Fun with Construction

When 4 trapeziums are joined together to form a 3D object - it looks like a square-base pyramid without the "tip" is called a truncated square pyramid.

You can make a hologram out of this when it's created with transparent sides.
From wikipedia (15 August 2015): 
Holography is the science and practice of making holograms. Normally, a hologram is a photographic recording of a light field, rather than of an image formed by a lens, and it is used to display a fully three-dimensional image of the holographed subject, which is seen without the aid of special glasses or other intermediate optics. The hologram itself is not an image and it is usually unintelligible when viewed under diffuse ambient light. It is an encoding of the light field as an interference pattern of seemingly random variations in the opacity, density, or surface profile of the photographic medium. When suitably lit, the interference pattern diffracts the light into a reproduction of the original light field and the objects that were in it appear to still be there, exhibiting visual depth cues such as parallax and perspective that change realistically with any change in the relative position of the observer.

There are several ways to create the truncated square pyramid.

Method 1: Construct 4 trapeziums and piece them together.

Method 2: Construct a hexagon - use 4 of its equilateral triangle and "trim" the tip.

What you need:
  • A sheet of transparency - that will form the pyramid
  • A graph paper - to draw the trapezium with exact dimensions
  • Scissors, Tape, Marker
  • Online resources 

METHOD 1: 4 Trapeziums

Step a: Draw a trapezium on a graph paper (well, this part can be done by construction, too!)
Step b: Trace the trapezium onto the transparency. You will need 4 identical trapeziums.
Step c: Cut the trapeziums and tape them together (along the slant edges) as shown.



METHOD 2: Transforming a regular Hexagon (length 7 cm) into 4 trapeziums



Step a: Construct a hexagon as shown above.
Step b: Trace the shape on a transparency.
Step c: "Trim" the tip and tape the 2 edges together.



Now, creating the hologram!




Step 1: Place the inverted truncated square pyramid on the screen as shown


Step 2: Now, play the clip against a dark background.







Online resource:
Hologram video (you can search for "hologram video" at Youtube)


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To answer the question,
Explain why, by constructing a circle, you are able to construct a hexagon? What are the key properties used.

Both Yi Ming and Norman have almost complete answer; and it would be a complete answer if we merge the points given by both of them:

from Yi Ming


from Norman



Consider the following: 

  • With reference to the centre of the circle,
    • A circle can be divided into 6 identical sectors, each with an angle 60 degrees
      • because angle at a point is 360 degrees. 
      • hence, 360 ÷ 6 = 60
    • Radii of the circle will form two sides of a triangle with the centre of the circle being the vertex. Hence resulting an isosceles triangle. 
      • To find the base angles of the isosceles triangle, we have (180 - 60) ÷ 2, which gives 60 degrees.
    • In other words, the triangle has all three angles = 60 degrees, and is therefore an equilateral triangle. 
      • Since it is an equilateral triangle, it means the length of the base would be same as the radius.
      • Hence, with a compass (without changing the length), we can use use it to mark arcs on the circumference. 
      • By doing this, we would be able to divide the circumference into 6 equal parts. 
    • By joining the adjacent markings on the circumference, we can form a regular hexagon.




















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