III-1. Abstract geometry and the architecture of death

Monuments to death

    The Egyptian pyramids have an extremely primitive geometrical shape of a square pyramid. Why did the tombstones to the kings of this ancient kingdom have to be such pure shapes? This is one of the mysteries. In any case, there is a history of development from the step-shaped pyramids, stacked with so called mastabas, towards even more pure shapes, and the ancient Egyptians exhibited a tendency to return towards primitive shapes. Thanks to the stone construction that gives such as stern impression, these have continued to survive a lengthy history and continue to exist today.

    Ancient tombs in Japan also employed square tombs, round tombs, or a combination of the two geometrical shapes with a square front and round back, where the shapes of the plans are the most primitive geometrical shapes. It can be thought that there is some fundamental link between gravestones and geometrical shapes regardless of country. But it is not possible to explain clearly the fact that ancient people represented pure geometrical forms by pure shapes because pure geometrical planes and shapes are extremely fundamental shapes that anyone can imagine from the start.

    Aldo Rossi designed the Monument to the Partisans (1965) in Segrate, using a combination of cylinders, triangular prisms, and cuboids (Figs. 27 and 28). The cylinders and triangular prisms in particular are intensely self-affirming. Although they are simple, these geometrical forms strike at the human heart. In fact, it could be said that it is because of their extreme purity that they evoke a strong monumentality. The 16 steps in the large cuboid rear scarcely produce an anticipation of a building in relation to the dimensions of the human body, but apart from such fact, the monument does not create any impressions of any scale and is a completely abstract form.

    The fact that a simple geometrical form was given for this kind of monument linked to death through the resistance and deaths of the partisans should draw attention, compared to the pyramids, etc. Rossi also drew the plans that combine several simple geometrical shapes including cone shapes, cubes, with squares, triangles, and rectangles (Fig. 29), and also here the simple geometrical forms created a cemetery scene with an extremely deep impression. This is also a building for dead people, and such a simple form also multiplies the evocation of the image of death.

    Of course, even though we say an image of death, these form a variety of manifestations, depending on each age and each societies view of life and death. For example, in Buddhism where there is a desire for Gokuraku-Joudo (Paradise), it is disputable as to whether these kinds of simple geometrical forms could be monuments. This kind of a simple geometrical form would evidently be viewed as appropriate for the world of ancient gods. For the kind of geometrical forms that do not in any way evoke any kind of human or living biological form, it is as if they represented the attitude of a harsh rejection of the natural world of living things. These can be thought of as laying in the most primitive level of consciousness in the human mind, and represent forms that leave a mark on a place as if to mark one’s own territory, which could be said to be the most ancient intellectual activity in the history of humankind.

    Among the natural scenery above the earth being filled with mountains and valleys, and overgrowing with vegetation, these kinds of simple yet stern forms clearly stand out as alien forms. This is vaguely shown even in the perspective drawing of Rossi. Within this picture, a monument is placed on the right side; a horizontal wall stretches out opposite this; in front are three workpieces that are abstractions of the bottom ends of circular pillars; left of this are outdoor steps that emphasize the perspective effect; on the left edge are relatively complex undulations in the surface of the ground; and several large trees are visible. In this scenery composed of an artificial structure in association with ancient ruins and an untouched landscape of natural trees of earth, actually a theme of contrast between abstract geometry and natural organic forms was employed. This effect appears as if diminished because the monument was constructed within a high-rise residential area that bears almost no resemblance to the scenery of this perspective drawing, but the designer obviously intended to tell something in the depth of our consciousness through this kind of an illusory perspective drawing.

    When you watch a single triangular pyramid among a solid truss structure being constructed from a combination of a lot of regular tetrahedrons, it does not evoke a strong visual impression. This is because among a sequence of identical geometrical forms, a single shape is easily accepted into the group. But among the natural scenery that presents complex and diverse shapes, the pure geometrical shape is sharply and aggressively antagonistic, creating a special spot in the landscape. The perspective view of Rossi is worth attention in terms of splendidly representing this point. One can’t help feeling sympathy for the time and mental effort expended by the person who drew this endlessly complex figures of the trees in contrast to the simple workpieces designed that did not need much work to draw, although labor may have been saved by the drawing technique of collage. The unbalance that more effort were put into the incidental details of the picture over the actual thing designed, does not evoke any feeling of unbalance to the eyes in this perspective drawing.

