There is a general math-related puzzle called the frog hop problem. I first heard about it from watching a Stand-up Maths video. What interested me most was trying to visualize the whole thing – to create a piece of art based off of a specific set of branching decisions. The artworks here all relate to this problem.
The usual method for descending a staircase is one step at a time.
Occasionally, one might get in a hurry and skip a step or two, especially at the end.
The most daring approach is to jump straight to the bottom, making a flight of stairs really live up to its name.
These are not the only choices.
There are many others.
In this particular case, there are ten stairs.
Allowing only downward movement and the freedom to skip any number of steps, the piece of art on each stair indicates the following: the number of future choice possibilities vs the number of declined paths; a color-coded guide for jumping to another step; your current position on a map of choices; and the final velocity your body will be traveling at the moment of landing.
Stairfall can be used as a guide for planning a trip down the stairs.
Even better, it can be used to visualize all possible trips down the stairs.
Since tripping or really even jumping down stairs is not always safe, sometimes itโs nice to see how much complexity there can be in something routine.
Stairfall is a set of eleven 20″ x 10″ panels designed for the Candoro Marble Building staircase. It will be on display through 25 January 2025.
You may wonder if this piece addresses the same idea as rana viam. It does!
These pieces are based on the problem: how can multiple lines pass through the centers of polyhedra and form a knot, so that each face is identical? The first part of this problem is: what happens when multiple lines intersect in a knot diagram? The answer to this is that it becomes a mess, so the easiest way to avoid multiple crossings in one spot is to have crossings subsequent to the first one simply dodge the intersection. This creates the arcs in the artwork. For lines to pass through the face centers they have to start out at certain points on the perimeter. They can’t meet at the vertices or you would have another mess.
The name comes from a figure in Jewish mythology: ืืืืจืื. An archangel and scribe, Metatron is associated with a diagram called “Metatron’s Cube,” a graph-like diagram which sort of represents the five Platonic solids, and maybe a tesseract. Upon close examination, the diagram becomes less meaningful from a math point of view. However, the idea of an angel using part of his soul to create a topological diagram is pretty epic. Additionally, I like the reference to ancient Jewish lore better than to ancient Greek math.
This is a wall tile recovered from the excavated temple. The room was a small enclosed space with no windows to the outside. The gas lamps we were using revealed the dimensional texture of the tiles. A relevant offering was placed in its stead upon removal.
This pattern tiles a plane divided by either triangles or rectangles
This piece is currently on display at the City County building in Knoxville, TN.
This paper explains the process for constructing a space-filling curve, segments of which tile the faces of a regular polyhedron; describes its application to a polyhedron net; and illustrates some of its interesting properties with physical examples.
There is a method for constructing a space-filling curve which positions its termini at points allowing copies of the same curve to be connected end to end and applied to the net of a regular polyhedron. The resulting polyhedral shell has faces tiled with segments of a unicursal path: a surface-tiling curve.
Each face of a polyhedron includes a space-filling curve which is one segment of the entire curve, referred to here as a tile. Figure 1(aโd) shows tiles for the first four iterations. Rotated and reflected tiles connect across dihedral angles of polyhedrons, wrapping the surface in a continuous loop and comprising the entire curve, as shown in Figure 1(eโh).
Curve tiles follow a basic pattern of fractal curve construction: each iteration is made up of transformed copies of the previous iteration, indicated as outlined paths in Figure 1(bโd). Each iteration also includes an adjusted version of the previous iteration (described below).
The construction method alternates for every other iteration, indicated in Figure 2 as A and B. Both methods consist of four steps, each indicated by thin lines with arrows, which place transformed copies of the previous iteration in order in quadrants IโIV. The two methods differ in the third step (โป 90ยฐ vs GLIDE). Construction for each iteration ends with a transformation (indicated by an outlined arrow) of the new tile for its orientation in quadrant I as the basic component of the following iteration. The two methods also differ here (FLIP โ vs โบ 90ยฐ).
