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NCERT Solutions Ganita Prakash (Part 2) Chapter 4 –102Drawing on Isometric Grids — Figure it Out

Book page 100 Updated on2026-09-05

Q1.
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?
Answer

Yes — three more, making 8 in all. The five in Fig. 4.8 are the flat ones; the extra three cannot be laid flat on a table, because they reach into all three directions.

PieceHow it is builtFlat?
IFour in a rowFlat
O2 × 2 squareFlat
L / JRow of three with one turning at the endFlat
S / ZZig-zag of two plus twoFlat
TRow of three with one at the middleFlat
Tripod (branch)Three cubes making an L on the table, plus one stacked on the corner cubeNot flat
Right screwTwo cubes side by side, then two more on top but turned a quarter-turnNot flat
Left screwIts mirror image — it cannot be turned into the right screwNot flat
5 flat + 3 non-flat = 8 ways
(If you count the two screws as one shape because they are mirror images, you get 7.)
Why the flat ones and the solid ones are counted differently: a flat piece such as L and its mirror image J can be turned into each other simply by picking the piece up and flipping it over in space. The two screws cannot — no rotation carries one to the other, exactly as your left hand cannot be turned into your right. Shapes like this are called chiral, and they are the reason the count is 8 rather than 7.
Did you know? Six of these 3- and 4-cube pieces make the Soma cube puzzle, in which seven pieces fit together into a perfect 3 × 3 × 3 cube — there are 240 essentially different ways of doing it.
Q2.
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.]
Answer

Work edge by edge, saying out loud which axis you are moving along and in which direction.

  1. Fix the three axes on the paper. | is the height; one slant is the depth; the other slant is the length. Mark them in a corner of the page so you do not lose track.
  2. Start at the nearest bottom corner of the solid — the corner closest to you in the picture.
  3. Trace the outline of the figure, one grid unit per cube, following the hint: for each edge decide the axis and whether you are going forwards or backwards along it.
  4. Add the internal edges that show where one cube ends and the next begins, on the visible faces only.
  5. Rub out any line that is hidden inside the solid, or draw faintly at first and darken only what can be seen.

The three figures set here are of the same family as the solids on page 95 — a step, a T-shaped block and a staircase. Take the staircase as an example: from the near bottom corner go 3 along the length, then 1 up, then 1 back along the depth, then 1 up, then 1 back, and so on. Each ‘up-then-back’ pair is one step of the stair.

Tip: Count the cubes in the figure first and check your drawing has exactly that many top faces showing plus the ones you know are hidden. It catches most mistakes before you have inked anything in.
Q3.
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.]
Answer

Yes — the ball rolls downhill the whole way round and yet arrives back where it started. That is impossible for any real staircase. The picture is a version of the famous ‘endless staircase’ illusion.

Follow the hint. Cover up part of the drawing with your hand. Whatever is left is a perfectly ordinary flight of steps, and you can see at once which grid direction is height, which is length and which is depth. Every small piece of the figure is realisable. It is only when the loop is closed that the trouble appears: the four flights each go down by the same amount, so after going round the total drop should be four steps — yet the drawing brings you back to the same point, where the drop must be zero.

Going round the loop: down + down + down + down = a definite drop
Back at the starting point: drop must be 0
Both cannot be true ⇒ no such solid exists

To recreate it, draw the four flights of steps as four ordinary flights, each stepping down as it goes. Then, at the last corner, join the top of the fourth flight to the bottom of the first as though they were at the same place on the paper — which they are, even though in space they would be four steps apart.

Why the drawing can lie: an isometric projection throws away all information about depth. Two points that are far apart in space can land on exactly the same spot on the paper, and the picture cannot tell you which is nearer. The artist uses this gap: the eye assumes that lines meeting on the paper meet in space, and the brain, which reads each corner as a perfectly ordinary corner, tries to piece the local truths into a whole that cannot exist.
Q4.
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?
Answer

(i) No, it cannot be built as drawn. This is the Penrose triangle. Its three arms are drawn as three bars at right angles to one another, and the picture shows each arm joined to the next — but three mutually perpendicular bars can never close into a triangle. Cover any one corner and what is left is a perfectly ordinary bent bar; it is only the third join that is a lie.

The profiles. Take the object that can be built — three bars along the three axes, meeting at right angles, with a gap left at one corner. Look along each axis in turn. You see the two bars that are across your line of sight, meeting at a right angle, while the third points straight at you and appears only as a small square. So:

ViewProfile
FrontAn L — two arms meeting at a right angle
TopAn L — the other two arms
SideAn L again

(ii) Recreating it. Draw one bar along each of the three grid directions, each a few units long and a unit or so thick. Bring the end of the first up to the start of the second and the second to the third in the usual way. Then, at the last corner, draw the third bar as if it ended flush against the first — on the paper the two ends coincide, even though in space one is far behind the other.

(iii) Why the illusion works. For the same reason as the staircase, and it is worth spelling out because it is the fact this whole section is built on:

  • An isometric projection loses depth entirely. Nothing in the picture says how far away anything is.
  • Parallel edges stay parallel and equal on the paper however far off they are — there is no perspective shrinking to give the game away.
  • So two ends that are metres apart in space can be drawn touching, and the eye reads them as joined.
The deeper point: every corner of the figure is locally correct, and our brain builds its picture of a solid corner by corner. Three correct corners are simply assumed to fit together. The illusion is not a trick of the ink — it is the price of the very property that makes isometric drawing so useful, that it says nothing about depth. Read that way, the impossible triangle is a demonstration of the chapter’s central warning: a projection does not determine the object.
Did you know? A sculpture in Perth, Australia, is built as three separate bars; from exactly one spot in the park a camera sees them close into the impossible triangle. The object is real; only the viewpoint is doing the lying.
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