Geometry in nature

Dear students,
I came across this slide show while searching ,something related to geometry in nature. I liked it and sharing it with you. Look how the geometrical shapes are hidden in the nature.

This album is powered by BubbleShare - Add to my blog

Exploring a dodecahedron

This Project is contributed by Kapil Khurana XF

Aim Exploring a dodecahedron

Material required

Procedure
Step 1. Take a blank sheet and draw the outline of a CD .

Step 2. Now find out the centre of the circle formed by drawing two chords and their perpendicular bisectors and draw one radius and produce it slightly out of the circle .

Step 3. Now taking the centre and with the help of a protector make an angle of 72 degree .

Step 4 . Repeat the process till the first radius and the last radius also make an angle of 72 degree .


Step 5 . Now put a CD over it and mark the meeting points of CD corners and these lines

Step 6 .Now join all five points on the CD obtained by the previous step
Step 7 . Now cut the CD along these lines
Step 8. Repeat all the previous steps to form 11 more polygons of CD and hence we have 12 CDs

Step 9 . Now paste the two sets CDs with the help of M seal in the following manner

Step 9 . Put the two sets over one another and paste them very carefully
Step 10. Cover all the boundaries with tape and then paint it with a permanent marker to give it a metallic look

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. It has twenty (20) vertices and thirty (30) edges.

The Dodecahedron has been a source of metaphysical interest for at least 2000 years. Like a crystal or gem, its facets and symmetries compel our eyes and hearts to observe life more deeply. Some have believed that the Dodecahedron represents an idealized form of Divine thought, will, or idea. To contemplate this symbol was to engage in meditation upon the Divine. Today many people believe there is a lost knowledge residing in the past, slowly being rediscovered. It is known that figures like Pythagoras, Kepler, and Leonardo, among many, were educated in these Sacred Geometries, and held many beliefs about them and their role in the Universe .
This is a very creative work done by Kapil. He has utilised e-waste for making such a useful mathematical model.I would like to thank him on behalf of the members of Planet Infinity.

Count the number of triangles

Mathematics in nature

Appreciate the presence of Mathematicsin nature. The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. It begins with the number 1, and each new term from there is the sum of the previous two.

Fact Map Probability

This is a mind map to learn the sample space in Probability.

Some Examination Tips

Dear Students,
Do remember the following...

1. Divide a chapter into smaller parts.

2. Prepare a revision time table.

3. Revise the difficult topics when your mind is fresh. Do not postpone the difficult topics.

4. Prepare short diagrams/mind maps for the end time revision.

5. Do not just read. Learn by writing. Practice the difficult questions.

6. Divide a question into steps. Try to remember the general methods.

7. Learn the statements of the theorems by writing.

8. Take a break after every 40 minutes of study.

9. Plan the syllabus according to time.

10. Eat well and sleep well.

11. On the examination day, do not worry, even if you feel that you remember nothing! Give it your best shot, you will be able to recall once the questions are in front of you.

12. Answer all questions. Put the correct question number.

13.Leave some space after each question.

14. Read the instructions on your exam paper and if you are unsure about anything, don’t hesitate to ask the invigilator .

15. Don’t waste time. If you get stuck on a question because it seems difficult or confusing, move on to the next and return to the tough questions after.

16. Make neat diagrams/figures in the geometry questions. Do not forget to label the figures.

All the Best.

Project on Sieves method.

This album is powered by BubbleShare - Add to my blog

This project is contributed by Harmeet X F.

Making a Fractal Card

Q. What is a fractal ?

Ans. A fractal is a shape which is repeated many times at different intervals . They are similar to each other . In other words , if we magnify one portion of the fractal card then it will look like the whole fractal.

Fractals were introduced by IBM scientists Benoit Mandelbrot while studying geometrical figures like structure of coastline and lightening flashes .

Trees branching, rivers meandering, lightening zigzagging, insects whirring and birds wheeling in the sky...flames leaping, metal fracturing, earthquakes rendering…all trace fractal paths through space and time. Processes in lungs, bile duct system, bowel and kidney, in the neutral networks in the brain, in the placenta and the heart, in the ways we walk and talk, are full of fractals. Each of the topics contains examples of fractals in the arts, humanities, or social sciences.

AIM:
To create paper sculptures of fractal shapes by cutting and folding paper.