The basic component for each iteration can be seen as having two parts. One of these parts, indicated by a gray background in Figure 2, is reflected in the last step of each iterationโs construction, and becomes a symmetrical version of the basic component. This last step creates the overall path direction necessary to apply the curve to a net.
A surface-tiling curve tileโs overall path direction allows an unbroken end-to-end path to be applied to a net by using alternating transformed tiles. Because the tileโs path does not branch, only non-branching nets will work. Figure 3 shows the transformation pattern for a tile applied to such a net.
When the net with the applied tiles (the whole curve) is folded into a polyhedral shell (Figure 4), the path traverses all the faces and connects with itself, becoming continuous.
Cube shells can be arranged into stacks that preserve the unicursality of the exterior, visible curve. Figure 4 shows a few of the many possible arrangements.
Figure 4: Various arrangements of stacked shells with surface-tiling curves (paint on steel).
Even numbers of tiles arranged in the basic net pattern used in Figure 3 can be bent into a continuous ring. Odd numbers of tiles can be bent and twisted into a Listing band (a.k.a. Mรถbius strip), shown in Figure 5. Both arrangements maintain unicursality.
When a shell is cut along the path of a surface-tiling curve, the shell is sliced into two mirrored pieces with mirrored, branching nets, shown in Figure 6. The two pieces are not locked in place and can be separated with two slide moves.
Platonic polyhedra were chosen for the exploration in this paper for two reasons: 1) each is composed of identical faces and 2) the shapes of those faces are easily mapped into Gaussian and Eisenstein domains, keeping surface-tiling curve construction simpler and more consistent with other curve construction methods [3].
In the examples given, the term โpolyhedronโ is used although all the examples above are cubes. Surface-tiling curves also exist for the triangle-faced regular polyhedra. These three polyhedra can use the same tile, iterations shown in Figure 7.
Non-branching nets of all three triangle-faced regular polyhedra can encompass surface-tiling curves (Figure 8). Bending / twisting nets and slicing shells work the same as for cubes.
There is a similarity between surface-tiling curve tiles and the Hilbert curve. However, the Hilbert curve cannot tile the faces of a polyhedron without adjustment. Its iterations alternate between entering / exiting a square containing the curve on the same edge and on opposite edges [2]. It is the feature of entering / exiting adjacent sides of containing shapes which allows the polyhedral face tiling of surface-tiling curve tiles for polyhedrons with square and triangle-faced polyhedra.
The collection would be complete with a surface-tiling curve for the dodecahedron. However, a continuous linear path with identical tiles on each face is not possible with a dodecahedron net because the pentagon requires two different general path directions across connecting faces: a path entering one side would need to exit an adjacent side sometimes and an opposite side other times (diagram in supplementary document). A look at the 340 non-branching dodecahedral nets shows that not one is composed of purely adjacent-side-exiting or opposite-side-exiting arrangements [1].
A curve with tiles spanning more than one face (such as a curve segment contained in an octagon made up of two connected pentagons) might address this issue and is an avenue of future investigation.
[1] E. Pegg Jr. Wolfram Demonstrations Project. Path Nets for Dodecahedron and Icosahedron. 2018. https://demonstrations.wolfram.com/PathNetsForDodecahedronAndIcosahedron/
[2} H. Sagan. Space-Filling Curves. Springer-Verlag, 1994.
[3] J. Ventrella. The Family Tree of Fractal Curves. Eyebrain Books, 2019.
The artworks shown were on display at Dalhousie University in Halifax, NS in July 2023 as a part of the Bridges Conference Mathematical Art Galleries.
The written content is a paper presented at the same conference.
The paper was also presented at the 103rd Annual MAA-SE Section Meeting at the University of Tennessee-Knoxville in March of 2024.
A frog rests on the bank of a pond. There are nine lily pads in a line across the pond. The frog may make 1) any size hop and 2) any number of hops in order to land on the opposite bank without overshooting. It can hop ten spaces (the whole pond), it can hop one at a time to every single pad all the way across, or it can take one of the other 510 combinations.