     This perspective view indicates something that has floated up through the deep layers of the mind of the designer. The scenery that gives a spark of life to the surroundings of the monument do not necessarily have to be actual trees, but could as easily be a place where the scent of human activity hangs like a corner of a residential block. The geometrical monument fractures the indefinite amorphous space within an unfocused wavering space and gives a nervous tension to the space. It is as if, in that moment, the wind slides across and away from the smooth surfaces of geometrical forms, and the space, where this feeling of life floats, loses the feeling of vitality in this point. The monument should not be interpreted as a geometrical form placed within a space that had no feeling of life from the beginning on the drawing board, but should be interpreted as a geometrical form placed within the feeling of vitality of nature in order to calm the surging surrounding space.

    Perhaps we should say that one of the reasons that the pyramids had to have spectacular square pyramids was because it gave a still place to this space above the earth for the dead. Even among the desert scenery of seemingly quiet Egypt, where the line of the horizon extends in a straight line, the large winding curves of the Nile that also sometimes changed created the surrounding scenery. And we are forced to admit the need for giant solemn geometric forms in sharp competition with this.

    The construction materials used here such as stone and concrete have an inorganic quality, which conflict with the organic nature of plants and animals. Although the awareness of this is rare, there are many cases where non-living organisms are used in the buildings (although in cases where wood is used, and particularly when used by Japanese, these may not appear as completely non-living organisms). If this global ecosystem is thought of as having a single life (or a pseudo-life), then humans are actually scattering a part of the natural order of life, while taking the materials to supply for their own uses and build structures. As construction materials, these materials are from the part of the global ecosystem, where the feeling of vitality is most rare. And also the geometrical forms that are constructed create a renewed systematic order of death (non living) within the global ecosystem.

    The square pyramids of the Egyptian pyramids create a visual statement on the vertical axis connecting the earth and the heavens. The idea that humans expressed an outlook on the universe through this came from the philosopher Hegel. This can be said to be the world of semantics represented by the architecture of death, and the simple forms are a media for expressing lucid concepts. There is quite a difference between this kind of idea and the idea of geometrical forms that are cut sternly form the surrounding space. In order to represent the vertical axis connecting heaven and earth, a mountain-shaped object is not necessarily suitable, and there is no reason why it necessarily has to be a crystalline solid. The difference between the pyramids and graves made from mounds of earth is that the abstract forms themselves already contain death, or at the very least, a form that rejects the phenomenon of nature and the phenomenon of life on earth.

    These kinds of geometrical forms that already have an intrinsic image of death came to be employed in general buildings only after buildings for the dead were conceived and created. In the history of architecture, buildings for the dead or the god also came to be used in general buildings much in the same way as, for example, sacred temple buildings were often used as models for residential buildings (for example, we can think of the Palladio and so on where the shape of a temple facade was frequently used in rural housing). For example, in the step-shaped pyramid placed on the apex of the Japanese Diet building, although this is not necessarily seen as having a strong meaning, it is actually a shape motif that evokes the ancient Egyptians and forces a return back to the buildings for the dead. Typically, even if there is in fact only the light feeling of using simple geometrical forms within our subconsciousness, the world today unexpectedly follows the images of the buildings for the dead.

    The skyscrapers in New York are buildings with eclectic forms, where the detailed areas are decorated with a variety of historical patterns. When looked at as a whole, they resemble the gravestones lined up in a cemetery and are mainly formed from three sections of a foundation, a square pillar, and a spire-shaped apex. We may consider that there is a process of several stages of the transformation of the meaning that intervenes between the gravestones and skyscrapers, but the shapes of the silhouettes of both are eerily similar. In general, many high-rise office buildings, including those constructed in the so-called “tombstone style” that are formed from a parallelepiped-shaped tower and a foundation, certainly fulfill some kind of role as having the shape of a gravestone within the deep levels of psyche of constructing the building in the first place. If we consider that the shapes of the most refined gravestones prefer abstract geometries (for example, Lenin’s Mausoleum in “Red Square” in Moscow is one type of example), then this will make clear the large meaning originally held by abstract geometry.

