4.1 Fractals — Sierpinski Carpet
Take a square, break it into 9 smaller squares, and remove the central one. Repeat the same procedure on each of the remaining 8 squares, forever. Drag the slider to step through the construction.
Every square that survives step n gives rise to 8 surviving squares at step n+1, so Rn+1 = 8Rn, giving Rn = 8n. Every surviving square also creates one new hole, so Hn+1 = Hn + Rn.
4.1 Fractals — Sierpinski Triangle / Gasket
An equilateral triangle is divided into 4 identical triangles by joining midpoints, and the central triangle is removed. Repeat on the 3 remaining triangles, and so on.
Since each remaining triangle is replaced by 3 smaller ones with 1/4 the area each step, the total remaining area shrinks by a factor of 3/4 every step — approaching zero, even though the perimeter of all triangle edges grows without bound!
4.1 Fractals — Koch Snowflake
Start with an equilateral triangle. Divide each side into 3 equal parts, raise an equilateral "bump" over the middle part, and remove the base of the bump. Repeat this on every side, forever.
Every side of length s is replaced by 4 sides of length s/3, so the number of sides multiplies by 4 and each side shrinks to 1/3 its length every step — the perimeter keeps growing without bound, even though the snowflake stays inside a fixed circle!
4.2 Visualising Solids — Nets
A net is a flat shape that can be folded to make a solid. Faces are the flat surfaces, edges are where two faces meet, and vertices are where edges meet.
Fold a cube from its net
A cube has 6 faces, 12 edges, and 8 vertices. This particular net is a "cross" shape: one square in the middle (front), one square hinged to each side (top, bottom, left, right), and one more hinged below (back). Click Fold to watch it assemble into a cube — a cube actually has 11 different possible nets in total!
Which of these are valid cube nets? (click to check)
Faces, Edges & Vertices of Prisms and Pyramids
| Solid | Faces | Edges | Vertices |
|---|---|---|---|
| Prism (n-gon bases) | |||
| Pyramid (n-gon base) |
Prism: F = n+2, E = 3n, V = 2n. Pyramid: F = n+1, E = 2n, V = n+1.
Nets of a Cylinder and a Cone
Shortest Paths on a Cuboid — The Ant & The Laddu
An ant can only crawl along the surface of a box. To find its shortest path to a laddu, we unfold the box into a flat net — the shortest path on the surface becomes a straight line on the net!
Case 1 — Ant and laddu at the centres of adjacent faces
The laddu sits at the centre of the top face; the ant sits at the centre of an end face. Unfolding the end face and the top face flat about their shared edge lines the two points up on the same vertical line, so the straight-line (shortest) distance is simply half the height plus half the length.
Case 2 — A trickier unfolding (worked example)
Box: 30 cm × 12 cm × 6 cm. The laddu is stuck to the back face, 1 cm from an edge; the ant sits at the centre of an end face. Depending on how we unfold the box, the straight-line distance between the two points changes! One unfolding gives a straight 42 cm path — but it is not the shortest. Unfolding differently gives a right triangle with legs 24 cm and 32 cm:
This shows that when hunting for the shortest path, we must carefully try every reasonable way of unfolding the box, since different unfoldings give different straight-line distances.
Representing Solids — Front, Top & Side Views
Build a stack of unit cubes on the grid below (click a cell to raise its height, shift-click to lower it). Watch the front view, top view, and side view — the projections onto the vertical, horizontal, and side planes — update automatically. This is the same grid used on the Isometric tab!
Click = raise height (max 3). Shift + click = lower height.
Top View (also the build grid)
Front View
Side View
Isometric Grids & Drawing
An isometric projection keeps the lengths of all cube edges equal in the drawing — that's why isometric grids are made of equilateral triangles / hexagons. The same cube stack you build here (or on the Projections tab) is rendered below in isometric view.
Isometric View of Your Stack
Build Grid (shared with Projections tab)
Click = raise height (max 3). Shift + click = lower height. The isometric grid uses three families of parallel lines — /, \, and | — matching the depth, length, and height axes of the solid.