The frog agrees to the rules but it’s a frog, so it’s going to do what it wants. At first, it enters the following in Excel:
| 1st hop | =RANDBETWEEN(1,10) |
| 2nd hop | =IF(B1<10,RANDBETWEEN(1,10-SUM(B1)),”—“) |
| 3rd hop | =IF(SUM(B1:B2)<10,RANDBETWEEN(1,10-SUM(B1:B2)),”—“) |
| 4th hop | =IF(SUM(B1:B3)<10,RANDBETWEEN(1,10-SUM(B1:B3)),”—“) |
| 5th hop | =IF(SUM(B1:B4)<10,RANDBETWEEN(1,10-SUM(B1:B4)),”—“) |
| 6th hop | =IF(SUM(B1:B5)<10,RANDBETWEEN(1,10-SUM(B1:B5)),”—“) |
| 7th hop | =IF(SUM(B1:B6)<10,RANDBETWEEN(1,10-SUM(B1:B6)),”—“) |
| 8th hop | =IF(SUM(B1:B7)<10,RANDBETWEEN(1,10-SUM(B1:B7)),”—“) |
| 9th hop | =IF(SUM(B1:B8)<10,RANDBETWEEN(1,10-SUM(B1:B8)),”—“) |
| 10th hop | =IF(SUM(B1:B9)<10,RANDBETWEEN(1,10-SUM(B1:B9)),”—“) |
I appreciate the logic but not the lack of visual appeal so I provide the two following visual representations:
Every possible path can be visualized, but it needs to be done symbolically to make any sense: an expanded (non-symbolic) version three inches high at the proportions shown here would need to be about sixty-five feet long.
The bottom edge is the starting bank and the upper edge is the target bank. The lower “row” (which is sort of diagonal) represents the first hop. The height of each shape corresponds (pun) to the number of pads hopped across. The shapes are also color-coded.
The upper row represents subsequent hops. Each pill shape is a symbol for the containing shape of the same color.
For example, the yellow pill represents a three pad hop which can be made up of a one pad hop (white) and a two pad hop (orange) in either order; each of the orange two pad hops can be made up of one two pad hops or two one pad hops.
The larger the hop after the first one, the more the symbolic version expands horizontally. The frog is overwhelmed by the vast number of possibilities in this diagram. That’s because this representation contains a lot of duplicates. The next representation removes all duplicates.
Again, the bottom row represents the first hop. The frog uses this as a map, plotting a course across the pond. Each color represents a hop distance.
The first hop has ten options, each represented by a different color. After the initial hop, the choices for the next hop are determined by the pill shapes penetrated by spikes. The top of that pill shape will lead horizontally to another rectangular shape. Rectangular shapes penetrate pill shapes, pill shapes lead to rectangular shapes, and so on, until a pill shape with a black outline is reached, indicating a landing at the other side of the pond.
A closer look at the condensed hop guide examples given above:
Satisfied that these maps work, I cut made art pieces out of cut paper.
These two artworks were on display at the 2024 Joint Mathematics Meetings in San Francisco.
The “deduped” version was on display at the d’Art Center in Norfolk, VA as a part of the June / July 2024 Fibrous exhibition.
This band is a larger version of the one used as the artwork for the surface-tiling curves project (found under the bending / twisting heading).
The structure is composed of twenty-one identical components. Each component traces a segment from the third iteration of a surface-tiling curve. They are joined in a way that preserves unicursality, then twisted and bent into a Listing band. I chose to make the curve go all the way through the metal to make it clear that a surface-tiling curve segment can be applied to a one-sided nonorientable surface.
On a clear day, the sun makes projections just blurry enough to give the appearance of unbroken, knocked out lines. If the components really did have unbroken (negative) lines, the entire structure would be in two pieces.
The surface is treated with dye oxide. If you don’t know what that is, it involves using a blow torch, which is always fun.
This is a larger version of the sliced cube appearing as the accompanied artwork to the surface-tiling curve project.
Each folded shell fragment is 10″ on each side, constructed from a single piece of steel, perforated at the edges. The inside is painted gold.