Representations of eternity

    In the two movies “2001: A Space Odyssey” (directed by S. Kubrick) and “2010” (directed by P. Hyams), which were both made into films based on the original novels by Arthur C. Clarke, a parallelepiped shaped slab called “monolith” that resembles a beautiful black steel plate plays an important role (Fig. 30). This mysterious slab, which is sometimes floating in space and sometimes buried in the ground, and continues to watch over the activities of human kind from the time of anthropoids through to the space age of the future, evokes the anticipation of playing some kind of role in the world of living creatures. But it is not clearly explained what kind of material it is made from or what kind of powers it possesses. This is literally a black box, and it seems that something is hidden within, but the external appearance is nothing more than a completely unresponsive parallelepiped.

    I dare say, the author established this as a catalyst and information source for the development of space and evolution of life that could be called the genome of both the global ecosystem and the entirety of space. However, at a glance it is reminiscent of an inorganic steel plate, and leaves an impression of the purest geometrical solid that appears completely unrelated from the phenomenon of life. The contrast between the phenomenon of life and non-living forms could also be said to be a technique for expressing a clear contrast. But it simultaneously offers a glimpse into the intentions of Clarke to remove the wall that divides living organisms from non-living things. In other words, it is something that exposes the profound morphology of life and death through the rhetoric of a simple film expression.

    Of course, because this parallelepiped was conceived by humans, it consequently is formed from this kind of a pure shape (the ratio of the sides are 1 : 4 : 9). If it was some genome that had originated in the evolution of the earth, it would not actually have taken this shape. Furthermore, from the viewpoint of religion, the thing that created the earth and living organisms one after the other was the God, and the elusiveness of God’s nature is expected to be hidden far away from the material world. In the end, this parallelepiped was conceived by Clarke as a representation of God, and therefore understandable why it appears is if an absolute entity that is mysterious and unknowable, and that rejects all inquiry by humans.

    Furthermore, when the “monolith” is half-buried in the earth and standing up vertically, it appears exactly like a marble or granite gravestone. Of course, there is not necessarily a direct connection between this geometrically shaped slab and gravestones, but there is some kind of commonality between the phenomenon of aura when a monument envelopes people in awe and the mysterious atmosphere of this parallelepiped slab. This is because it is positioned as being packed with conclusive information that could alter the fate of humankind.

    The parallelepiped slab appears as if an artificial creation, and yet, it also appears to have a deep connection with the phenomenon of life in nature as if integral within nature. However, this does not simply represent a fractured character, such as simply having ambiguity or having two faces, but provides a single overall character that is unified at a deeper level. Within the vagueness of not knowing if this is a factor inherent to nature or something that was intentionally created by someone lies an new image of material and simultaneously organic body, which has actually been overlooked by modern rationalism. This is the concept of one manifestation as something that has the two vectors of, on the one hand, tending towards pure geometrical forms that are reminiscent of artificial objects, and, on the other hand, of tending towards an elusive great power shrouded in mystery.

    When this inorganic object surpasses time and attends the evolution of humankind and advancement of space, it evokes the feeling that it is equipped with eternalness that transcends the ebbs and flows of this world. This black slab was at least depicted as a symbol of such thing. This actually seeks the aspiration towards eternalness that often appears in the ideals of neoclassical architecture. The neoclassical trend to prefer abstract forms, in particular, gives the feeling of something in common. This is not necessarily anything more than a human psychological phenomenon, but abstract geometrical forms often manifest as refined trends and preferential expressions in each of the ages through history. The parallelepiped slab originally drawn by Clarke is not something that transcends time, but is a catalyst that contributes to the history of evolution at a point in time for each point in time, and has also a character opposite to transcendental eternalness.

    Even here the parallelepiped slab is not organically linked within the natural world of advanced complexity and deep development, and creates aggressive conflict, independence, and tension. As a visual image of an entity that changes from moment to moment and transcends the flow of time, a geometrical solid having smooth surfaces and sharp ridgelines can be said to be appropriate.

