Fractals and Visualising Solids | FIO

Question 22

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?

Solution

Understand the Question

An impossible triangle (or Penrose triangle) is an optical illusion: a two-dimensional drawing that the brain perceives as a three-dimensional solid, but which cannot physically exist in three-dimensional space.
The illusion relies on contradictory depth cues drawn along isometric grid lines ( $30^\circ$ angles and vertical axes).
Viewing the figure orthographically (from front, top, or side) projects the outer $5 \times 5$ shape with an inner $2 \times 2$ cutout.

(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?

Step 1 · Feasibility and Orthographic Profiles

Diagram 1

1. Physical Feasibility:

No, it is impossible to build this model with actual cubes in 3D space.
The 2D drawing connects corners that exist at mutually incompatible depths, creating a geometric contradiction.

2. Profiles (Orthographic Views): Each arm is $5\text{ units}$ long and $1\text{ unit}$ thick, with an inner opening of $2 \times 2\text{ units}$ .

Front Profile: A right-angled triangle with base $= 5\text{ units}$ and height $= 5\text{ units}$ , containing a $2 \times 2\text{ unit}$ square hole at the right-angle corner.
Top Profile: Identical to the front profile ( $5 \times 5\text{ unit}$ right-angled triangle with a $2 \times 2\text{ unit}$ square hole).
Side Profile (Right Side): Identical to the front and top profiles ( $5 \times 5\text{ unit}$ right-angled triangle with a $2 \times 2\text{ unit}$ square hole).

Answer

(i) No, it cannot be built with actual cubes. The front, top, and side profiles are all identical: a right-angled triangle of base $5\text{ units}$ and height $5\text{ units}$ with a $2 \times 2\text{ unit}$ square opening at the corner.

(ii) Recreate this on an isometric grid.

Step 1 · Recreate on an Isometric Grid

Diagram 2

To draw the impossible triangle on an isometric grid:

Draw the bottom horizontal arm ( $5\text{ cubes}$ long) along one of the $30^\circ$ isometric axes.
Draw the vertical arm ( $5\text{ cubes}$ high) upwards from one end along the vertical grid line.
Draw the third diagonal arm ( $5\text{ cubes}$ long) along the opposite $30^\circ$ axis to complete the triangular loop.
Add a thickness of $1\text{ cube}$ to each arm using parallel grid lines.
Leave a central $2 \times 2\text{ cube}$ opening inside the triangle.

Answer

(ii) Draw three mutually intersecting arms each $5\text{ cubes}$ long and $1\text{ cube}$ thick along the isometric grid axes ( $30^\circ$ and vertical), leaving a central $2 \times 2$ cube opening.

(iii) Why does the illusion work?

Step 1 · Reason for the Optical Illusion

The human brain interprets 2D images by assuming standard rules of 3D depth, perspective, and continuity.
In this drawing, each individual joint appears locally plausible and three-dimensional.
However, globally the drawing joins foreground and background elements continuously, forcing contradictory depth cues that the brain cannot reconcile into a single valid 3D solid.

Answer

(iii) The illusion works because each corner looks locally valid in 3D, but together they present contradictory depth cues that the brain cannot resolve into a physically possible 3D object.

Common Mistakes

Assuming Different Profiles: Assuming that the front, top, and side profiles look different; due to the symmetrical cubic structure, all three 2D orthographic projections are identical.
2D Drawing vs. 3D Reality: Confusing the ability to draw a figure on 2D isometric paper with the physical possibility of constructing it in 3D space.

Question 22

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