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 it’s 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: מטטרון. 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(a–d) 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(e–h).
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(b–d). 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 I–IV. The two methods differ in the third step (↻ 90° 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 ↔ vs ↺ 90°).
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 iteration’s 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 tile’s overall path direction allows an unbroken end-to-end path to be applied to a net by using alternating transformed tiles. Because the tile’s 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. Möbius 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 “polyhedron” 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.
This is a slight departure from my idea of doing printed pieces based off of my other artwork. It started off as a pretty straightforward multicolor print, but I got an opportunity to display it in an unusual spot in Relay Ridge’s gallery as a part of a Printmakers Anonymous (Knoxville Chapter) show.
The main idea was first realized with my piece negative single twist components. Since the print was going in a window, I wanted to keep some transparency. And, since this particular window is several inches deep, there was a great chance to play around with substantial depth differences between layers / colors.
As usual, it may be a stretch to call this sculpture. In this case, it may also be a stretch to call it printmaking. But, since it’s 3d and it used silk screening, technically it’s both.
This piece looks much cooler in person. Getting a good photo of something that’s backlit and detailed proved more difficult than I anticipated.
This piece was recently on display at Relay Ridge’s gallery as a part of a Printmakers Anonymous (Knoxville Chapter) show.
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.
Probably despite his better judgment, Thomas Zachary has once again invited me to appear on his podcast, The Knoxville Area Artist Networking Platform, a.k.a. the KAANP.
Listen as he, cloverfinearts, and I destroy pieces of our own art, practice ephemerality awareness, make fart noises, and discuss Batman. Oh, we talk about art too.
For reference, this is the piece of art I destroyed for this episode. Read the post and it makes more sense.
This short film accompanies an abbreviated version of the album Pool of the Black Star by New York / California guitar duo KillDry.
The film incorporates cymatics, time lapse imagery, experiments with nanoparticles, and nature scenes from surrounding areas, including Cumberland Falls state park.
This video is scheduled to be shown publicly at the Candoro Marble Building in November of 2024.
My work is about hidden worlds. This show provides a glimpse into these worlds by answering such questions as: What would overlapping 2-dimensional objects on the same plane look like? Can a single component form wildly different structures? How can a viewer’s perspective reveal a greater range of meaning? Geometric and abstractive works in metal sculpture, painting, woodworking, papercraft and other media illustrate basic principles of topology and tiling.
There is in all things a pattern that is part of our universe. It has symmetry, elegance, and grace – those qualities you find always in that which the true artist captures. You can find it in the turning of the seasons, in the way sand trails along a ridge, in the branch clusters of the creosote bush or the pattern of its leaves. We try to copy these patterns in our lives and our society, seeking the rhythms, the dances, the forms that comfort. Yet, it is possible to see peril in the finding of ultimate perfection. It is clear that the ultimate pattern contains its own fixity. In such perfection, all things move toward death.
From “Collected Sayings of Muad’Dib” by the Princess Irulan
Beauty is for me absolutely fundamental. And it is important. It’s not just an accent. Without beauty we would not do any good science.
Sir Michael Atiyah
short film to accompany music by KillDry
Lattices of three different colors intersect over a field of triangles. The large hexagon contains the basic unit for the pattern, made up of transformed copies.
Two knots based on 90° radial symmetry intertwine. The same material (steel) is heat treated with different processes to create two different colors.
Listing bands with odd and even numbers of twists
A helix and a hexahedron are both composed from a single component, a negative single twist knot.
Rotated copies of two identical outlined knots weave together.
Five knots are bent into the shape of a larger knot, the symbol for infinity.
Eight knotted hexagons cover the surface of a cube in a pattern which is identical on each face of the solid.
Two unknots are arranged vertically above a charred wood base. The shape of the metal makes self-supporting convex and concave arrangements possible.
Two sets of interlocked hexagonal rings traverse three color fields.