    Even for the same geometrical solid, in contrast to a building for death, here it is created as if it should be called a monument to life that watches over the phenomenon of life. This also does not follow any gloomy shadow, and has a fairly different aura from the concept of fear. Johann Wolfgang van Goethe had a house on the outskirts of Weimar and in the garden he erected an altar to good luck called “Agathe Tyche,” which consisted of a sphere atop a cube (Fig. 31).This combination of the two most perfect geometrical solids, in a way, conveyed the aesthetic sensibility of the Neoclassicism of the time, and yet was enlightened with a single geometrical existence as a lighthearted monument being eternal without evoking an image of death. For example, simple geometrical forms can be said to have a mysterious unpredictable power called “pyramid power.” But even though these may be commonly believed, we obviously cannot really expect any kind of power from this kind of lump of stone, and it holds nothing more than some kind of effect on the human psyche. Nonetheless, Goethe’s idea of hoping for eternal happiness was well expressed in this abstractly shaped unremarkable stone. This kind of geometry has some kind of cultural power over humans.

    The “Casa Rotonda” designed by Mario Botta (Stabio, 1980 - 1982) is a house based on a simple cylinder (Fig. 32). The house that has three floors above ground and one floor underground with a diameter of approximately 10 meters presents the architect's feeling for geometry, with axes of symmetry that are emphasized even inside the house and a feeling of tightness when going about everyday life. There are large cutouts in the walls, and entrances and windows were created only when required, and yet the basic form of a pure geometrical solid exhibits an unmistakably strong self-assertion. Botta also used square pillars in the Riva San Vitale residence (1972 – 1973) exhibits similar modeling methods.

    For Japanese, who prefer change and complexity, this kind of a plainly geometrically shaped house is viewed as intolerable. This could be said to be something that is imagined by Europeans who traditionally come from focus on strongly geometrically formed ideas, but this trend towards geometry that transcends racial frames should attract focus as a universal human trend. The background to this cylindrical form should be viewed as human history that admits eternalness and transcendence in geometry. And on the other hand it should also be thought of as an expression of the human desire for eternity that aspires to transcend time and place. This geometrically shaped building by Botta is so simple that it evokes the feeling that a similar building must have been built previously. However, it simultaneously needs to be grasped as a contemporary housing form that surpasses often criticized functional housing. While it continues to be a house, as a monument it must aspire to pure geometrical solids. Therefore, it was not designed for autonomous living lifestyle, but had the intention of living within the geometry of a monument. This is because providing an eternal building is already prioritized ahead of a single person’s or a single family’s living in a house.

    Nonetheless, this is not the geometry given by God. In the age of the Renaissance, circles and squares were preferred and often used in churches and chapels, but modern geometrical ideas have changed greatly from these ideas of Neoplatonism of the Renaissance. Even in the age of the Renaissance, there were houses like the residence that Mantegna designed for himself (Mantua), where a central courtyard was formed by cutting a cylinder out of the house. However, although the geometrical forms demanded by these ideas also continued to exhibit aspects in common with the geometry of Botta, the houses created by Botta do not seem to have the sense of something holy, rather have a dialog with undulating natural terrain and are a great deal human. In this way, today's geometrical ideas need to be reconsidered and repositioned within today’s paradigms.

System geometry of space

    When pure geometrical shapes, such as cylinders, parallelepipeds, square pyramids, and spheres are used independently, these carry the meaning of establishing a nucleus or focal point within a space. This kind of structure, therefore, necessarily takes on the character of a monument. Conversely, there are also geometrical methods that do not create a focal point. These are used when geometrically arranging a wide space that envelopes humans. The purist form of these that has been used historically and economically is the square grid (lattice) pattern. If we attempt to create a hole within a rigid-frame structure of columns and beams over a wide space, it is as if the nature of humans were to draw a grid-shaped pattern on the surface of the ground, and then erect columns at the intersections and mount horizontal beams onto these. As we would expect, this kind of technique had already been discovered in the “hypostyle halls” of the sacred structures of the ancient Egyptians. Further evidence that the architects were clearly aware of the nature of these design methods comes from plan drawings from Rome during the Renaissance. A clear statement of the grid methodology appeared in “Précis des leçons d’architecture données à l’École royale polytechnique” written by J. N. L. Durand at the start of the 19th century. That book was a precursor that grids were already not only being used on horizontal surfaces, but were also being employed on vertical surfaces (Fig. 33). Furthermore, the appearance of three-dimensional grids began to be used in earnest from the turn of the 20th century, forming the groundwork of the design of high-rise buildings.

