Why Draw It

Drawn Mathematics teaches mathematics by construction: every figure starts from a stated rule, and every step can be redone on paper.

This is a teaching site, not a library of math clip art or stock drawings. The diagrams are here to be redrawn, measured and checked, and each one comes with the rule and the numbers that produced it.

What you will find here

The tagline is Mathematics you can see, draw, and fold. Five categories carry it:

• Ornamental Symmetry: wallpaper groups, kaleidoscopes and star patterns, each drawn from its symmetry rules. Patterns are described by their geometry, and any claim about where a pattern came from carries a citation.
• Fractal Construction: classic fractals built step by step from their rules, how their dimension is measured, and where nature comes close to them, over a measured range of scales only.
• Crease Patterns: the fold in the tagline. How to read an origami crease pattern, the theorems of flat folding, facts you can fold from a single sheet, and the structure of traditional models.
• Perspective Drawing: linear perspective and anamorphosis drawn step by step, with vanishing-point overlays on public-domain paintings that state the assumptions behind every line.
• Visual Puzzles: dissection puzzles, tangrams and visual paradoxes such as the missing square, each drawn to scale, with every answer checked by exhaustive search or a proof.

Near the top of every article, a line marked Rule: states the rule behind its figures in one sentence. Every article closes with Draw it yourself:, three to five steps you can repeat with paper and a few simple tools.

Who it is for

The main readers arrive with a how-to question. They are people teaching themselves to draw, origami folders, puzzle solvers and readers of popular mathematics who would rather make the thing than read about it.

Math and art teachers come for classroom material. Designers and crafters, including quilters, embroiderers and tattoo artists, come for patterns they can generate from a rule and diagrams they can credit and reuse.

Some readers will not find what they want here. The site does not rate or recommend origami paper, drawing tools or apps. It does not coach for exams or contests, and it does not supply homework answers. It also does not derive group notation for researchers: symmetry and fractals start from pictures, and notation appears only as a label.

What the site stands for

A mathematical fact is finished only when it comes with a drawing that a reader can understand and check step by step. That is a position about how this site chooses and writes its topics, not a mathematical claim.

The drawing is the way in; it is not the proof. A finite drawing, or a sheet folded on a table, is a demonstration. A general statement needs its assumptions and a proof, or a pointer to where the proof was published.

Three positions follow from that stance:

• A good drawing beats a page of algebra. If a claim cannot be drawn in a way that convinces, the site would rather leave it out.
• Popular mathematics that gives no construction steps is sightseeing. An article that leaves you impressed but with nothing to do counts as a failure here.
• Where a famous painting departs from a single vanishing point, the departure deserves to be laid out with overlay lines, not covered for. How far the lines miss is geometry; why they miss is interpretation.

Each diagram can be read as the rule, caught in the act: what is left on the page after the rule has run. That is a figure of speech. A diagram carries reasoning you can recompute, but it is not the proof of a general statement.

How the facts are checked

The full method is on Checking the construction. In short:

• Statements of fact are compared with original publications and standard references, and anything that can be calculated is calculated again.
• Every geometric diagram is generated directly from its mathematical definition or coordinates, and its caption gives the parameters, so you can redraw it.
• Every puzzle answer is checked by exhaustive search or by a proof before it appears, and the article explains how the check works.
• Crease patterns are checked by computation for the local flat-folding conditions of Maekawa and Kawasaki at every interior vertex. That check is necessary for a sheet to fold flat but does not prove that it will, and each pattern is labelled with the level of checking it has actually reached. A folding-simulator test on 2026-10-26 found that a simulator settles even patterns that cannot fold flat into a nearly flat shape, so any simulator render on this site will be labelled as an approximation, not proof.
• Scene illustrations are marked one by one as "Generated scene illustration — not a photograph, historical image or researched reconstruction", and no geometric point rests on one.

Found a mistake?

Write to contact@drawnmathematics.com, the address at the foot of every page. Give the page address and the sentence or figure, say what is wrong, and add a counterexample or a reference if you have one.

A correction is dated and recorded at the end of the article and on Corrections and redraws, together with the original wording. A mathematical error also means the check behind it is repeated and the diagram is redrawn. What to send, and what the address does not handle, is on the contact page.

Reusing the drawings

The site's own diagrams and CSV files are released under the Creative Commons Attribution 4.0 license (CC BY 4.0). You may copy, adapt and republish them as long as you credit Drawn Mathematics. Paintings, plates and photographs from other sources keep their own licenses, and each one is credited in full under the image. Scene illustrations have separate use terms. The exact credit lines are on Borrow these drawings.

Every article carries a Cite this page block giving the site name, the title, the publication date, the date of the last substantial revision and the page address. The site does not issue DOIs.

Three terms used here

These three phrases are the site's own coinages, not standard mathematical terms:

• the seventeen ceiling: the limit of exactly seventeen symmetry types for flat patterns that repeat in two independent directions, a ceiling that pattern-makers in every culture work under (strips that repeat along one line have their own list of seven, and patterns that never repeat sit outside both lists).
• the vanishing audit: computing where a painting's receding lines meet and laying those lines back over the painting, to test its perspective against a stated assumption. Its first full run is Vermeer's Milkmaid: 11 Traced Edges Aim Just Above Her Hand.
• paper theorem: a section for mathematical facts you can construct by folding one sheet of paper, where a demonstration comes first and the statement with its proof, or a pointer to the proof, comes second. The section opens with A4 Paper Size: Built So Each Half Keeps the Same Shape.