Daring Diagonal Virtual Museum

The Daring Diagonal: The Right Angle

Gallery 1.8C

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“A prehistoric tomb found in the sands at Hieraconpolis,
is already right angled.”

“Very early the square became an emblem of truth,
justice and righteousness,
and so it remains to this day,
though uncountable ages have passed.”

IN THIS CHAPTER, I focus on the right angle: how it has dominated architecture and design for so many centuries, why it makes sense, why it doesn’t always make sense, why it is rooted in the biology of our very beings and even in our language, and why it finally lost its dominant grip on world culture. It took courageous individuals to break the mold and do things that were not only unconventional but also shunned.

Throughout history and for multiple reasons, the right angle made great sense in the construction of buildings. It derived directly from a response to the force of gravity, from the need for stable structures, from the ease and efficiency of cutting and joining structural members at right angles, and from the economy and functionality of the spaces constructed. In ancient Egypt it was also useful in setting out plots of land, calculating taxes on those plots, and reestablishing legal borders after a flood. On a more abstract level, the qualities of order, rationality, uniformity and reliability were inherent in the employment of the right angle. On a psychological level, an intrinsic quality that one might call spiritual, divine, or perhaps simply good was associated with the practical ease of using the 90-degree right angle in constructing buildings and creating cities.


Figure 2.1 Cape Cod Style House

Although most of the earliest temporary structures, such as tents, tepees, and yurts were circular, the construction of permanent buildings throughout most of recorded history was dominated by the right angle. Because the right angle fixed the shape of countless individual rooms, it also determined the massing (overall external shaping) of innumerable buildings, forcing them into simple shoebox-shaped volumes or combinations thereof (Fig. 2.1). This geometry dominated the floor plans of most buildings constructed prior to the 20th century, particularly grand buildings such as the three-story Elizabethan manor house in Somerset, England called Longleat (Fig. 2.2). If you overlay the plan of a typical building built before the 20th century with a transparent film marked with graph-paper-like grid lines, it is nearly certain that almost all walls of the buildings will align with the lines on the grid. This geometry of right angles is called orthogonality.

Figure 2.2  Plan of a Longleat House, near Somerset, England. 1580

Moreover, throughout history, multiple buildings in a single complex have often been laid out in an orthogonal pattern like the black tiles in the board game dominoes. The Forbidden City in Beijing (Fig. 2.3) is an excellent example of this arrangement. Not only are the footprints of individual buildings based on right angles, but the underlying orthogonal format also extends to the site plan of the complex; the exterior walls (and perhaps all internal partitions) of every building are set parallel to or perpendicular to the walls of other buildings.

It is almost as if vertical walls and square corners were predetermined by an immutable biological imperative lodged in the brains of the workers who dressed the stones, cut the timbers, and more recently, erected the steel beams. Surprisingly, this poetic sounding notion is probably correct.

What justified this steadfast faith in the right angle during the past seventy centuries of building construction? What was it about the right angle that contributed to its prevalence and perhaps idolization in building construction and site planning? And what impact did that have on the emergence of Diagonality in the modern era?

Figure 2.3  Beijing’s Forbidden City, construction started in 1406


There is a simple yet pervasive fact of the natural world that can go a long way toward explaining why the right angle dominated design and construction for so many centuries and why the arrival of Diagonality was so long delayed. That fact is gravity.

Figure 2.4  Rushing stream, Plitvici Lakes, Bosnia. Photo: Joel Levinson

Gravity is a fundamental force of nature that acts throughout the universe. Briefly stated: Everything attracts everything else. According to physicists, gravity is the weakest of all forces. This means that the pull of another person or of a table, or even of a building, is negligible. But the pull of our whole, massive Earth is very noticeable indeed. It affects all terrestrial features such as tectonic plates, mountains, and rivers. Gravity holds the ocean in its seabed, leashes our atmosphere to the earth’s crust, carries rainwater down from mountain tops (Fig. 2.4), and carries loose rocks from high ground to valleys. It also affects the growth of plants and the movement of animals. It forces most plants and trees to grow in a vertical direction (Fig. 2.5). Indeed, its overall significance for us humans and for what we design, and build is that it defines vertical and, by extension, horizontal. It tells us what is up or down or sideways, what is steep or gentle or flat.


