Exploring Some Geometric Themes

Step 0 is one solid square.

Divide it into a 3×3 grid and remove the centre square.

At the next step, repeat this in each of the 8 remaining squares.

Each square gives 8 squares at the next step, so R_n=8^n.

At each new step, each previously remaining square creates one new hole.

H_0=0 and H_n=1+8+…+8^(n−1)=(8^n−1)/7.

Start with one filled equilateral triangle.

Join the side midpoints and remove the central triangle; 3 triangles remain.

Repeat the same operation in each remaining triangle.

Each triangle produces 3 smaller triangles, so R_n=3^n.

At step k, 3^(k−1) new holes are created.

Total holes through step n: 1+3+…+3^(n−1)=(3^n−1)/2.

In the carpet, each step keeps 8 of 9 equal-area parts: A_n=(8/9)^n.

In the gasket, each step keeps 3 of 4 equal-area parts: A_n=(3/4)^n.

Each side is split into two equal segments.

A segment joining midpoints is parallel to the third side and has half its length.

Each of the four small triangles therefore has three sides of length half the original side.

Start with an equilateral triangle.

Divide each side into three equal parts, raise an outward equilateral bump on the middle third, and remove that third.

Repeat the same replacement on every new side.

There are 3 sides initially.

Each step replaces every side by 4 sides.

S_n=3×4^n.

At each step the number of segments is multiplied by 4 and each segment becomes one-third as long.

P_n=3×4^n×(1/3^n).

Look for a large motif containing smaller copies with a similar arrangement.

Check whether the pattern repeats at more than one scale.

Joining adjacent side midpoints makes a smaller square.

The four removed pieces are congruent right-isosceles triangles.

The four triangles can be rearranged to form a square equal in area to the middle square.

Each side keeps its middle third.

Each of the three corner cuts has length one-third of the original side.

At each corner, join the one-third points on the two adjacent sides.

The four cuts are congruent and symmetrically placed.

A cube viewed straight at a face has a square profile.

A cylinder viewed along its axis has a circular profile.

A triangular prism viewed at an end has a triangular profile.

A cylinder gives rectangle from the side and circle from an end.

A cone gives circle from above and triangle from a side through its axis.

A triangular prism has triangular end and rectangular side views.

A circular frustum gives a trapezoidal side profile and circular top; a pentagonal prism gives pentagon end and rectangle side.

There are 2 polygon faces and n side faces: F=n+2.

There are n edges on each end and n joining edges: E=3n.

There are n vertices on each end: V=2n.

There is 1 base and n triangular side faces: F=n+1.

There are n base edges and n edges to the apex: E=2n.

There are n base vertices and one apex: V=n+1.

The chapter states there are 11 distinct arrangements after rotations and flips are identified.

A cuboid net has two faces of each pair of dimensions.

List the three rectangles for each set of dimensions.

A tetrahedron has four congruent triangular faces.

The source identifies two distinct foldable arrangements, up to rotation/flip.

The curved surface unfolds to a rectangle of height h.

Its width equals the base circumference 2πr.

The two ends are circles of radius r.

The curved surface opens into a sector of radius l, the slant height.

Its arc length equals the base circumference 2πr.

Sector angle=360°×(2πr)/(2πl)=360°r/l.

Each square pyramid has 4 triangular side faces.

The two square bases are joined internally and are not exterior faces.

A sphere has continuously curved surface and no flat faces.

Flattening it onto a plane without distortion is not possible.

Look at the written letters from right to left.

Example: RAVI becomes IVAR when read backwards.

Follow which square would cover each of the cube’s six faces.

Reject arrangements that overlap or leave a face uncovered.

Treat ■ as a square and · as an empty grid position.

Each row of symbols is one row of the net; read from top to bottom.

A tetrahedron has four triangular faces.

Fold the triangles and check whether all four faces meet without overlap.

Draw four equilateral triangles with side 6 cm.