MATERIAL REQUIRED :
Geometry box , white ivory sheet , black pastel sheet , big ruler , scissors , cutter , fevicol , box board sheet .

Step 1: Take a rectangular white ivory sheet with dimensions 24x11 inches.

Step 2: Draw lines on both sides at a distance of ½ inches. Leave 1 inch as a border.

Step 3: Draw diagram of the design of fractal card we are willing to represent. Draw the lines darker than the previous ones.

Step 4: According to the diagram, cut the vertical lines. Use a cutter to cut the lines.

Step 5: Draw crease on horizontal lines according to the diagram. Use a scale and an empty refill of gel pen to draw creases.

Step 6: Fold along the creases properly. Wrinkles should not be there on the card.

Step 7: Similarly, complete other folds according to the diagram .


Step 8: Take a box board sheet. Cut to squares of 13 inches. Box board sheet should be very hard.

Step 9: Take a black pastel sheet. Cut a rectangle of dimensions of 26x13 inches.

Step 10: Paste the black rectangular sheet (26x13”) on the two square sheets (13”). Keep the two square sheets adjacent to one another, so that they are joined together through the black sheet. This becomes a folder.

Step 11: Paste the final fractal card on this folder.The final Fractal Card looks like…..This is really a beautiful fractal card.

Mathematics Around

Pictures clicked by Jigyasu X F

Pascal Triangle Link

Dear Students,
Use this link for your project on Pascal's triangle.
http://ptri1.tripod.com/

Making Platonic Solids

This Project is contributed by Sahil XB

• Aim:To make Platonic solids and verification of Euler's formula , F - E + V = 2
  where , F represents number of faces , E represents the number of edges and V represents number of vertices .

Introduction
A platonic solid is a polyhedron all of whose faces are congruent regular polygons and where the same number of faces meet at every vertex.
Platonic solids are 3 – dimensional solids bounded by regular polygons which are equilateral triangle , square and regular Pentagon . They are only five in number , namely cube , Tetrahedron , Octahedron , Dodecahedron and Icosahedron .

• There are 5 platonic solids
  Why??

The Greeks recognized that there are only five platonic solids.

The key observation is that the interior angles of the polygons meeting at a vertex of a polyhedron add to less than 360 degrees. To see this note that if such polygons met in a plane, the interior angles of all the polygons meeting at a vertex would add to exactly 360 degrees.

• About Leonard Euler...
  Born April 15, 1707Basel, Switzerland
  Died September 18 ,1783St Petersburg, Russia
  Nationality Swiss
  Field Mathematics and physics
• Pre requisite Knowledge...
  
  Knowledge of a regular polygon
  and vertices , faces and edges of 3D solids
• Material Required
  Nets of Platonic solids ,a pair of scissors , transparent sheets (OHP sheets) ,a marker , cello tape , compass and a ruler.
• What is a net of a solid?
  A net is a form of skeleton-outline in 2-D which when folded results in a 3-D shape .
  For example . A tetrahedron net a dodecahedron net
  an icosahedron net

an octahedron net

a cube net


My observations

The interior angle of an equilateral triangle is 60 degrees. Thus on a regular polyhedron, only 3, 4, or 5 triangles can meet a vertex. If there were more than 6 their angles would add up to at least 360 degrees which they can't. Consider the possibilities:3 triangles meet at each vertex. This gives rise to a Tetrahedron .

The tetrahedron also has a beautiful and unique property ... all the four vertices are the same distance from each other! And it is the only Platonic Solid with no parallel faces.

Tetrahedron Facts
•It has 4 Faces
•Each face has 3 edges, and is actually an Equilateral Triangle
•It has 6 Edges
•It has 4 Vertices (corner points) and at each vertex 3 edges meet

cube Facts
•It has 6 Faces
•Each face has 3 edges, and is actually a Square
•It has 12 Edges
•It has 8 Vertices (corner points) and at each vertex 3 edges meet
Since the interior angle of a square is 90 degrees,therefore atmost 3 squares meet at a vertex.

A cube is called a hexahedron because it is a polyhedron that has 6 (hexa- means 6) faces.
Cubes make nice 6-sided dice, because they are regular in shape, and each face is the same size.