Copies of a single metal tile are woven together to form a cube with a concave vertex.
A three-hexagon structure repeats to form a lattice made of knots.
A single shape shows four negative paths curving around the center where they would otherwise intersect.
Outlined knot segments are inlaid with copper wire. Each segment follows a path which eventually returns to its origin.
Stippled paint creates a unicursal knot design. The aggregation of dots suggests straight lines not present in the artwork. This single component is connected end to end with two rotated copies to create the knot. Doubling the individual segments of the knot design does not disrupt the over / under pattern that knot diagrams have.
album art and some extras for guitar duo KillDry
Intersecting spirals become one line. The line can be traced from any point back to its origin.
This piece is an exploration of knots as lattices.
Each line segment traverses four triangles. The pattern contained in each group of four triangles includes two pairs of rotated copies.
I chose cut paper as the medium. Physically lining up a grid of cut triangles is a fool’s errand, so I chamfered the corners of the larger triangles. This meant that whatever substrate I used as a background would show through the gaps, so I started out by painting a panel gold.
I constructed the panel out of 1/8″ material with 2″ strips on the side.
Starting from the middle was the easiest approach since I could line up the edges of the first triangles perpendicularly to the panel. Then, I worked my way outward.
I let the edge pieces overhang and used a little extra glue to add rigidity to make it easier to trim the edges at the end.
This is a two-part piece. The small section indicated in the diagram here makes up the entire lattice. There are several versions of the fundamental component, such as the one mentioned earlier, so this is just the one I chose. However, with the use of six colors, I’m pretty sure this is as basic as it gets.
The second part was constructed much larger, since one at the same scale seems a little puny. This way, they end up being more similar in size.
I almost started out with cutting a hexagon first, but decided to be much more cavalier…
…with a circular saw.
You may be wondering if a saw blade just shreds the edges of paper like this. It does, but they were fairly easy to clean up with a knife.
Also, my plan was to seal both of the pieces with mulberry paper and varnish which gives a really cool look. The paper alone looks almost like it was just printed out. The final result looks a lot more natural.
This piece was on display at the City County Building in Knoxville, TN in 2024.
It was most recently on display at the d’Art Center in Norfolk, VA as a part of the Fibrous exhibition.
This knot is (my version of) very straight forward. Two unicursal knots with four-fold rotational symmetry intersect, interacting with one another as they do. In this way, each knot acts like a line since the lines making up the individual knots also have to weave in and out of one another.
The metal is thin steel. The two different colors of steel were achieved by heating until the desired color appeared and quenching in linseed oil. The pieces of one knot were first lightly oxidized, so that the rust was burnt into the metal.
The backing is solid maple rubbed with Danish oil and polished to a glass-like finish.
Limited edition serigraph prints of this design are available to order from my online store.
This piece demonstrates the difference between Listing bands with odd and even numbers of twists, as well as band components with two types of asymmetry. One photo on its own doesn’t really give enough visual information to understand the shape. Multiple photos just end up being confusing because different angles look like completely different pieces of art. Videos make this much easier.
The numbers punched through the bands show that a cycle of numbers will repeat as long as the cycle is an odd number. (Even numbers only work with band components that are completely symmetrical.) In a design like this, where it is easy to view numbers from many angles and orientations, most Hindu-Arabic numerals aren’t ideal because they will often look backwards. Because of the twist in the loop, each number would need to appear forward and backward at the same time anyway.
Cistercian numbers were an inspiration for the symbols I created for this piece. Someone never having seen these symbols could figure out what value each one has, because each numeral’s value is just the number of line segments.
The two types of symmetry used for the band scales are shown here. In order for the ends to meet at any point in the band, a single twist is necessary for one, and a double twist is necessary for the other.
Can you tell which is which? You can always cheat by looking at the pictures and video.
One added bonus of bands this size is that the structures are self-supporting.
This piece uses a single component in two configurations to form a hexahedron (cube) and helix.