The structures are mirrors of one another, whether folded or unfolded. They fit into one another (without bending anything) and form a cube shell.
When painting the surfaces, the cardboard they were placed on ended up looking pretty cool.
A single shape shows four negative paths curving around the center where they would otherwise intersect. The resulting arcs weave over and under one another.
Copies of the projected shape can be rotated and aligned to form a hexahedral shell on which the paths combine to form four knots.
This sculpture (metatronic solid) is a hexahedron shell formed from the same face pattern.
This piece was on display at the City County Building in Knoxville, TN in 2024.
Four lines make a reversed knot pattern covering all faces of a hexahedron, regardless of rotation (this same pattern can tile a plane).
One of my most basic rules for knot creation is that knot lines only cross one another in pairs. Any more than two lines appearing to overlap in the same place doesn’t allow for clearly implying that one line is over another. And it’s confusing.
In this design, a diagonal line cuts a square in half. Three more lines attempt to converge in the center of the square. A ripple radiates from this center point, forcing these subsequent lines to bend around the point they would all intersect. The positioning of the lines allows a cube to be formed.
This is the second (or maybe third) in a series of hollow polyhedra based on this concept, taking its name from Metatron, a figure associated with the geometry of regular polyhedra.
After cutting the lines and carefully (read: dangerously) beveling the edges, I glued the faces together.
It looks great when light is shining through it, but displaying it resting on one of the faces seems like a waste.
The stand obscures much less of the finished piece and presents a much more interesting viewing angle.
This piece was a part of the The Arts & Culture Alliance’s 16th annual National Juried Exhibition and displayed at the Emporium in 2022.
A single line bisects a hollow cube into two mirrored sections, revealing a second, smaller solid cube inside which supports the three pieces when suspended from the top corner. The bisecting line (curve) is plane-filling, and each face of the exploded hexahedron is identical.
A single line bisects a hollow cube into two mirrored sections, revealing a second, smaller solid cube inside which supports the three pieces when suspended from the top corner. The bisecting line (curve) is plane-filling, and each face of the exploded hexahedron is identical.
The diagram here shows the first 4 iterations of this curve although only the first one is used in this piece. It may look like the Hilbert curve at first, but while it’s ok for a space-filling curve to start at the left and end at the right, to use a curve to tile the faces of a polyhedron, it will have to follow a more circuitous path.
This piece evolved into the surface-tiling curve project.
The metatronic knot on a solid surface has to be “knocked out,” but is there a way to show the inverse? The not-knot? With an opaque material, not really, but with a translucent medium, not only could you show the knot counter-spaces, but you could view the inside and outside simultaneously.
This piece is best viewed with abundant sharp light such as sunlight.
Sculpture usually has to make a compromise because in anything but zero gravity, it has to rest on something. This means you don’t get to see what’s on the bottom. An artist isn’t likely to put something interesting in a place no one can see. In this case, I wanted to make sure a viewer saw how the knot pattern worked out on all six sides, leaving nothing to doubt.
Something of added interest: the pattern used in metatronic knots works regardless of rotation; any of the faces can be rotated and the knot pattern still works fine.
Although the main challenge of creating the metatronic solids was applying knot patterns to regular polyhedra, each face was interesting itself.
The first iteration of the faces was the hexahedron. I thought would be fun to use the face of a hexahedron, a square, to create a cube, but by stacking copies of the face instead of rotating them along their edges.
For the next version I focused on just one face, a triangle from a tetrahedron. I used gold leaf poles to elevate the shape off of the background.
The radial lines in the knot design were emphasized by darker wood in the background.
The dodecahedron face, a pentagon has many more radial lines. They were inlaid with a contrasting wood.
There’s one more of the square / hexahedron. This one was a little more involved, so there’s a separate post devoted to it.
Three lines make a reversed knot pattern covering all faces of a tetrahedron, regardless of rotation (this same pattern can tile a plane).
The first in a series of polyhedra shells based on this concept, taking its name from Metatron, a figure associated with the geometry of regular polyhedra.
This piece was included in the 21st Master Woodworkers Show in Knoxville, TN.
