11.2 Visualising Solids

Designing in the mind’s eye Long before a single line is drawn on paper, a good engineer assembles the object inside the imagination, rotating it, testing how the parts meet, redesigning the awkward joints, all without lifting a pencil. The American architect and inventor Buckminster Fuller (1895 to 1983), who popularised the geodesic dome, was famous for turning structures over in his head until they were sound. This section is an invitation to work that way: to build, fold and spin shapes in your imagination before paper ever enters the picture.

Build it in your imagination

Try This: eyes closed, no paper! For each prompt, picture the answer by sight, point, describe it aloud, but do not draw.

  1. Picture a word, then read off its letters backwards by sight (not by sound). Try it next with a classmate’s name.
  2. Snip the four corners off an imaginary square, each cut running between the midpoints of two neighbouring edges. What shape is left? How would you piece the four corners back into another square?
  3. Mark the sides of an equilateral triangle into thirds and snip off each corner up to the marks. What shape appears?
  4. Mark the sides of a square into thirds and snip off each corner up to the marks. What shape is left?
  1. Cutting at the midpoints removes four right-angled corner triangles and leaves a smaller square (tipped 45 degrees, its corners sitting at the midpoints of the original). The four cut-off triangles are congruent right isosceles triangles; two of them join along their long sides to form a small square, and two such small squares rebuild a square equal to the inner one.
  2. Trimming an equilateral triangle’s corners at the third-marks turns each 60-degree corner into a short edge, giving a regular hexagon.
  3. Trimming a square’s corners at the third-marks turns each 90-degree corner into a short edge, giving a regular octagon.

Profiles: a solid from different viewpoints

When you look at a solid object, what you actually catch is its profile, its outline from one particular direction. Swing round to a new direction and that outline can change beyond recognition. (Picture a cartoon character bursting through a wall and leaving a hole shaped like their silhouette.)

A single solid can offer wildly different profiles:

Solid A possible profile
Cube square (face-on), hexagon (corner-on)
Sphere a circle, from every direction
Cylinder rectangle from the side, circle from the end
Cone triangle from the side, circle from below
Square pyramid triangle from the side, square from below

Figure it Out: match a solid to its profile

For each profile, describe a solid (and a viewpoint) that produces it. Then look for solids that show two contrasting profiles from two different directions.

  1. A profile with a square outline.
  2. A profile with a circular outline.
  3. A profile with a triangular outline.
  4. Rectangular from one viewpoint, circular from another.
  5. Circular from one viewpoint, triangular from another.
  6. Rectangular from one viewpoint, triangular from another.
  7. Trapezium-shaped from one viewpoint, circular from another.
  8. Hexagonal from one viewpoint, rectangular from another.

Are the solids one-of-a-kind, or can you list several possibilities for each?

  1. A cube or cuboid seen face-on; a square pyramid seen from directly below also works.
  2. A sphere (any direction), or a cylinder/cone seen end-on.
  3. A cone seen from the side, or a triangular prism seen end-on, or a square pyramid from the side.
  4. A cylinder: side view is a rectangle, end view is a circle.
  5. A cone: side view is a triangle, the view from below is a circle.
  6. A triangular prism: a rectangle from one side, a triangle from the end. (A box-with-a-roof shape works too.)
  7. A cone with its tip sliced off (a frustum): side view is a trapezium, top/bottom view is a circle.
  8. A hexagonal prism: the end view is a hexagon, the side view is a rectangle.

None are unique, many solids share the same profile, which is exactly why a single profile can never pin down a solid. (We patch this below using three views.)

Faces, edges and vertices

For solids bounded by flat surfaces we name three features:

  • Faces, the flat surfaces forming the boundary.
  • Edges, the line segments where two faces meet.
  • Vertices, the points where edges meet.

A cube or cuboid has 6 faces, 12 edges, 8 vertices.

Prisms. A prism carries two congruent polygons as its opposite faces, joined by edges between matching vertices; the remaining faces are parallelograms. A prism takes the name of its end polygon (triangular prism, pentagonal prism, …).

Pyramids. A pyramid has a polygonal base and one apex point above it; edges run from the apex down to each base vertex. A pyramid takes the name of its base (triangular pyramid, square pyramid, …). A triangular pyramid is also called a tetrahedron.

Math Talk: counting for an n-sided figure Work out the faces, edges and vertices for a prism and a pyramid whose polygon has 7 sides, then for a general n-sided polygon.