The component itself is a negative knot (it is knocked out of the overall shape). The knot is a simple twist with one rounded side and one angular side: a circle and a square.
I love the almost optical illusion of this kind of knot: looking at either the light or dark ink paths, at a glance, it appears as if there are two offset triangular figures. Tracing that path quickly reveals that something much weirder is going on. The same basic knot is doubled, with its copy being rotated in place.
At the time I drew this, my mind was preoccupied with the sculptures of Wenzel Jamnitzer (or at least Jost Amman’s woodcuts of them, assuming the sculptures ever existed).
Limited edition serigraph prints of this design are available on the online shop of the Arts & Culture Alliance.
Five knots are bent into the shape of a larger knot, the symbol for infinity. The five knots are the same, with different degrees of distortion and smoothing:
The infinity symbol is really just a loop with one twist. The smaller component knots are loops with three twists.
I charred the fir (the dark backing behind the copper) and brushed it with a brass wire brush several times. This transfers a tiny amount of the brass into the wood and gives it an unusual shine.
Eight knotted hexagons cover the surface of a cube in a pattern which is identical on each face of the solid.
The cube is positioned so that observing the sculpture from a certain angle makes the presence of a hexagon centered around each vertex more obvious.
One hexagon is emphasized further with silver leaf, which was done after airbrushing the knot pattern on all sides.
In order to have a cube propped up on an edge, I needed to build a stand for the piece.
I recessed a mirror below where the sculpture would go because I wanted the viewer to be able to see that all the lines continue around the piece, and that all the faces are identical.
Finally, I made a viewfinder so that someone encountering the piece would know exactly what to do.
sort of pretending the viewport is a wizard’s staff
Here is a video of the piece installed. It’s just not the same with still photos.
One fun thing to watch is people seeing the icon, the viewport, and figuring out what to do.
Two unknots are arranged vertically above a charred wood base. The shape of the metal (painted aluminum) makes self-supporting convex and concave arrangements possible.
An unknot is a simple, closed loop. This particular one has sharp angles, but it still unfolds to something equivalent to a circle.
My interest in the unknot relates to the rules you have to follow to make a knot diagram. A branching path presents a problem when trying to represent the over- / under-lapping of crossing paths: what happens when a path branches? Turning a path into an unknot by using a copy of itself as negative space centered in the branching path removes this problem.
The charred wood of the base resembles embers in light, and glows in the absence of light.
Two sets of interlocked hexagonal rings traverse three color fields.
The flat painting and the sculptural painting are versions of the same knot using 2d and 3d grids. However, the 3d knot is not a projection of the 2d knot on a solid object. It’s more like the 3d knot is wrapped with a version of the knot that conforms to a grid on its surface.
I did two small versions of the 2d knot. The more black and white one is spray paint; the one with more colors is acrylic paint and pencil.
I have used a for similar to this one as a base to hold up a few other smaller sculptures. It’s great because it can hold a cube, sits flat on a surface, and is easy to construct. This one was much larger than previous ones.
Limited edition serigraph prints of this design are available.
A small version of the 2d painting was on display in the City County Building in Knoxville in 2023.
As if there weren’t enough ideas in this piece, here’s one more: the colored shapes behind the knot, either on the flat painting or as seen isometrically on the sculptural one, are part of the progression of the aperiodic monotile transformations. In other words, even though the arrangement of the red, green, and blue shapes can be arranged to form a hexagon as they are here, they can also be arranged to aperiodically tile a plane.
“Oh, cool.”
Craig Kaplan
(I was lucky enough to share this piece and its explanation with Craig Kaplan in person, but he is welcome to correct or retract his statement which was part of our discussion.)
Copies of a single metal tile are woven together to form a cube with a concave vertex.
Any form based on connected cubes can be created with this tile.
