Question 8
A cube has 11 possible net structures in total. In this count, two nets are considered the same if one can be obtained from the other by a rotation or a flip. For example, the following nets are all considered the same —
Find all the 11 nets of a cube.
A cube net is a 2D shape made of 6 squares joined edge-to-edge that can be folded to form a 3D cube without any overlaps or gaps.
Step 1 — Understanding Cube Nets
We are looking for all unique ways to arrange 6 squares, connected by their edges, such that they can form a cube. Two nets are considered the same if we can rotate or flip one to make it look like the other. There are 11 such distinct nets in total. We will categorize them based on the longest row of squares they contain.
Step 2 — Nets with a 1x4 Spine (4 squares in a line)
These nets have a central row of four squares. The remaining two squares are attached to this central row. There are 6 distinct nets in this category.
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This net looks like a cross.
<DIAGRAM: Net 1 of a cube. A horizontal row of 4 squares. A fifth square is attached above the second square from the left. A sixth square is attached below the third square from the left.>
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This net has one square attached above the first square of the row and another below the third square of the row.
<DIAGRAM: Net 2 of a cube. A horizontal row of 4 squares. A fifth square is attached above the first square from the left. A sixth square is attached below the third square from the left.>
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This net has one square attached above the first square of the row and another below the fourth square of the row.
<DIAGRAM: Net 3 of a cube. A horizontal row of 4 squares. A fifth square is attached above the first square from the left. A sixth square is attached below the fourth square from the left.>
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This net has two squares attached below the third and fourth squares of the row, forming a 1x2 block.
<DIAGRAM: Net 4 of a cube. A horizontal row of 4 squares. A fifth square is attached below the third square from the left. A sixth square is attached below the fourth square from the left.>
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This net has two squares attached above the first and second squares of the row, forming a 1x2 block.
<DIAGRAM: Net 5 of a cube. A horizontal row of 4 squares. A fifth square is attached above the first square from the left. A sixth square is attached above the second square from the left.>
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This net has one square attached below the second square of the row and another below the third square of the row.
<DIAGRAM: Net 6 of a cube. A horizontal row of 4 squares. A fifth square is attached below the second square from the left. A sixth square is attached below the third square from the left.>
Step 3 — Nets with a 1x3 Spine (3 squares in a line)
These nets have a central row of three squares. The remaining three squares are attached to this central row. There are 3 distinct nets in this category.
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This net forms a 2x3 rectangle.
<DIAGRAM: Net 7 of a cube. A 2x3 rectangle made of 6 squares.>
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This net has a 1x3 row. Two squares are attached below the first two squares of the row, and a third square is attached below the second of these two.
<DIAGRAM: Net 8 of a cube. A horizontal row of 3 squares. A fourth square is attached below the first square from the left. A fifth square is attached below the second square from the left. A sixth square is attached below the fifth square.>
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This net has a 1x3 row. One square is attached below the first square of the row. Two other squares are attached below the second and third squares of the row, forming a 1x2 block.
<DIAGRAM: Net 9 of a cube. A horizontal row of 3 squares. A fourth square is attached below the first square from the left. A fifth square is attached below the second square from the left. A sixth square is attached below the third square from the left.>
Step 4 — Nets with a 2x2 Block (no 1x4 or 1x3 spine)
These nets have a 2x2 block of squares as their base. The remaining two squares are attached to this block. There are 2 distinct nets in this category.
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This net has a 2x2 block of squares, with two additional squares attached to one side of this block, forming a 2x3 shape with an extra square.
<DIAGRAM: Net 10 of a cube. A 2x2 block of squares. Two additional squares are attached to one side of this block, forming a 2x3 shape with an extra square.>
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This net has a 2x2 block of squares, with two additional squares attached to adjacent sides of this block, forming an L-shape extension.