Prism (n-gon ends): 2 end faces +\ n side faces = n+2 faces; \ 2n vertices; \ 3n edges (n on top, n on the bottom, n joining them). For n=7: 9 faces, 14 vertices, 21 edges.

Pyramid (n-gon base): 1 base +\ n triangular side faces = n+1 faces; \ n+1 vertices (base corners plus apex); \ 2n edges (n around the base, n up to the apex). For n=7: 8 faces, 8 vertices, 14 edges.

Notice that each of these obeys Euler’s formula F + V = E + 2.

Exam Tip The commonest slip with pyramids is to forget the slant edges. An n-gon pyramid has n edges around the base and n more running up to the apex, giving 2n edges in total, not n. A quick safety net: count F, V and E separately, then check that F + V = E + 2. If Euler’s formula fails, you have miscounted something.

Worked Example 1

A hexagonal pyramid. How many faces, edges and vertices does a hexagonal pyramid have, and does it satisfy Euler’s formula?

A hexagonal pyramid has an n=6 base. Using the pyramid counts:

  • Faces: n+1 = 6+1 = \mathbf{7} (one hexagonal base plus 6 triangular sides).
  • Vertices: n+1 = 6+1 = \mathbf{7} (six base corners plus the apex).
  • Edges: 2n = 2\times 6 = \mathbf{12} (six around the base, six up to the apex).

Check Euler: F + V = 7 + 7 = 14 and E + 2 = 12 + 2 = 14. They agree, so the formula holds.

Worked Example 2

Reading Euler’s formula backwards. A convex solid has 20 faces and 12 vertices. How many edges does it have?

Euler’s formula says F + V = E + 2, so E = F + V - 2.

E = 20 + 12 - 2 = 30.

The solid has \mathbf{30} edges. (This is in fact the icosahedron, the solid with twenty triangular faces.)

Nets: unfolding a solid

A net is a flat shape that can be folded up into a solid, equivalently, what you get by unfolding a solid onto a plane. (To build a real model you add extra flaps for gluing faces together, but the net itself is only the unfolded faces.)

Worked Example 3

How many nets does a cube have?

A cube has 11 different nets, counting two as the same when one is a rotation or flip of the other. A well-known valid net is the “cross”: a column of four squares with one extra square stuck to the left and one to the right of the second square in the column. Six squares joined edge-to-edge fold up into the cube’s six faces.

A flat strip of six squares in one straight row is not a net, folding it would stack two faces in the same place and leave gaps elsewhere. Arrangement matters, not merely the head-count of squares.

Other handy net facts:

  • A regular tetrahedron (every face an equilateral triangle) has exactly 2 nets.
  • A cylinder unrolls into a rectangle plus two circles. With a cut along the height, one side of the rectangle equals the height of the cylinder and the other equals the circumference of the base circle, 2\pi r.
  • A cone unrolls into a circle (the base) plus a sector of a larger circle. Because every point on the base circle sits the same slant distance \ell from the apex O, the curved edge of the net is an arc of a circle centred at O with radius \ell.
  • An octahedron (two square pyramids glued base-to-base) is built from 8 equilateral triangles and, like the cube, has 11 nets.
  • A dodecahedron (12 pentagonal faces) has a stunning 43,380 nets.
  • A sphere has no flat net whatsoever, you cannot wrap a flat sheet round a ball without creases, gaps or overlaps. (That is why every flat world map warps the globe.)

Figure it Out: Nets

  1. Which of these are nets of a cube? Decide first by visualising, then check with cut-outs. (A dependable test: it must hold 6 squares, and when folded no two squares should land on the same face and no face should be left open.)
  2. A cube has 11 distinct nets (rotations/flips counted as the same). Try to find all 11.
  3. Draw a net of a cuboid with side lengths (i) 4\text{ cm},\ 3\text{ cm},\ 2\text{ cm} and (ii) 6\text{ cm},\ 4\text{ cm},\ 1\text{ cm}.
  1. A shape is a cube net only when it has exactly 6 squares and folds with no overlap and no missing face. The “cross” (plus-sign) shape and the various staircase/T/Z layouts work; a single straight row of 6 squares, or any layout containing a 2\times 2 block of four squares, does not fold into a cube.
  2. The 11 nets sort into families: six of the “1-4-1” kind (a row of four squares with one square above and one below, the two extras placed in different spots), three of the “2-3-1” kind, one “2-2-2” staircase, and one “3-3” kind. Together: 6 + 3 + 1 + 1 = 11.
    1. Lay out the three pairs of rectangles, two 4\times3, two 4\times2 and two 3\times2, in a cross so that opposite faces sit across from each other. (ii) Same idea with two 6\times4, two 6\times1 and two 4\times1 rectangles. Folding either cross gives the cuboid.