Octahedron Facts
•It has 8 Faces
•Each face has 3 edges, and is actually an Equilateral Triangle
•It has 12 Edges
•It has 6 Vertices (corner points) and at each vertex 4 edges meet
4 triangles meeting at a vertex,gives rise to an octahedron.

It is called an octahedron because it is a polyhedron that has 8 (octa-) faces, (like an octopus has 8 tentacles)

Icosahedron Facts
•It has 20 Faces
•Each face has 3 edges, and is actually an Equilateral Triangle
•It has 30 Edges
•It has 12 Vertices (corner points) and at each vertex 5 edges meet
5 triangles meeting at a vertex gives rise to an icosahedron.

It is called an icosahedron because it is a polyhedron that has 20 faces (from Greek icosa- meaning 20)

Dodecahedron Facts
•It has 12 Faces
•Each face has 5 edges, and is actually an Equilateral Triangle
•It has 30 Edges
•It has 20 Vertices (corner points) and at each vertex 3 edges meet
3 pentagons meeting at a vertex, gives rise to a dodecahedron.

It is called a dodecahedron because it is a polyhedron that has 12 faces (from Greek dodeca- meaning 12)

• Experimentation with Platonic Solids.

Now I will verify the Euler’s formula .
* F - E + V = 2
where , F represents number of faces , E represents the number of edges and V represents number of vertices .
Firstly, take a solid e.g. an octahedron. Using a pin and thermocol balls , mark the vertices.

It is observed that there are 6 vertices.

Now, count the number of faces.Using a marker ,number them.
It is observed that there are 8 faces.

Now, count the number of edges.

It is observed that there are 12 edges.

Now ,In an octahedron V= 6

F= 8

E= 12

Therefore, V+F-E= 6+8-12 =2

Thus Euler's formula is verified.
Similarly the formula is verified for other solids also.

This is a creative project done by Sahil.He has made beautiful Platonic Solids and experimented on them.

Tangram My Way

This work is contributed by Khilesh X F.
It was a wonderful experience for me working on such as interesting topic.I also taught my elder brother and sisters about tangram and how to play with it. I enjoyed working on this topic.
Firstly, On a square board I made the tangram lines according to the instructions given .
1. Draw a square ABCD of side 5 inches.
2. Joint BD
3. Mark mid point of BC as E and CD as F.
4. Joint EF.
5. Mark mid point of EF as G and BD as H.
6. Join AHG.
7. Mark mid point of DH as I and HB as J.
8. Join FI and GJ.
9. Cut all 7 pieces and make the figures.



Tangram is the oldest Chinese puzzle. It consists of seven pieces called tans.
5 right isosceles triangles( 2 small ,1 medium size ,2 large size )
1 square
1 parallelogram

Then I made various shapes and figures using the 7 shapes.

Geoboard is Fun!

This project is contributed by Tushar Bansal X B

Aim To make a geoboard and verify properties of geometrical shapes on it.

Introduction
A geoboard is a device often used to explore basic concepts in plane geometry such as perimeter, area or the characteristics of geometrical figures.
It was invented and popularized by Egyptian mathematician Caleb Gattegno in the 1950's

Material Required
•A wooden board
•A white chart paper
•Fevicol
•Sketch pen
•Nails
•Hammer
•Rubber bands

Procedure
20th June 2007(starting date of project)
Step 1- Take a wooden board of size 12inches x 12inches

Step 2- Take a white sheet of paper and cover one side of the board with it

Step 3-Now, with a sketch pen, draw 11 lines horizontally (parallel to length) and 11 lines vertically (parallel to breadth) leaving a gap of 1 inch between every line

After completing step 3, the board will look like this

Step 4-Now,hammer the nails on the points where vertical and horizontal lines intersect each other.
Now the geoboard is ready.

Experiment
Take a rubber band and stretch it along 6 horizontal and 4 vertical nails, so as to form a rectangle of size 5in.x3in.Now calculate the number of squares(1inchx1inch), inside the rectangular figure.
The number of squares inside the rectangular figure i.e. 15 is equal to area of that rectangle.
Proof
Area of rectangle= length x breadth
Area of rectangle= 5 x 3 sq.inches
Area of the rectangle= 15 sq.inches

Utility
It can be used to calculate areas of regular and irregular shapes.

Its a very useful device to learn various geometrical results and verify them .

Good Job!