<DIAGRAM: Net 11 of a cube. A 2x2 block of squares. Two additional squares are attached to adjacent sides of this block, forming an L-shape extension.>
Answer
The 11 distinct nets of a cube are:
(i)
<DIAGRAM: Net 1 of a cube. A horizontal row of 4 squares. A fifth square is attached above the second square from the left. A sixth square is attached below the third square from the left.>
(ii)
<DIAGRAM: Net 2 of a cube. A horizontal row of 4 squares. A fifth square is attached above the first square from the left. A sixth square is attached below the third square from the left.>
(iii)
<DIAGRAM: Net 3 of a cube. A horizontal row of 4 squares. A fifth square is attached above the first square from the left. A sixth square is attached below the fourth square from the left.>
(iv)
<DIAGRAM: Net 4 of a cube. A horizontal row of 4 squares. A fifth square is attached below the third square from the left. A sixth square is attached below the fourth square from the left.>
(v)
<DIAGRAM: Net 5 of a cube. A horizontal row of 4 squares. A fifth square is attached above the first square from the left. A sixth square is attached above the second square from the left.>
(vi)
<DIAGRAM: Net 6 of a cube. A horizontal row of 4 squares. A fifth square is attached below the second square from the left. A sixth square is attached below the third square from the left.>
(vii)
<DIAGRAM: Net 7 of a cube. A 2x3 rectangle made of 6 squares.>
(viii)
<DIAGRAM: Net 8 of a cube. A horizontal row of 3 squares. A fourth square is attached below the first square from the left. A fifth square is attached below the second square from the left. A sixth square is attached below the fifth square.>
(ix)
<DIAGRAM: Net 9 of a cube. A horizontal row of 3 squares. A fourth square is attached below the first square from the left. A fifth square is attached below the second square from the left. A sixth square is attached below the third square from the left.>
(x)
<DIAGRAM: Net 10 of a cube. A 2x2 block of squares. Two additional squares are attached to one side of this block, forming a 2x3 shape with an extra square.>
(xi)
<DIAGRAM: Net 11 of a cube. A 2x2 block of squares. Two additional squares are attached to adjacent sides of this block, forming an L-shape extension.>
More questions in FIO
Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Sierpinski Triangle.
Find the number of holes, and the triangles that remain at each step of the shape sequence that leads to the Sierpinski Triangle.
Find the area of the region remaining at the th step in each of the shape sequences that lead to the Sierpinski fractals. Take the area of the starting square/triangle to be 1 sq. unit.
Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Koch Snowflake.
Find the number of sides in the nth step of the shape sequence that leads to the Koch Snowflake.
Find the perimeter of the shape at the nth step of the sequence. Take the starting equilateral triangle to have a sidelength of 1 unit.
Figure it Out
- Which of the following are the nets of a cube? First, try to answer by visualisation. Then, you may use cutouts and try.
A cube has 11 possible net structures in total. In this count, two nets are considered the same if one can be obtained from the other by a rotation or a flip. For example, the following nets are all considered the same —
Find all the 11 nets of a cube.
Draw a net of a cuboid having sidelengths:
(i) 5 cm, 3 cm, and 1 cm
(ii) 6 cm, 3 cm, and 2 cm
Observe the front view, top view and side view of the different lines in Fig. 4.6. Is there any relation between their lengths?
Find the front view, top view and side view of each of the following solids, fixing its orientation with respect to the vertical, horizontal and side planes: cube, cuboid, parallelepiped, cylinder, cone, prism, and pyramid. If needed, see the next problem for clues.
Match each of the following objects with its projections.
Draw the top view, front view and the side view of each of the following combinations of identical cubes.
Imagine eight identical cubes, glued together along faces to form the letter 'C'.
Which solid corresponds to the given top view, front view, and side view?
Solids to choose from:
Using identical cubes, make a solid that gives the following projections:
Find the number of cubes in this stack of identical cubes.
What are the different shapes the projection of a cube can make under different orientations?
In addition to the 5 ways shown in Fig. 4.8, are there any additional ways of gluing four cubes together along faces? Can you visualise and draw these as well?
Draw the following figures on the isometric grid.
[Hint: It may be useful to determine whether the edge to be currently drawn — say, along the height — goes from down to up or up to down. Accordingly, draw the line segment on the grid either in the direction of the height axis or opposite to it.]
Is there anything strange about the path of this ball? Recreate it on the isometric grid.
[Hint: Consider a portion of this figure that is physically realisable and identify the 3 primary directions.]
Observe this triangle.
(i) Would it be possible to build a model out of actual cubes? What are the front, top, and side profiles of this impossible triangle? (ii) Recreate this on an isometric grid. (iii) Why does the illusion work?