SandstoneGeometryon the Colorado Plateau

Stan Wagon, Dept. of Mathematics, Macalester College, St. Paul, Minnesota 55105, USA [email protected]

H. Allen Curran, Dept. of Geosciences, Smith College, Northampton, Massachusetts 01063, USA [email protected]

Mathematicians often encounter interesting geometrical shapes in their travels, in both developed or natural areas. One region that has continually presented the rst author with diverse geometric surprises is the Colorado Plateau, a vast semi-desert region of largely Late Paleozoic to Mesozoic sandstones ranging from western Colorado through southern Utah and northern Arizona. In this article, we present some examples that cover the gamut, from blocks to spheres to other diverse geometrical forms. See [3] for the big picture of geologic evolution in this area. All photographs were taken by Stan Wagon during his two decades exploring the region.

1. Tiling by Cubes

At a location overlooking the famous goosenecks of the San Juan River in Utah (eight km NW of Goosenecks State Park and 450 meters higher), there are hundreds of huge right-angled blocks falling out of the upper layer of Cedar Mesa Sandstone (Figs. 1, 2). This sandstone is of Early Permian age, formed about 270 million years ago. Some blocks are close to perfect cubes, while others are rectilinear boxes. An aerial view should show the orthogonal structure more clearly. Such a view is easy to get using Google Earth and indeed one can see an almost perfect geometrical sawtooth made from right angles (Fig. 3). Further examination of the region leads to a nearby location where the aerial view shows an almost perfect tiling by rectangles (Fig. 4).

Figure 1. Large blocks eroding out of the uppermost layer of Cedar Mesa Sandstone. 2

Figure 2. The dihedral angles of the blocks are right angles. Their side dimensions are generally more than 10 meters.

Figure 3. An aerial view of the falling blocks in Figures 1 and 2 shows the sawtooth pattern (in cyan) at the cliff edge. Google Earth image ©2021 Google.

Figure 4. An aerial view of a nearby location (37° 13 18 N, 109° 57 37 W) reveals the regional orthogonal jointing pattern that yields the blocks. Google Earth image ©2021 Google.

The pattern in this area arises from an orthogonal joint set, i.e., oriented fractures in this at-lying, evenly-textured, and well cemented quartz sandstone with dihedral angles close to 90°. The fractures formed in response to regional exten- sional stress within the plateau (see [1]). The top view in Figure 4 shows the part of the pattern that is evident on the surface, but there is also jointing within the horizontal beds of the sandstone below the surface (see Fig. 1). So it is a true three-dimensional tiling, although we cannot be certain of the depth to which the vertical planes penetrate the layers. Comparable examples of systematic joint sets of different ages and localities from around the world were 3 layers. Comparable examples systematic joint ages analyzed in detail in a recent study [9]. A rather different collection of cube-like objects is shown in Figure 5: the remarkable Cobblestone Bridge. This struc- ture is in Arches National Park, 0.3 miles NNE of the end of the parking lot for Delicate Arch. Arches often gain strength from catenary shapes; this one does the direct opposite, as it appears to be a collection of cubes supporting themselves in a purely horizontal alignment. The bridge has likely stood in this form for thousands of years.

Figure 5. Cobblestone Bridge in Arches National Park.

2. Spheres

While spheres in nature are not unusual, the sand and iron concretions known as Moqui Marbles are very striking. They can be near-perfect spheres. Before the Navajo Sandstone (deposited near the beginning of the Jurassic period, 200 million years ago) was cemented into rock, groundwater owed through these sands and iron would leach out into the water [6, 12]. That iron would eventually reprecipitate out and form an iron-cemented sand concretion that very often takes a spherical form. The iron-rich cement is primarily hematite. The balls then see the light of day as the overlying layers are eroded, and they eventually fall out of the sandstone. They can roll down over the millennia, gathering into collections of thousands (Fig. 6). Figure 7 shows a half-buried concretion with a mysterious spiral structure (combined with concentric rings). The surface texture of individual spheres can vary, but some examples are very smooth and symmetrically round (Fig. 8).

Figure 6. Moqui Marbles in the Navajo Sandstone. 4

Figure 7. The lower half of an iron concretion (diameter = ten cm); this one shows a rare spiral structure in the iron bands.

Figure 8. A small iron concretion (diameter = three cm) that has a remarkably uniform spherical shape.

When the Moqui Marbles gather in a sloping area having a polygonal tiling, they can roll into the edges, delineating the polygon boundaries (Fig. 9).

Figure 9. Here the Moqui Marbles rolled into the boundaries of the pentagons and hexagons on the surface.

While beyond the reach of a mathematical tourist, structures similar to Moqui Marbles have been found on Mars; this has led to increased interest in the study of these objects on Earth. The accepted origin theory for the spheres on Mars (called “Martian Blueberries”) is that they arose for reasons similar to the water-based iron-leaching explanation for the ones in Utah. An argument for a cosmic origin for the Martian Blueberries — remnants of meteorites breaking up in the atmosphere — has been proposed [11], but experts generally prefer the iron-leaching hypothesis [2, 5, 12]. 5

3. Center of Gravity

Near the Utah–Arizona border there are thousands of hoodoos: towers of rock where a resistant caprock lies atop a softer, more easily eroded layer. So long as the erosion is uniform in a circular sense, the caprock’s center of gravity will remain balanced, if delicately, above the base, and the center of gravity of the whole mass will be positioned over a very small base. An extreme example is the Twisted Hoodoo (Fig. 10).

Figure 10. The Twisted Hoodoo, north of Church Wells, Utah.

