4. Exploring Some Geometric Themes

Remaining squares: 1, 8, 64, … so R_n = 8ⁿ.

Holes: 0, 1, 1 + 8 = 9, 1 + 8 + 64 = 73, … so H_n = 1 + 8 + 8² + … + 8^{n − 1}.

Each corner triangle has two sides equal to half the side of the big triangle, with a 60° angle between them. So it is isosceles with a 60° top angle, and its other two angles are also 60°. So each corner triangle is equilateral with side half of the big side.

The middle triangle has all three sides equal to these half-sides, so it is also equilateral and of the same size. Hence all 4 triangles are identical.

Each remaining triangle gives 3 triangles in the next step. So R_n = 3ⁿ (1, 3, 9, 27, …).

Each remaining triangle also gives one new hole. So H_{n + 1} = H_n + R_n, and H_n = 1 + 3 + 3² + … + 3^{n − 1} (0, 1, 4, 13, 40, …).

Carpet: each step keeps 89 of the area. Area = (89)ⁿ

Triangle: each step keeps 34 of the area. Area = (34)ⁿ

The area keeps getting smaller and smaller as n increases.

Each side becomes 4 sides. Step 0 has 3 sides, so Step n has 3 × 4ⁿ sides (3, 12, 48, 192, …).

At each step, the number of sides is multiplied by 4 and each side becomes 13 as long. So the perimeter is multiplied by 43.

Perimeter at Step n = 3 × (43)ⁿ units. It keeps increasing without end.

A smaller tilted square is left (half the area).

The four corners are right isosceles triangles. Place them with their right angles together at one point; they form another square of the same size.

A regular hexagon. Each cut-off corner is an equilateral triangle, so all six sides are one-third of the original side and all angles are 120°.

An octagon. It is not regular: four sides are one-third of the square’s side and the four cut sides are longer (diagonals of small squares).

• Square outline: a cube seen face-on.
• Circular outline: a sphere from any side, or a cylinder seen from the top.
• Triangular outline: a cone or a square pyramid seen from the side.
• Rectangle and circle: a cylinder (side view rectangle, top view circle).
• Circle and triangle: a cone (top view circle, side view triangle).
• Rectangle and triangle: a triangular prism lying on its side (or a tent).
• Trapezium and circle: a bucket or a frustum of a cone.
• Pentagon and rectangle: a pentagonal prism (end view pentagon, side view rectangle).

The solids are not unique. For example, a cylinder and a sphere both have a circular top view.

10 sides: faces = 12, edges = 30, vertices = 20.

n sides: faces = n + 2, edges = 3n, vertices = 2n

10 sides: faces = 11, edges = 20, vertices = 11.

n sides: faces = n + 1, edges = 2n, vertices = n + 1

Nets of a cube: (ii), (iii), (iv) and (vi)

(i) and (v) are not nets. In both, two squares fold onto the same face and one face is left open.

Draw a row of four rectangles for the sides around: for (i) 5 × 1, 5 × 3, 5 × 1, 5 × 3 (widths 1, 3, 1, 3 cm, all 5 cm long). Attach two 3 × 1 rectangles at the ends of one 5 × 3 rectangle for the top and bottom.

For (ii): four rectangles 6 × 2, 6 × 3, 6 × 2, 6 × 3 in a row, and two 3 × 2 rectangles as top and bottom.

The first (big triangle made of 4 triangles) and the third (a strip of 4 triangles) are nets.

The second is a bent strip of 4 triangles; two faces overlap on folding, so it is not a net. The fourth has 5 triangles, but a tetrahedron has only 4 faces.

There are no other nets. A regular tetrahedron has only these 2 nets.

Tetrahedron: Draw an equilateral triangle of side 8 cm. Join the midpoints of its sides. Fold along these lines.

Square pyramid: Draw a square of side 4 cm. On each side draw an isosceles triangle with equal sides 5 cm. Fold the triangles up so that their tips meet.