The shortest path on a cuboid

On a flat plane the shortest path between two points is the straight line. But on the surface of a box, where you may only crawl along the surface, the answer stays hidden, until you unfold the box.

The big idea: unfold the cuboid into a net. A surface path becomes a path of the same length on the net, and the shortest path on the (flat) net is simply the straight line. Then fold the net back up to read off the path on the box.

There is a catch: a cuboid can be unfolded in several different ways, and the straight line is shortest only on the right unfolding. So we must list the candidate unfoldings and compare.

Worked Example 4

The beetle and the honey drop. A beetle rests on one face of a cuboidal carton, and a drop of honey clings to another face. After choosing the correct unfolding, the flat picture places the beetle and the honey at the two ends of a right triangle with legs 27\text{ cm} and 36\text{ cm}. By the Baudhāyana (Pythagoras) Theorem, the straight-line distance d is

d^2 = 27^2 + 36^2 = 729 + 1296 = 2025 \quad\Rightarrow\quad d = \sqrt{2025} = 45\text{ cm}.

A different unfolding of the same carton stretches the straight line longer, so it is not the shortest. The moral: the way you unfold matters, try every unfolding and keep the smallest distance.

Worked Example 5

A second unfolding. For another carton, the correct unfolding sets the beetle and the honey at the ends of a right triangle with legs 16\text{ cm} and 12\text{ cm}. Find the shortest surface path.

d^2 = 16^2 + 12^2 = 256 + 144 = 400 \quad\Rightarrow\quad d = \sqrt{400} = 20\text{ cm}.

So the beetle’s shortest crawl is \mathbf{20}\text{ cm}. Always re-check that the straight segment stays inside the net, if it strays outside, that unfolding does not match any real surface path, and you must pick a different unfolding.

Math Talk Picture the beetle at the centre of one face and the honey at the centre of the top face, or stuck at the centre of an edge. For each placement, sketch a net, mark the straight line, and measure it. Why can we be certain a given path is shortest? Because once it is the straight segment on a net, no shorter path exists on a plane, and every surface path matches some net path of equal length.

Representing solids on paper: projections

Designers need to record a 3-D object on flat paper precisely enough to rebuild it. Nets are one route, but they are hard to read at a glance. A second route is to use projections.

What is a projection? Take a point P in space and a plane M. Drop a perpendicular from P to the plane, hitting it at O (so OP makes a 90-degree angle with every line through O that lies in the plane). Then O is the projection of P onto M. Project every point of an object and you get the projection of the object.

What becomes of a line’s length under projection? A projected line is usually shorter than the original (think of a slanted stick and its shadow on the floor). Set up the standard figure: if l is the true length of a segment and p the length of its projection, then drawing a perpendicular forms a right triangle in which p is a leg and l is the hypotenuse, hence

p \le l.

The projection equals the original length, p = l, only when the line runs parallel to the plane (it lies flat, so nothing is foreshortened).

Parallel lines stay parallel A key fact that makes engineering drawings trustworthy: the projection of a pair of parallel lines is again a pair of parallel lines. So the projection (or perpendicular shadow) of a parallelogram is always a parallelogram, however you tilt it; the projection of an n-sided polygon is assembled from the projections of its sides.

One projection is not enough. A single projection can come from many different solids (a long stick seen end-on projects to a single point, just as a lone point does). To clear the ambiguity, we take three mutually perpendicular projections:

View Projected onto What you see
Front view vertical plane (in front) the object seen from the front
Top view horizontal plane (below) the object seen from directly above
Side view side plane (to the side) the object seen from the side

These three views together usually fix the solid completely.