Figure 11 shows an elegant hoodoo known as the Tower of Silence, located in Wahweap Creek near Big Water, Utah. What is unusual about the white-and-red towers in this region is that two sandstone formations that are usually nowhere near each other are represented. The white rock is the Gunsight Butte Member of Entrada Sandstone, while the reddish caprock is Dakota Sandstone. The contact between the two units represents a signicant unconformity because typically other formations (Morrison, Summerville) representing signicant time (millions of years) would be present between these two sandstones. For some reason those other layers are absent here, and this gap leads to the abundance of hoodoos in the area west of Page, Arizona. Figure 12 shows an example where the caprock was tilted to a ridiculous angle, but remains balanced on the pillar. See [13] for an explanation of how water can cause the erosion necessary for these towers to form and [4] for an argument that the weight of the caprock can cause the base to become more resistant to erosion, thus leading to longevity of the structure. 6

Figure 11. The Tower of Silence is one of the most elegant hoodoos in Utah.

Figure 12. A Wahweap hoodoo with an eccentric cap.

4. Intersecting Sets of Parallel Planes

Differential erosion combined with a complex structural deformational history can lead to remarkable patterns of thin, intersecting, and parallel sheets of modied quartz sand within the Navajo Sandstone. Such layers are termed “co- mpaction bands” [8, 14] and are commonly called “lace rock” or “boxworks.” These compaction bands form under a complex set of circumstances. When initially deposited, dune-sand layers will have different characteristics of grain size and porosity due to deposition by winds of varying strengths and directions. Eventually, these sands were covered by an approximately1.5-km thick layer of sediments of other formations, and the great pressure of burial caused the Navajo layers to harden in different ways, depending on their grain size and porosity, resulting in the formation of vertical compaction bands. In addition, tectonic forces acting on these layers can result in compression, generating shear-enhanced compaction bands that formed at high angles to the more common near-vertical bands. Details, including microscopic investigations of grain reorganization processes, can be found in [8, 14]. The bottom line is that some thin compaction bands of sandstone are more resistant to erosion, and when the softer surrounding sandstone erodes away, very thin layers of modied sandstone will remain. These form exceptionally delicate and beautiful patterns; the example in Figure 15 7 exceptionally patterns; example Figure seems especially fragile; it is remarkable that it has existed in this form for a signicant period of time. The examples of Figures 13 and 14 are from an area known as Edmaier's Secret near the Buckskin Gulch trailhead in south-central Utah. The example of Figure 15 is in the White Pockets, in Arizona near the Utah border.

Figure 13. An extreme example of “lace rock.”

Figure 14. A spectacular stand-alone example of intersecting parallel sheets of modied and compacted quartz sands revealed by differential erosion. 8

Figure 15. Another extreme example of lace rock arising from a very delicate compaction band.

5. Saddles: Double and Triple

Thin, horizontal beds can be present in the Navajo Sandstone, owing to different wind strengths that carried different- sized grains of sand into dunes. When these layers solidify under pressure from overlying layers and water ows through, the grain size and porosity affect the amount of ow and hence the amount of iron that leaches out of the particles [10, 15]. The less iron that leaches out, the redder the layer. A famous example of this occurs at The Wave, in the Vermilion Cliffs National Monument near the Arizona–Utah border (Fig. 16).

Figure 16. The remarkable swirling and colored layers of The Wave.

Although beds at The Wave are not all horizontal, when they are, the curves correspond to the contour lines on a surface. Figure 17 shows how such structures can look like the contours of a classic saddle surface. The shape here has a striking resemblance to the standard saddle given by z = y2 - x2, with the different shades of sandstone providing near perfect contour lines. 9

Figure 17. A near perfect, classical saddle point, with the contours above and below the saddle point etched in the Navajo Sandstone. The climber is Phil Hage.

A type of saddle that might hold more interest for the mathematical tourist is the monkey saddle. These are rare but do occur naturally on a large scale. At a monkey saddle point there are three regions of descent and three of ascent, the classic example (Fig. 18) being z = x3 - 3 xy2 or, in cylindrical coordinates, z = r3 cos(3).

Figure 18. A monkey saddle has three directions in which water drains and three ridges where the terrain rises.

There is a well-known triple saddle near Broad Cairn in Scotland (Fig. 19, left). And perusing USGS topographic maps, Coté, Elgersma, Kline, and Wagon found several in the United States [7]. The most striking one is in Washington State near the Canadian border north of Spokane (Fig. 19, right). The sculpting of the land by glacial ice seems relevant to the occurrence of such saddles in the northern U.S. states.

Figure 19. Left: A monkey saddle in Scotland. Right: A monkey saddle in Washington State with the region between elevations 2680 feet and 2720 feet shaded in brown. 10

Editor: We used side-by-side here with 3.75 inches each. Seems efcient and I think 3.75 inches is adequate for legibility. Legibility is critical. We could use side-by-side layouts for some other pairs as well (11 and 12? 7&8?) These saddles are related to the idea of a bifurcation lake, or trifurcation lake. Bifurcation lakes drain out of two ends and are not uncommon; a famous one is Yellowstone National Park’s Isa Lake, which drains into both the Pacic and Atlantic Oceans. A lake at a monkey saddle would be a trifurcation lake, but we know of no such example

6. Conclusion

The arid Colorado Plateau is one of the most spectacular geological regions in the world. Here water is in short supply, which means that the growth of plants is minimal and rocks that formed over a half-billion years are on full display. The paradox is that ash ooding is the primary agent of erosion for sculpting the canyons and the many other fantastic sandstone structures of the plateau, with wind erosion being an important secondary agent. The interesting and varied properties of these sandstones and their geologic histories lead to formations that have value to photographers, geolo- gists, and perhaps even mathematicians.

Acknowledgements. We thank several geologists for valuable comments and references: John Craddock (Macalester College), Greg Pope (Montclair State Univ.), and Winston Seiler (KCSI Aerial Patrol).

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