Place one triangle in the centre and attach one along each of its three sides.

Draw a 4 cm × 4 cm square base.

Attach four congruent isosceles triangles, each with base 4 cm and equal sides 3 cm.

Use two equilateral triangles with side 2 cm.

Use three rectangles measuring 6 cm × 2 cm to join their corresponding sides.

Unfold adjacent faces into one plane.

Mark the start and end points on the unfolded faces.

Draw a straight segment between them; test different valid unfoldings.

Every surface route through the same sequence of faces becomes a path of equal length on the net.

The straight segment is the shortest planar path between the corresponding endpoints.

Use Pythagoras: d²=24²+32².

d²=576+1024=1600.

d=√1600.

Different unfoldings place the two marked points at different planar separations.

Some straight segments may cross outside the unfolded faces and therefore are not valid.

The perpendicular component to the plane is removed.

The in-plane component has length p=l cos θ, where θ is the angle between the segment and the plane.

If the square lies parallel to the plane, its projection is a congruent square.

Tilting it can shorten one direction and produce a rectangle or, in a general orientation, a parallelogram.

If viewed edge-on, its projection can collapse to a line segment.

Each pair of opposite sides is parallel before projection.

Parallel projection preserves parallelism.

Project each vertex and each straight side.

Each projected side is a line segment, and opposite parallel sides (when present) remain parallel.

Projection records only the footprint on one plane.

Depth and hidden structure can change without changing that footprint.

The vertical plane gives the front view.

The horizontal plane gives the top view.

The side plane gives the side view.

First choose a fixed upright position for each solid.

Draw the outline seen straight from the front, directly above, and from the side.

A nearby light sends rays that spread out, scaling the shadow.

As the light moves far away, its rays become nearly parallel.

Perpendicular parallel rays reproduce the object’s orthogonal projection on the wall.

Treat each cube as one grid cell with a position across, a position in depth, and a height. In a top view, show a cell if any cube is in that position. In a front or side view, show a cell if at least one cube lies along that line of sight.

(i) Front: 3 in a row; top: an L of 3; side: 2 in a row. (ii) Front: an L of 3; top and side: 2 in a column.

(iii) Front: an L of 3; top: 3 in a row; side: 2 high. (iv) Front heights 1,2; top: a 2×2 square; side heights 2,1.

Count complete layers from the top: 1, 3, 6 and 10 cubes.

1+3+6+10=20.

Face parallel to plane → square; tilted positions → rectangle, parallelogram or rhombus.

Balanced on a vertex → hexagon; edge-on limit → line segment.

In this orientation, the three perpendicular edge directions are equally inclined to the projection plane.

Their projected lengths are therefore equal.

The visible outline is a regular hexagon when all edges are shown.

Join cubes face-to-face and require the whole shape to stay connected.

Rotate or flip arrangements to compare them.

Do not count two arrangements as different if a rotation makes them coincide.

Use the vertical grid direction for height.

Use the two slanting directions for length and depth.

Draw equal unit steps along each axis.

Compare the three corner connections: each visible segment suggests a different depth relation.

Those depth relations cannot all hold at once in one consistent 3D arrangement.

Compare the silhouette of each object from the shown directions.

Match corresponding front, top and side outlines.

The three silhouettes together determine which candidate can fit.

Check each candidate in all three directions.

It appears to descend continuously and return to its starting point.

A real staircase cannot do both; the drawing is an optical illusion.

A cone viewed from the side has a triangular outline.

An upright pyramid can also have a triangular side outline.

Closer torch → larger shadow; farther torch → smaller shadow.

A front view is the silhouette seen straight on.

For side and top views, look perpendicular to those faces and record which unit squares are visible.

Top view fixes the footprint.

Front and side views give the greatest height in each row and column.

Use the three grid directions for the cube’s three edge directions.

Keep parallel cube edges parallel and use equal grid-step lengths.