Children learn to stand and, after wobbling  a bit,  to walk. We adults know that we can lean a little but that if we lean more, we topple over. Our response to gravity is etched into the very biological “marrow” of our beings. We have an ingrained sense of what is upright when we stand and we unconsciously sense what is truly vertical or truly horizontal. From this sense, we intuitively grasp perpendicularity. A newborn will look into the face of its parent and see an alignment of eyes that is perpendicular to the axis of the nose (Fig. 2.6). Thanks, no doubt, to the effect of gravity on evolution, we also have shoulders and hips aligned perpendicular to our spine (Fig. 2.7). All these experiences and alignments contribute to our sense of perpendicularity and our appreciation of the right angle.

One consequence of this pervasive perpendicularity in our very nature is that there are cells that exist along the route from eye to brain that respond only to energy pulses from vertical lines. Other cells respond only to horizontal lines, and still others, surprisingly, discern only lines that have an oblique orientation. Moreover, your inner ear helps to keep you balanced, sending warning signals when your head tilts. And in our dreams, what we know to be vertical and horizontal orientations with eyes open are pictured as such even when we are lying on our side. This suggests that our mind-made images of verticality in our dreams override our body-formed sense of what is upright—at least during sleep. The force of gravity has clearly generated in humans a deeply embedded sense of the horizontal and vertical that is an essential component of our consciousness.

The innate directionality that we all possess is related, in turn, to symmetry, a balancing of parts, that in general we find pleasing.  That internal sense of symmetry has found expression in art and in the geometric ordering of our buildings, whether the architectural symmetry made sense or not.

Sometimes the aesthetic appeal of symmetry overrides practicality. In a building with a central room or complex of rooms flanked by wings, the space needs in a wing on the right, for example, may not be the same as those of a wing on the left, yet the desire for “beauty,” expressed through symmetry and balance, will override functional concerns.

Figure 2.5  A commonplace stand of trees Photo: Joel Levinson

Figure 2.6  Perpendicularity in the human face.

Figure 2.7 Perpendicularity in the human body as illustrated in this drawing by Leonardo da Vinci.


“Foursquare” is a term applied to boxy, unadorned homes built in the early part of the 20th century, in a reaction to decorated, asymmetrical homes of the Victorian era. I adopt the term here to mean, more generally, the approach to construction that was dominant for eons: vertical walls, right-angled corners, and rectangular floor plans. Such construction surely resonated deep within builders on an unconscious level. This inner sense of perpendicularity became a powerful symbol of the hidden order of things—an order that literally, figuratively, and structurally proved useful and enduring.

To achieve safe and permanent structures, humans discovered long ago that walls must be built as close to plumb (vertical) as possible (Figure 2.8). A builder could sight by eye alone whether a wall “looked” vertical. But this method was not perfectly reliable so builders discovered many centuries ago that hanging a heavy, arrow-shaped object, point down, from a string could precisely establish a vertical line: gravity in action This arrow-shaped object came to be known as a plumb bob (Fig. 2.9) , gaining its name from the Latin word for the heavy element lead. Modern technology has provided no better way than the ancient plumb bob to establish the vertical direction.

Architects and builders also came to realize that walls that supported beams should be erected parallel to one another because it was more efficient to cut timbers and lengths of stone for the beams to a uniform length. This applied also to the end walls of a room, for if they were at something other than right angles to the supporting walls, the timbers at the end of the room had to be individually measured and cut to different lengths. Consequently, foursquare construction became the bedrock geometry of construction. A good example is the Temple of Chefren in Giza (Fig. 2.10).

Figure 2.11 Giza, Valley temple of Chefren

Another factor favoring rectangular rooms is that cutting acute angles in stone is risky and unpredictable. To cut the end of a stone beam at a 45-degree angle requires much more time, energy and care than cutting it at a 90-degree angle because so much more material must be sawed through in the angled cut, and breakage is possible at the sharp corner. Those sharp corners are also vulnerable to fracture by external assault or internal structural forces. Consequently, builders learned to avoid angle-cut beam-ends whenever possible (Figure 2.11)

Figure 2.8 A wall is usually considered unsafe if it leans to such an extent that a plumb line passing through its center of gravity does not fall inside the middle one-third of its base (called the V-3 rule).

Figure 2.9 Ancient Egyptian form of plumb bob.

Figure 2.11 A structural bay with uniform framing members producing a space with right angles.


Among the earliest examples of right-angle construction are the “fortifications and partly excavated rectangular houses at Jericho that can be dated back to pre-pottery ages, and, according to carbon 14 testing, to between the middle of the sixth and the middle of the seventh millennium [BC].” [Gideon, 1964] P. 183. Further evidence of prehistoric right-angle construction are the postholes and trenches of the long timber houses, circa 5000 BC, in the area of Sittard, in the Netherlands.