One side is the height of the cylinder. The other side is the circumference of the base (2πr).

The paper still rolls into a cone shape, but the bottom edge is not a flat circle. The cone becomes slanted and will not stand evenly on a table.

Draw three rectangles side by side, each 6 cm × 3 cm (sharing the 6 cm sides). On the top and bottom of the middle rectangle, draw an equilateral triangle of side 3 cm. Fold to get the prism.

No. A flat sheet cannot cover a sphere exactly. There will always be some wrinkles, gaps or overlaps. A sphere has no net.

We must check different unfoldings. The straight path over the top gives 1 + 30 + 11 = 42 cm.

In the unfolding that goes over the top, a side and the bottom, the path is the hypotenuse of a right triangle with legs 24 cm and 32 cm.

d = √(24² + 32²) = √1600 = 40 cm. This is the shortest path.

In the figure, p = AE and l = AB. ∆AEB has a right angle at E, so AB is its hypotenuse. Hence p ≤ l.

The lengths are equal when the line is parallel to the plane.

A square can project to a square, a rectangle, a rhombus, a parallelogram, or a line segment (when it is edge-on).

A parallelogram always projects to a parallelogram (or a segment), because parallel lines project to parallel lines. It can never be a quadrilateral that is not a parallelogram.

It is an n-sided polygon (or a line segment). Its opposite parallel sides stay parallel, but the sides and angles need not stay equal.

Top view (circle): a cylinder or a sphere gives the same circle. Side view (triangle): a square pyramid or a triangular wedge gives the same triangle.

Each view is never longer than the actual line. If a line is parallel to a plane, its view on that plane has its full length. If it is perpendicular to a plane, its view on that plane is just a point.

Car → row 1; Pressure cooker → row 2; Slide → row 3; Chair → row 4; Ceiling fan → row 5; Funnel → row 6; Hammer → row 7; Mug → row 8.

When the torch moves closer to the object, the shadow becomes bigger. The light rays spread out from the torch, so nearer the source they spread over a wider area.

Look at the solid along each arrow. Draw only the outline of the squares you see from that direction. A step or gap in the solid shows as a line or missing square in the view.

Front view: the left column is 3 cubes high and the right column 2 cubes high. Side view: the back row is 3 high and the front two rows are 1 high. Top view: the back two rows are full and the front row has only the left cube.

So it is a stepped wall at the back (3 high and 2 high) with an L-shaped single layer in front. This is solid (ii).

Start with the top view and place one column of cubes on each square. Then use the front and side views to decide the height of each column. Check all three views after building.

The stack is a stepped corner pyramid of 4 layers. The layers have 1, 3, 6 and 10 cubes.

Total = 1 + 3 + 6 + 10 = 20 cubes

A square (face-on), a rectangle (tilted about one edge), or a hexagon (tilted about a corner). When balanced on a corner, it is a regular hexagon.

When balanced on a corner, the three edges from the top corner are tilted equally to the floor. By symmetry, all 12 edges make the same angle with the floor, so their projections are all equal.

Yes. There are 3 more shapes that are not flat: two “twisted” shapes (mirror images of each other) and one “tripod” shape (three cubes attached to one corner cube in three directions).

Draw edge by edge. Use the vertical lines for height and the two slanting directions for length and depth. Count the units along each direction and draw only the visible edges.

Yes. The path seems to go only downhill (or only uphill) all the way round and still come back to the start. This is impossible in real space. Each small part of the drawing is possible, but the parts cannot be joined in three dimensions.

(i) No. Each corner looks like a right angle, but three bars meeting at right angles cannot close up into a triangle. It can be built only as a model that looks right from one special viewpoint.

(ii) Draw three L-shaped bars along the three directions of the isometric grid and join them.

(iii) In an isometric drawing, near and far parts look the same size. So our eyes join edges that are actually at different depths.