A sponge with no volume and infinite surface: the Menger sponge

AA SLAI SLAA HL Accessible Also in: Pure maths

Research question

How do the volume and surface area of the Menger sponge change at each stage, why does the volume tend to 0 while the surface grows without limit, and what happens if you remove a different pattern of cubes?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

Visual, countable and surprising. Careful counting of faces is the challenge, and designing your own removal rule makes it original.

The mathematics you'll need

Geometric sequences for volume
Counting exposed faces at each stage (the tricky part)
Limits of sequences
Fractal dimension log 20 / log 3
Generalising to your own rule

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

No data needed; build stage 1 from sugar cubes or card to check face counts.

GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Derive the volume sequence.
Count surface area at stages 1 and 2 carefully.
Find a recurrence and solve it.
Find the dimension.
Design your own rule and compare.

Pitfalls that cost marks

Miscounting internal faces.
Copying a surface formula without deriving it.
Ignoring your own variation.

Showing personal engagement

Build a stage-1 model.
Invent a removal rule.
Compare with the 2D Sierpiński carpet.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Geometric sequences for volume; Counting exposed faces at each stage (the tricky part)
AA HL	Fits, but add an HL technique	Geometric sequences for volume; Counting exposed faces at each stage (the tricky part)
AI SL	Good fit	Geometric sequences for volume; Counting exposed faces at each stage (the tricky part)
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

Level: Accessible. A good first extended piece of maths, with room to go deeper. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Generate the fractal or the iteration yourself, choose your own variation (a different rule, angle or starting value), and pursue a question you raised while exploring.

Reflection (D)

Reflect on the limits of the model: real coastlines and plants are only self-similar over a range of scales, and computer iterations carry rounding error. Say how that affects your numbers. For this idea, start with: miscounting internal faces — say how it affects your answer.

Use of mathematics (E)

SL: Geometric sequences and series for lengths and areas, logarithms for dimension, and iteration of functions, each calculated and checked numerically.

HL: Proof of a limit or dimension, complex-number iteration with a derived condition, or analysis of fixed points and their stability using derivatives.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Find the rule that gives a chosen dimension, or investigate cross-sections of the sponge.

Extending it for HL

Find fixed points and decide their stability with derivatives, or prove the limit you found numerically.

See a complete IA, marked

Our annotated exemplar How far can a stack of books lean over the edge of a table? (AA HL) asks a different question, but shows how a complete fractals & chaos exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).