicated. This document may be used, reproduced, communicated and adapted free of charge for non-commercial educational purposes provided all acknowledgements associated with the material are retained. Maths Inside is a UTS project in collaboration with CSIRO and AAMT.
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Investigating the maths inside: Stargazing with the SKA ACTIVITY 4: SEEKING SPIRALS
Drawing spirals 1
To draw spirals we need to use polar graph paper, which will be provided.
What do you notice about this kind of graph paper?
Use the information in the table below to plot the points on polar graph paper.
Angle (in degrees) 0 30 60 90 120 150 180
Distance from centre 0 1 2 3 4 5 6
Connect the points in order with a smooth curve.
Is this an Archimedean or logarithmic spiral? How do you know? Did you need the graph to
determine this?
What patterns can you see in the table?
Use the pattern to extend the table. How many times around do you need to go for you to decide
what kind of spiral it is?
© Maths Inside 2016 except where otherwise indicated. This document may be used, reproduced, communicated and adapted free of charge for non-commercial educational purposes provided all acknowledgements associated with the material are retained. Maths Inside is a UTS project in collaboration with CSIRO and AAMT.
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Investigating the maths inside: Stargazing with the SKA ACTIVITY 4: SEEKING SPIRALS
Drawing spirals 2
Plot the points on polar graph paper. Connect the points in order with a smooth curve.
Angle in degrees 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420
Distance from centre 0.5 0.55 0.6 0.65 0.7 0.8 0.85 0.95 1.05 1.15 1.25 1.35 1.5 1.65 1.8
Angle in degrees 450 480 510 540 570 600 630 660 690 720 750 780 810 840 870
Distance from centre 1.95 2.15 2.35 2.6 2.85 3.2 3.4 3.75 4.1 4.5 4.95 5.4 5.9 6.5 7.2
Is this an Archimedean or Logarithmic spiral? How do you know?
© Maths Inside 2016 except where otherwise indicated. This document may be used, reproduced, communicated and adapted free of charge for non-commercial educational purposes provided all acknowledgements associated with the material are retained. Maths Inside is a UTS project in collaboration with CSIRO and AAMT.
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Investigating the maths inside: Stargazing with the SKA ACTIVITY 4: SEEKING SPIRALS
Drawing spirals 3 (using technology)
Change the shape of a logarithmic spiral at. https://www.geogebra.org/m/PCdxBP84
Change the shape of an Archimedian spiral at https://www.geogebra.org/m/f8MDV6YS or
https://www.geogebra.org/m/dZuH5hWa
This spiral was drawn using the software The Geometer’s Sketchpad. A trial version can be
downloaded for free from: http://info.mheducation.com/sketchpad.trial.html.
Placing the telescopes
See this webpage re telescope arrangement in the SKA: https://www.skatelescope.org/layout/
The antennas of the SKA will extend across a continent. Credit: SKA Organisation
More information can be found in this document, especially the diagram on page 34.
https://www.skatelescope.org/wp-content/uploads/2012/07/SKA-TEL-SKO-DD-001-
1_BaselineDesign1.pdf
© Maths Inside 2016 except where otherwise indicated. This document may be used, reproduced, communicated and adapted free of charge for non-commercial educational purposes provided all acknowledgements associated with the material are retained. Maths Inside is a UTS project in collaboration with CSIRO and AAMT.
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