    If a square grid is viewed by looking at each square individually, it becomes pure geometrical forms. In this case, despite a single morphological element that becomes self-asserting, when many of them are gathered together and arranged in a grid, it produces a systematic geometry. This could just as easily be equilateral triangles or hexagons (honeycomb shaped) because they also form a system, but square and rectangular grids are the most suitable in a variety of ways for the constructed spaces created by humans, and these have become the most familiar grids for us. We feel that arranging and creating housing interior products and construction materials to have right angles is the most familiar and most convenient for us. The system of intersections in a grid is as self-evidently natural to us as water and air.

    At this point, I do not want to explain grids simply as a convenient and useful systematic geometry. What I want to focus on is that the grid pattern itself carries some kind of character of monumentality. The monumental space of the “hypostyle halls” cannot necessarily be said to have been functional, and the purpose appears to have been to create a monument in itself. This is not only because each of the circular columns was enormous yet beautiful, but it is also the methodical arrangement of many circular columns into a regularly aligned grid that is a source of deep impression. Even if an ordinary person feels moved upon seeing an actual space with regularly arranged circular columns, they are probably not aware of the mechanism behind this. When seen through the eyes of an architect, however, the horizontal grid pattern can probably be visualized mentally.

    Oswald Mathias Ungers used grid patterns repeatedly in a way that seemed dogmatic. For example, in Pavilion 9 at the Frankfurter Messehalle (1983, Fig. 34), the giant expansive walls were faithfully divided by a square grid measuring 108 horizontally and 25 vertically. The same grid pattern was also used in the corridor of the neighboring semi-cylindrical glass ceiling (“Galleria”), and later in a super-high-rise tower at the same trade fair (“Frankfurter Messe Torhaus”, 1980). Moreover, the apartment block on Schiller Avenue in Berlin is fenced in by walls arranged in a square grid, exhibiting a metaphysical internal space image (Fig. 35). His forms completely rejected complex free forms and rigidified his own abundance of sensitivities, as if shackling himself to an overall cold systematic geometry.

    Many other architects and designers in America, Europe, and Japan used square grid forms from their own intentions and feelings. The shades of a deep psychological meaning held by the square grid, therefore, became rather vague. Furthermore, they did not use the grid in a positive way. There are many cases where the grid is incorporated rather negatively or critically. Ungers took a unique standpoint amongst this, and added a new monumentality to the grid as an orthodox design method. For people with a refined sense of taste, this was taken as sadistic behavior based on a cold weapon. For example, between the grid patterns used by Richard Meier as if embodying the urban refinement of New York and those of Ungers, we pass through the contrast from aesthetics that are pleasing to the eye to aesthetics of exhibitionism.

    The age where the grid is preferred simply as a convenient design technique has passed. And today, when the grid is drawn upon and used, it is often used as nothing more than a metaphor, i.e. as rhetoric in a design. In fact, the systematic geometry today does not have the time to contribute to such a fundamental thing. If asked why the grid is to be taken onto the stage for the sake of architectural expression, one answer is that the interpretation of monumentality of abstract geometry as described earlier is useful.

    Difference from a geometrical solid that can be grasped by the eye as a single form (Gestalt) a grid is nothing but an method to arrange the void spaces. Although the interpretation of spaces as a systemat has taken a variety of forms since humankind first began, modern rationalism in particular has come to attach lucid theories to these vague and elusive spaces and outer space. The grid was the most useful for the interpretations of space and means of dominating space that human beings had nearly instinctively. In other words, the grid is, as it were,  an archeological monument speaking a history of interpretation of space by human beings, and that should be compared to the pure geometrical solid of the pyramid. In fact, it is known that square and rectangular grid patterns appeared very early in the networks of roads in ancient Chinese cities and the city plans of ancient Greece. The human mind has advanced by continually overlapping new layers. If we consider that the process of advancement of human thinking abilities is written within those, then information about the grids within the old layers can be brought out. Conversely, there may also be an effect on the modern human system of thinking from the depths of the brain.

    The 1970s architectural trend called “rationalism” reignited a trend toward reviving the geometry that appeared once in Neoclassicism. But the role of this kind of age of geometrical revival can be said to stimulate the old layers and, once again, review the direction that was lost in the process of diversifying the new layers. Among the methods of historicism that reference and borrow from the various historical forms, the meaning of referring to ancient forms is a distinguished case. The role of re-erecting a vertical axis extending infinitely through the old ages and new ages is that of neoclassicism and rationalism.

