Three facets, three solids, one family: Skwedge.

The three Skwedge species: curvum, convexum, and projectivum

A skwedge is a solid with a circle on one end and a straight line on the other.

Skwedge curvum
Skwedge convexum
Skwedge projectivum
The Shape

The Wonderful World of Skwedges

Pick up a standard tube of toothpaste. On one end is the screw-off cap, the other is where the tube becomes a line. The shape where the tube becomes a line is a skwedge.

You have seen or held this approximate shape multiple times without thinking about it.

Toothpaste tube Toothpaste tube
Hair dryer concentrator Hair dryer concentrator
Otter Pop Otter Pop
Drill bit Drill bit
Pastry tip Pastry tip

Over the centuries, different people gave different names to these similar shapes. Guarino Guarini wrote about "cones terminating in a line segment" in 1671. John Wallis in 1684 called his version the conocuneus, meaning "cone-wedge." In 1790, Peter Friedrich Catel created a puzzle that challenged buyers to find one solid shape that could pass perfectly through a circle, triangle, and square hole. Martin Gardner in 1958 repeated the puzzle, calling his version the "cork plug." By 1994, computer vision had identified a broader construction, the "Visual Hull," within which the skwedge appears as a special case. The indigenous language Miluk, from the Oregon Coast, made the term ptsi·nɬ, combining its ancestral terms for three, psənɬ, and the word for wedge tool, tsi·nɬ, to represent a specific type of skwedge. Yet none of these varieties had ever been classified as a family of shapes.

In 2026, a set of companion papers gave these solids their mathematical foundation for their new skwedge home. From these founding skwedge papers, we know there are three variations to these solids, genuinely different from each other.

Why bother naming things?

Think about triangles. There are triangles where all three sides are the same length. Triangles with one square corner. Triangles where one angle opens up wide. People worked out names for all of these because once you name something well, you know things about it right away. An equilateral triangle? All angles are 60 degrees. You did not have to measure.

The skwedge works the same way. Once you have names for the three variations, you know things about each one automatically: which holds the most space, which casts a specific shadow, which is built from straight lines. The name is the beginning of the knowledge.

Read more: Why Taxonomy Matters →

The Three Variations

Same ends. Three different shapes.

Each version of the skwedge comes from a different way of filling the space between the round end and the flat edge. The circular bottom and the linear top don't change between shapes. It's what changes in the middle that makes them different.

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  Skwedge curvum

John Wallis first described this shape in a 1662 Latin letter to Sir Robert Moray, calling it the conocuneus. He named it by combining conus (cone, circular base) and cuneus (wedge, linear apex). The motivation was practical: a shipwright proposed the form as relevant to ship hull design.

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  Skwedge convexum

This is the version modern CAD software produces when you ask for a convex loft between a circle and a line. Engineers and industrial designers work with it constantly. See how the two flat sides make a triangle to the base.

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  Skwedge projectivum

Martin Gardner presented it as a puzzle in Scientific American in 1958: one shape that passes through a circular hole, a triangular hole, and a square hole. In Miluk, the word ptsi·nɬ describes a tool that casts three shadows. See how the two flat sides curve.

Build one yourself

The interactive Block Builder is made possible by the generous open source contribution of Plotz (plotz.co.uk), a beloved tool in the Minecraft® and Lego® communities for building geometric shapes block by block. We adapted their code to let you stack blocks into all three skwedge species.

Open Block Builder →

3D print your own

If you have a 3D printer, you can also download each of the skwedge shapes by clicking on the following STL files. STL files for all three species, released under CC BY-SA 4.0. Print, share, and modify freely with attribution.

Skwedge curvum Species I · the conocuneus Download STL
Skwedge convexum Species II Download STL
Skwedge projectivum Species III · the puzzle piece Download STL
The Name

From the Miluk ptsi·nɬ to the math of Skwedge

ptsi·nɬ

The Miluk language is a language of ancestry for the Coquille Indian Tribe, the Confederated Tribes of Coos, Lower Umpqua, and Siuslaw Indians, the Confederated Tribes of Siletz Indians, and the Confederated Tribes of Grand Ronde, among possibly elsewhere. The Miluk word ptsi·nɬ describes a specific kind of tool: hold it one way and its shadow is a triangle. Turn it and the shadow becomes a square. Turn it again: a circle.