More “recent” examples of complexes laid out according to an orthogonal grid are the Citadel of Sargon erected in 742 BC (Fig. 2.12), and Persepolis, built in 518 BC in what is now Iran.

Figure 2.14 Indian house plan based on Vatsu Shastra tradition.

In modern India, houses laid out according to the tradition of Vatsu Shastra follow an orthogonal plan based on 81 squares, nine to a side (Fig. 2.13). The center of the house, a courtyard consisting of nine squares, is likened to human lungs, and is therefore left open to the sky (Fig. 2.14).

Figure 2.12 Citadel of Sargon.

Figure 2.15 A Vatsu Purusha Mandal

Fig. 2.17  An angled street corner in Paris. Photo: Joel Levinson


Although the large majority of buildings constructed throughout history have orthogonal plans, angled walls have also appeared—largely by necessity, not choice.

Builders have been forced on occasion to shape a building to accommodate site conditions such as a winding road or another building or wall set at an angle. The Fortress or Uronarti (Figs. 2.15 A&B), built about 2,000 BC on an island in the Nile in what is now Sudan, illustrates how the overriding orthogonal geometry of a complex can be “forced” to include triangular rooms—the result of site conditions, not design intent. This 500-foot long fortress, which served as a prototype for most fortresses built in Europe and Asia during the next 3,000 years, reveals an orthogonal interior geometry that simply had to be “sliced off” at an angle along the massive north wall. Although this results in some angle-cut rooms, it does not influence the geometry of the remaining spaces, which remain true to the orthogonal grid. This building, despite its shape, does not qualify as an example of Diagonality.

Villages or urban settlements are understandably less constrained by orthogonal thinking than are individual buildings. With rare exceptions, human settlements evolved through the accretion of individual buildings rather than by a plan drawn in advance. The accretion process is likely to occur haphazardly, or what is sometimes called organically. The typical medieval city might have originated at a spot along a river with flat banks, or on a hill with a grand view, or with defensible terrain, or at a particular dwelling where food or wares were sold or bartered. People were attracted to the site and arrived along routes that were scenic or safe or easy to navigate. The footpaths became well-worn trails that wound around boulders and followed the contours of the land. Structures were erected along these winding trails, and over time these structures formed a village. Individual buildings, for the sake of efficiency and/or order, were composed of a combination of rectangles. However, their setting may well have been irregular, that is to say not orthogonal. The village or settlement reflects accommodation, not design intent. Paris has many examples, including the angled corner shown in Fig. 2.16.

Figure 2.15 A Fortress of Uronarti

Figure 2.15 B Fortress of Uronarti

Fig. 2.16  An angled street corner in Paris. Photo: Joel Levinson


The reliance upon the right angle by architects and builders through the ages may well be attributed to the right angle’s inextricable association with two important factors: one physical (gravity), the other abstract (symmetry). From symmetry it’s a short step to the notion of beauty, and another short step to the more abstract notion of good.

I don’t know if in languages other than English (or in prehistoric tongues) the “right” in “right angle” carries meanings other than 90 degrees or perpendicular—meanings such as correct, proper, virtuous, appropriate, lovely. Nor do I know whether, across the globe and across history, “square” may carry meanings other than a four-sided regular polygon—meanings such as honest, fair, ethical, unbiased, just (or naïve, darkish). Yet given the breadth of meanings of these terms in English, it is tempting to speculate that lodged in the brains of architects and builders throughout history has been the idea that orthogonal construction is good, proper, even beautiful.

Extending this thought, may we not also surmise that “oblique” was viewed, perhaps subconsciously, as more than just slanting or at an angle—perhaps wrong, bad, questionable, devious.

Below is a selection of quotations that convey the power of language associated with the right angle to both reflect and shape human activity. In the 16th century, “square” sometimes denoted dispute, wrangling, discord, or dissension. Again in the 20th century, it sometimes had negative connotations, as when a person who was not “hip” or was socially awkward was called square. But throughout recent centuries, “square” mostly had positive meanings, as the quotations below illustrate. Some 20th-century quotations appear in later chapters.