    Peter Eisenman, who is excelled in philosophical analysis of architectural forms, focused particularly on the process of gradual morphological transformation from primitive forms to more complexity. It was, as it were, an attempt to simulate the process of evolution within the brain from old layers to new layers. Executing such attempts he proposed several models of houses and even actually constructed some. For example, the model called “House X” (1975 – 1978, Fig. 36) was a joyous structure consisting only of right angles through a complex combination of rectangles and squares that is reminiscent of constructivism. It is suggested as the result of the transformation process starting from a single cube or parallelepiped and carrying out several transformation methods in steps.

    Although it can also be seen in the creations of Ungers that used textbookish square grids, there is a method of accentuating a diagonal line in the middle of a square grid to give a feeling of tension through breaking order. The “Sprengel Museum” proposal (1972, Fig. 37) designed by Josef Paul Kleihues is also an example that presents this seed of an idea. Indeed in the Baroque period there were transformation methods in some degree, where the central axis of a grid was emphasized more than any other axis lines and give the buildings a strong symmetry that showed distaste for homogenous, monotonous exteriors, but the grid transformation method here was fundamentally different. This was an attempt to overall break the consistency formed by the grid and the individual uniformity as a whole, and demonstrated a rejection of the trend of aspiring towards logical consistency of modern rationalism. Diagonal lines are recognized as foreign bodies, or looking at it conversely, if we trace along a diagonal line then the grid are recognized as a foreign body, and then grids are no longer a tool for humans to dominate spaces.

    This feeling of misalignment and foreign bodies represents, accepting indeed the monumentality of grids, a viewpoint that is critical of grids. By introducing this new attitude of the critic, architects believed that a fresh sense of humanity could be recovered. Conversely, people who believed in the monumentality of grids from the bottoms of their hearts were slaves of modern rationalism, must also bear responsibility for the sinful aspects held by modern rationalism. In contrast, the critical usage of grids showed dislike to the placing of a yoke on the diverse abundance of humanity and attempted to make grid system vague.

    Ungers placed grids directly in front without striking at vague escapes. However, at the same time, he drove the grids from the system of the world of human activity towards the system of metaphysics. In other words, he attempted to separate the concrete human space from the abstract metaphysical space without rejecting but exposing ideas, which were intrinsic to the spaces inhabited by humans. Put another way, if we are on the lookout for a dangerous person, there are two contrasting methods, one which is to constantly face into the line of sight of that person, and the other which is to constantly attempt to escape the line of sight. Within the vigilant nature of grids, these are both common.

    The grid, in other words, the abstract systematic geometry, can be thought of as one of the architecture of death in terms of killing the sensitivities of humans. This does not mean the death of the actual living organism, but refers to the death of the formal cultural activities of the human within their head. The various behaviors of artists who embrace grids can be thought of as a metaphorical solution to the question of how well it suits the effectiveness of this cold geometry. Even if we speak of abstract geometry as a form of death, the living culture created by humans could not be established without it. In contrast to the phenomenon of life in the natural world that has successively given birth to the “living form” from the beginning, humans hold nothing more than the ability to use inorganic materials and construct them together into structures with abstract geometrical forms. Humans are therefore befallen with the fate of constantly starting from the forms of death, and even the liveliest culture has lived under the shadow of death from the beginning. Among living cultures, the form of death was not recognized as what it was, and the non-living character of that form becomes clear beginning with the death of culture. This kind of abstract geometry is necessarily deeply related to the theme of life and death of culture.

    If we draw a picture that imagines the surface of the earth surface, where the global ecosystem lives as working as a large scale life that blends together inorganic material and organic material, then forming a city plan based on the grid pattern of the road network or placing the geometrical solids of structures within this has the meaning of introducing the inorganic foreign bodies of forms of death into this organic space where the phenomenon of life acts. This resembles, for example, embedding artificial organs into a human body or connecting bones with plastics and metals, and thought as supplementing the organic system of the living organism with replacement parts of inorganic materials or forms which are basically different from the composition of the organism. If we think in this way, then the pyramids should not be grasped as structures that surpass the size of human bodies, but could also be said to be small cultural information organs embedded within the global ecosystem.