The effort to translate ptsi·nɬ into precise mathematical language led to a simple question: how are these shapes classified using the terms of math? While these shapes had been made and studied for centuries, they remained unclassified as a family. By putting these shapes into a clear mathematical family, now there's an efficient way to say which shape one wants.

3D-printed skwedge projectivum in hand
The skwedge held in sunlight casting a triangular shadow on a white surface
The skwedge rotated to cast a circular shadow
The skwedge rotated to cast a square shadow

Here, the Coquille Indian Tribe Chief, Justin Futch, photographs his handmade ptsi·nɬ demonstrating its three shadows.

The Storybook

The Bear Becomes Ptsi·nɬ

A children's book written in Miluk first, following Miluk storytelling conventions, then translated into English. A bear wandering at night is named differently by each animal met along the way. Bear discovers more than one thing.

A page from The Bear Becomes Ptsi·nɬ

Read more and download free editions →

History

Two million years of solids resembling skwedges.

For millennia, people have been naming and making skwedge-like shapes. The skwedge shape turns up independently across every continent and every era of tool-making. That is not a coincidence. It is simply the right shape for splitting, shearing, and directing force. The physics keeps arriving at the same answer.

~1.75 million BCE
Acheulean hand axes appear across Africa, Europe, and Asia. The working edge tapers from a rounded base to a straight cutting edge. The same shape appears independently on every continent among groups with no contact with each other. It is not cultural borrowing. It is the right answer to the problem of cutting.
Since timeimmemorial
Northwest Coast peoples developed named sets of wedges for specific tasks: hollowing canoes, splitting planks, driving stakes. Over 1,160 wedge artifacts were recovered from the Ozette site alone (Makah territory, Washington State), preserved by a mudslide around 500 years ago. The form was not accidental. It was engineered, named, and passed down.
1671
Guarino Guarini, a Theatine priest and architect in Turin, computes the volume of "cones terminating in a line segment" in his geometry textbook. He is among the first to work out the math of this shape in print. His work and John Wallis's appear to have been independent of each other.
1684
John Wallis, the English mathematician who introduced the infinity symbol (∞), publishes his calculation of the conocuneus, Latin for "cone-wedge." He proves its volume is exactly (π/2) × R² × h. This is Skwedge curvum, and Wallis's result stands unchanged in the 2026 framework.
1790
Wilhelm Catel's puzzle book in Germany includes the earliest known printed version of the circle-square-triangle silhouette puzzle: can you make one shape that fits all three holes? This is the puzzle that Skwedge projectivum solves. The question was in circulation 168 years before Gardner gave it wide attention.
1958
Martin Gardner presents the "cork plug" in Scientific American: a shape that fits a round hole, a square hole, and a triangular hole. He describes two versions which our 2026 taxonomy would call the Skwedge curvum, the smaller one, and the Skwedge projectivum, the biggest version.
1979
Douglas Hofstadter publishes Gödel, Escher, Bach, and on the cover are the letters G, E, and B casting three shadows from a single carved object. The idea that one object carries multiple identities depending on the angle of the light is a central theme of the book. It is very skwedge-like.
1994
Computer vision researchers independently arrive at constructing the "Visual Hull," using a method for reconstructing a 3D object from its shadows. Resulting shapes can often be skwedges. Mathematicians, puzzle designers, and computer scientists all found the same shape by completely different paths.
1986 – 2026
The Coquille Tribe began revitalization efforts in earnest even during its termination from federal recognition. For four decades, work on Miluk has led to its own font, curricula, and classes across tribal boundaries. This work translating ptsi·nɬ into the terms of mathematics led to this indigenous representation in mathematics for the whole family of related shapes formerly not unified. Companion papers introducing the skwedge and the Miluk term hewel are published in February 2026.
Seven independent discoveries across four centuries, multiple disciplines, and every populated continent. The taxonomy that brings them under one roof was published in 2026.

For the full mathematical treatment, including proofs, formal classification, and the open problem of the fourth species variation, visit The Science. The companion papers are freely available on Zenodo under CC BY-SA 4.0.