Hidden geometry

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2D-3D | line > plane

hidden geo

Contents

*William Blake, The Ancient of Days, 1794*

Hidden geometry

As you are now familiar with coordinate systems, structural geometries, and orthographic projection systems through your readings in parallelUniverses, let’s expand on those systems by seeing how they are expressed in a drawing. The samples below run across history, and are visual descriptions of such hidden geometries as the datum, various axes of symmetry, orthographic drawing, and systems for drawing irregular, curving forms.

Expressing a datum

The Axis Mundi we learned about above is the first example of a fanciful, metaphorical datum. A datum is an arbitrary but conventionally agreed-upon point of reference, something used as the basis for measure or calculation. Among many others, some historical expressions of a datum include:

*The Axis Mundi expressed as the **World Tree**: Oluf Olufsen Bagge, **Yggdrasil, the World Tree** of Norse mythology, from Northern Antiquities, an English translation of the Prose Edda, 1847.*

**Sea level** expressing height elevation on Earth: **USGS Bench Mark**, installed at Battery Point Light, Crescent City, California, 1915.

The **Equator** and **Poles** expressing division of hemispheres and the axis of rotation for Earth: Oronce Fine, **Mappemonde en forme de cœur montrant la Terre australe. Recens et integra orbis descriptio**, Paris, 1536.

*The center line of a human form expressing biological mirror symmetry: Heinrich Cornelius Agrippa, illustration from Libri tres de occulta philosophia, 1533.*

The water line expressing the line of displacement of water by a vessel: Robert Fulton drawings for **USS Demilogos**, the first steam-powered warship, later renamed after its designer to USS Fulton, 1813.

Expressing axis of symmetry

One datum in particular that is often hidden but important to express is the axis of symmetry. The object you choose to model will likely contain an axis of symmetry or near-symmetry.

Bilateral symmetry

As we see da Vinci’s drawing of the Vitruvian Man above, humans along with many other living beings are biologically organized along an axis of bilateral symmetry. Many machines and architectural constructions also harbor bilateral axes of symmetry.

*T. de Bry, line engraving frontispiece featuring symmetrical human similar to Agrippa and da Vinci, **Utriusque cosmi maioris scilicet et minoris …** history by Robert Fludd, 1617.*

*Albrecht Dürer, **Studies on the Proportions of the Female Body**, 1528.*

Albrecht Dürer, **Anatomical and Geometrical Proportions**, ca 1528.

*Henry Ford, front view of **Ford Model T**, 1928.*

*Andrea Palladio, **Villa Capra**, also known as **La Rotonda**, 1570.*

Radial symmetry

In the Villa Rotonda facade we see bilateral symmetry. In its floor plan, we see axes of symmetry in multiple directions, creating the condition of radial symmetry. Many types exist:

Andrea Palladio, plan showing 4-fold symmetry for **La Rotonda**, 1570.

*Walter Coleman, illustration for starfish showing both bilateral symmetry and 5-fold radial symmetry, for **Beginner’s Zoology**, 1922.*

*6-fold radial symmetry in William “Snowflake” Bentley’s **Snow Flake No. 20**, ca. late 19th or early 20th century.*

*16-fold symmetry in the domed ceiling of Persian poet **Hafez’s Tomb**, 1935.*

Self-similarity: symmetry in scale

When a part is similar to its whole, such an object exhibits self-similarity. It is a scalar complement to symmetry.

**Diagram** by architect Aldo van Eyck, stating “tree is leaf and leaf is tree – house is city and city is house,” 1962.

**Koch snowflake fractal**, work by Wikimedia contributor Leofun01.

**Barnsley fern** — not a real fern, but an expression of a fractal algorithm by Wikimedia contributor Surt91.

“Near” symmetry

Certain objects display a tendency toward symmetry. We may even recognize an axis, although it (or the object controlled by it) exhibits distortion.

*Hannah Cohoon, **Tree of Life**, 1845, displays more symmetry as a drawing than an actual tree, but see if you can spot the anomalies.*

*A forensic study of a **shoe print** with an S-shape axial distortion.*

**An x-ray of a hand** exhibits a symmetry of number, but not of form.

Expressing coordinate systems

Orthographic and 3-view drawings

The orthographic 3-view drawing remains an important tool to visualize coordinate systems, even in a post-mechanical-drafting workflow. We’ll drill much deeper into orthographic projection in another chapter.

As a practical tool, it verifies that information seen in one coordinate view correlates with information seen in others, guaranteeing the precision of data.

*Scuderia Ferrari **Bimotore** racing car, 1935. What is the **datum** or **data** in this drawing? Can you relate these drawings to one another? These are in a proper 3-view relationship.*

J. D. Carrick or F. Yeoman, **SPAD S.7**, illustration in **Fighter Aircraft of the 1914-1918 War** compiled by W. M. Lamberton, published by Harleyford Publications Ltd., 1960. What is the **datum** or **data**? Can you find **centerlines**? **Sections**? Only two drawings are in the correct 3-view relationship — which ones?

*3 | Peter Kaboldy, **Mitsubishi A6M2 Zero** WWII fighter plane. This is a classic, correct **3-view**.*

*Albrecht Dürer, **Orthographic projection of a Foot**, 1528. Another classically correct **3-view**, including a pair of **sections**.*

Section drawings

If orthographic drawings show us the surface, sections oriented to particular coordinate planes will reveal the interior of the object: the topological division between the mass of the object and the void that surrounds it. Let’s explore Dürer’s head analysis visualizing formal slices, along with other examples.

*Albrecht Dürer, **Sections through the human head**, ca. 1520s. Showing the relationship between **orthogonal drawing** and spatial form.*

*Anonymous student, **cross contour drawing**, date unknown. **Cross contour** is a kind of analytical drawing that suggests multiple slicing planes through a mass.*

Anonymous student, **stacked-slice model of a shoe** using laser-cut corrugated cardboard sectioning. Where the drawing at 2 is sectioned horizontally, this model is sectioned at a 30˚ tilt. From a studio by the author.

**Pantheon, Rome**, cross-section. Wilhelm Lübke, Max Semrau: **Outlines of the History of Art**. Paul Neff Verlag, Esslingen, 14th edition, 1908. The section in gray contains a rich rendering of the interior **elevation**.

Frederick Newton Willson, **Cross-Section of Standard Rail**, one hundred pounds to the yard, as used on the Pennsylvania Railroad, 1898. The material is extruded, much like an extrusion operation can create form in a model.

Generating irregular form

Topographic lines

A topographic map uses a stack of horizontal sections: a regular division of elevation to describe the height of land above sea level. Think of the topographic levels as slicing planes parallel to the plane of sea level. This modulates an irregular surface, making it easy (ok, easier) to describe. Here are three topographic descriptions of the island of Hawaii for comparison, the bottom one interactive.

Animated SVG: drag or tap to turn

Boat lofting

Like topographic lines, the drafting technique of lofting can generate complex curving forms. Lofting is used in boat building to describe the streamlined shape of the hull. Usually, a profile was divided into 10 or so equal parts, creating a series of parallel sections that describe the shape of the hull. Because the boat is bilaterally symmetrical from the front coordinate orientation, you can see how the drawing is “halved” to describe the rear, then the front, of the boat. The deck plan is also a half-drawing, cut at the centerline as a datum.