Figure it Out: Views

  1. Draw the top, front and side views of various stacks of identical cubes. Is there a relation between the lengths of one line as it appears in the three views?
  2. Eight identical cubes are glued into the letter L. From the front it reads L; what does it look like from the side and from the top? Can you add cubes so it shows different letters from front, top and side?
  3. Given a front view, top view and side view, decide which solid produces them.
  4. Using identical cubes, build a solid that produces three given views.
  5. Find the number of cubes in a given stack (remember some cubes may be hidden behind others).
  6. What different shapes can the projection of a single cube make as you reorient it? (Square, rectangle, hexagon, …)
  1. The three views of one straight edge can have different lengths: a view shows the true length only when the edge runs parallel to that view’s plane, and a shorter length otherwise (shrinking to a single point if the edge is perpendicular to the plane).
  2. An L of cubes: from the side you see a single column (a vertical bar), and from the top you see the L’s footprint, the row of base cubes plus the foot. By adding cubes you can engineer the three views to read as different letters (for instance arrange cubes that spell one letter from the front and another from above).
  3. & 4. Read the three views together: the top view gives the floor footprint, while the front and side views give the heights along two directions. The one stack consistent with all three is the answer; build it cube by cube and re-check each view.
  4. Count layer by layer, and add the hidden cubes needed to support the visible ones. A stack that looks like a 3\times3 pyramid, for example, must hide cubes underneath.
  5. A cube’s perpendicular projection can be a square (face-on), a rectangle (tilted about one edge), or a regular hexagon (balanced on a corner).

Shadows are projections

Shine a torch perpendicular to a wall and an object’s shadow looks much like its projection, though a nearby torch makes the shadow larger and a little warped. The further back the light, the closer the shadow creeps to a true projection. The ultimate “infinitely distant torch” is the Sun: when sunlight meets a plane at right angles, the shadow it throws is indistinguishable from the projection. Cut out a parallelogram and check in sunlight, its shadow stays a parallelogram, confirming that parallel lines project to parallel lines.

Isometric projection

Projection usually loses information. But with the right orientation we can hang on to most of it. Balance a cube on one corner vertex and project it straight down: now all twelve edges project to segments of equal length. This is the isometric projection (“isometric” is Greek for equal measure). The cube’s outline becomes a regular hexagon, and if the cube were glass you would see all its edges at once.

Math Talk: why equal lengths? Hold a model cube balanced on one corner. The three edges meeting at the top vertex fan out symmetrically around the vertical axis, every one tilted by the same angle to the projection plane. Equal tilt means equal foreshortening, so all three (and by parallelism, all twelve) edges project to the same length. That symmetry is precisely what “isometric” names.

Drawing on an isometric grid. Tile the plane with the three edge directions of the hexagon and you get an isometric grid. On it, the three principal axes of a solid, length, depth and height, show up as three families of parallel lines, and unit distances along each axis project to equal lengths. That is why an isometric drawing conveys a solid so faithfully: parallel edges stay parallel and equal edges stay equal.

To draw a solid, march edge by edge, counting units along the height, length and depth directions. A 1\times1\times1 cube becomes a small hexagon of three rhombi; a 2\times2\times2 cube is the same drawing enlarged. A row of four cubes (one of the five Tetris tetromino shapes) can be oriented along any of the three axes.

Figure it Out: Isometric grids

  1. Beyond the 5 Tetris shapes (the ways to join four squares edge-to-edge in a plane), are there extra ways to glue four cubes together along faces in 3-D? Visualise and draw them.
  2. Draw the given figures on the isometric grid, deciding for each edge whether it goes “up” or “down” along the height axis.
  3. Recreate the path of a “strange” bouncing ball on the isometric grid, is anything odd about it?
  4. The impossible triangle (Penrose triangle): (i) Could you build it from real cubes? What are its front, top and side profiles? (ii) Recreate it on an isometric grid. (iii) Why does the illusion work?
  1. In a flat plane there are 5 ways (the Tetris tetrominoes). In 3-D there are extra non-flat arrangements of four cubes, for example an L-tromino with the fourth cube stacked on top rather than kept in the same plane. Counting all distinct solid four-cube shapes gives 8 in total (these are the 3-D “tetracubes”).
  2. Map each solid edge to one of the three grid directions (length, depth, height) and trace it unit by unit, lifting “up” or dropping “down” along the height axis as the figure demands.
  3. The ball’s path looks consistent on the isometric grid yet is physically impossible as a continuous 3-D trajectory, the isometric view buries depth, so a path can appear to connect levels that could never really meet.
    1. You cannot build a true Penrose triangle from real cubes; each corner looks fine on its own, but the three corners cannot all join up consistently in 3-D. Its separate profiles each look like ordinary right-angled bent bars. (ii) Draw three bars along the three isometric directions and join them so each junction looks correct locally. (iii) The illusion works because isometric drawing throws away depth information, your eye assumes the nearer-looking bar is genuinely nearer at every corner, but those assumptions clash with one another globally.