1549 E. Allen Par. Leo Jude Rev. 9. As the Christen religion shall be restored and reformed after the rule and square of holy scripture.
1549 L. Coxe Erasm. Par. Titus 28. Teaching shamefull thinges and far square from the veritie of the gospell.
1555 Eden Decades 346. Wherin he speaketh not greatly owt of square.
1556 Olde Antichrist 25 b. How great (and how out of square) this errour of ye world is.
1567 Jewel Def. Apol. v. 556. Syluester Prierias saith, that the Romishe Church is the Squier, and Rule of Truthe.
1579 W. Wilkinson Confut. Fam. Love B ij, J. Judge all thinges according to the ballance of equitye, and trying squaire or measure line of righteousness.
1589 Puttenham Eng. Poesie ii. xi[i]. (Arb.) 113. [Aristotle] termeth a constant minded man..a square man.
1594 West 2nd Pt. Symbol., Chancerie Sect.23. Lawes appointed to be rules and squares of mens actions.
1596 Spenser F.Q. v. Introd. i. Me seemes the world is runne quite out of square.


1604 T. Wright Passions i. iii. 13. To governe the body..by the square of prudence, and rule of reason.
1622 Peacham Compl. Gent. v. (1906) 39 The least disorder or ranknesse of any one flower, putteth a beautifull bed or well contrived knot out of square.
1632 Lithgow Trav. ix. 415. Fit to gouerne others, and to direct him selfe with the square rules of wisdome and iudgement.
1650 Howell Giraffi’s Rev. Naples i. 50 Had not a secret Treaty..against Masaniello, and his followers, bin discovered, which put all things again out of square.
1654 Whitlock Zootomia 24. Let thy Actions be justified by the Square of Religion and Justice.
1682 Penn in Dixon Life xxiii. (1872) 207. Keep upon the square, for God sees you.
1684 Leighton Wks. (1868) 675. Framing them to an external and square carriage whereby the world..is much advantaged.
1688 Bunyan Jerusalem Sinner Saved (1886) 75. Upon the square, as I may call it, of the worthiness of the blood of Christ, grace acts.


1710 S. Sewall Diary 3 Apr. I did not think him so square and stable a man.
1713 Steele Guard. No. 20 p.2. The oblique glance with which hatred doth always see things.
1716 Blackall Wks. (1723) I. 165. When he sees that those Christians with whom he trades, are not..so square and honest in their Dealings.
1718 Freethinker No. 135. In an Age, wherein it is almost become the Glory of States to circumvent each other, who does not see the Necessity of playing upon the Square?
1728 Chambers Cycl. s.v. Geometry. They observ’d, that God and Nature affect Perpendiculars, Parallels, Circles, Triangles, Squares, and harmonical Proportions.
1756 Burke Subl. & B; Wks. I. 238. Perfectly beautiful bodies are not composed of angular parts.
1770 Jortin Serm. (1771) I. vii. 128. There are persons to be found…who grow rich and great…by various oblique and scandalous ways.
1781 Cowper Charity 559All disguises shall be rent away That square not truly with the scripture plan.
1791 Washington Lett. Writ. 1892 XII. 90. It is of great importance..that the city should be laid out into squares and lots.


1811 Lexicon-Balatronicum. Square, honest, not roguish.
1821 Lamb Elia i. Old & New Schoolm. He is awkward, and out of place, in the society of his equals… He cannot meet you on the square.
1846 Trench Mirac. 255. Awaiting the great day when all things shall be set on the square.
1852 Mrs. Stowe Uncle Tom’s C. i. 2. I’ve trusted him, since then, with everything I have-money, house, horses,-..and I always found him true and square in everything.
1852 Bailey Festus 493. Peace, piety, and innocence, and joy Made up the square of Being.
1853 Dickens Bleak Ho. Xx. I had confident expectations that things would come round and be all square.
1867 Trollope Chron. Barset I. xxxvii. 326. He’s only going to give me my little bit of money..and then he and I will be all square.
1870 Dickens E. Drood 62. As a particularly angular man, I do not fit smoothly into the social circle.
1873 Hale In His Name vi. 57. [He] could not answer the square question put to him.
1878 Green Coal ii. 55. Nearly all the grains of quartz are angular.
1880 McCarthy Own Time IV. I. 61. His gestures were angular and ungraceful.
1884 J. Parker Apost. Life II. 153. This is a square Gospel; it will have all things at right angles. d. Straight, direct. This is a square Gospel; it will have all things at right angles.
1884 Harper’s Mag. June 56/2. Reputation for integrity and square dealing.

Is it any wonder that it took so many centuries for the square’s grip on the global cultural consciousness to be loosened? Because of the roles of gravity, human perception, habit—and even the Church—anything that was out of square was, for a long time, unacceptable. Finally, at the end of the 19th century, a multitude of forces united to bring change. The Daring Diagonal finally came to be and now we see if everywhere. Look around. It has become the signature geometry of the modern era.