Assembly is a matter of bending the tabs just enough to they curve into the next tile’s slot. The ones on the corners are more difficult, but putting everything together took less than an hour (the way I remember it).
This piece was recently on display at RED Gallery as part of an A1 Lab Arts group show.
Three rectangular solids make up the base. Brass pins hold the three pieces together without glue.
This piece was recently on view at the McGhee Tyson Airport Gallery.
A three-hexagon structure repeats to form a lattice made of knots.
A painted wood panel makes up the background with cut paper pieces as the knots and lines. I used templates to align the paper on the panel.
A much smaller detail version of this piece was on view at the City County building in Knoxville in 2023. It now lives in New Hampshire.
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.
A unicursal decagonal star intersects five chained quadrilaterals. The chain gives the impression of a five-pointed star.
This thing is big: 41″ x 41″ x 4″. It’s also heavy. It hasn’t been shown in public because it’s too cumbersome to move. And it’s hard to photograph.
I’ll be honest. I had a really big piece of glass and I wanted to try making enormous lap joints, so I made this.
The design combines a unicursal knot with some extra chained shapes.
After starting the design for this one, I wanted to go bigger than I have before. I printed the knot out on one large piece of paper so the alignment wouldn’t be dependent on my ability to tape tiled sheets of paper together.
Then I needed a giant hexagon for it to go on. After planing some boards down and gluing them up, I made a giant rectangle and cut the ends off at angles.
I really do love it when I get to use a circular saw to make something that counts as fine art.
This is one of those stages where I think, “I mean, it looks pretty cool. Maybe I could just stop here.” But, no.
Actually before I got to this point, I practiced using the same 1/8″ copper strip I used for my tercet chain knot. The technique I used on that piece was hammering a modified X-acto blade to make the grooves. It took forever, so I thought I might be able to just sort of cut it into a softer wood just using the knife without hammering. This was a terrible idea because the wood I used had a really open grain and my design was mostly curves. Utter disaster. Laughably so.
That’s when I completely changed everything, working much larger and using a design with only straight lines. Still not wanting to take an actual year to cut all those little lines, I further simplified the design. I also got my hands on a cute little router base for my Dremel tool. Finally, I used wire instead of strips.
Matching the depth of cut to the thickness of the wire was a little tedious. Each line took several passes because the bit was so tiny.
And now everything was set up to actually being really working. Hammering the wire in really is so satisfying. Unless you mess it up, in which case it’s terrifying.
Stippled paint creates a unicursal knot design. The aggregation of dots suggests straight lines not present in the artwork.
This single component is connected end to end with two rotated copies to create the knot.
Doubling the individual segments of the knot design does not disrupt the over / under pattern that knot diagrams have.
This artwork was created for the Knoxville-Oak Ridge chapter Hadassah membership directory.
The design is a detail of a much larger lattice. The knots, cut out of different colors and thicknesses of paper, form tiles which weave into one another. A gradient of a diamond pattern gradually appears in the lower section.
The piece is 11″ x 12″ x 1.5″ and can be hung or stand on its own.
It was donated to Hadassah, to be auctioned with all proceeds benefitting the organization.
A collection of 14 short pieces for traditional and electronic instruments, CUSP was written entirely during February, 2020 when the pandemic was just visible on the horizon. For me, it’s a reminder of how recently things went from normal to sideways.
On a less dramatic note, CUSP is my first large-scale attempt at blending classical and electronic styles. The word “cusp” is familiar to me in one context as a graphic designer, as a cusp node is a dramatic change in direction of a Bézier curve. A cusp has a similar appearance in medieval architecture. The usual definition of the word relates to being right on the edge of something, at the border between worlds.
CUSP was created using Sibelius and Ableton.
For me, making a new artwork begins with a question I don’t immediately know the answer to. It ends with the artwork as the proof that the answer I came up with is true. The middle part, figuring it out, is often more time consuming than shaping the actual piece, although it is the part I enjoy the most.
The most direct example of this is stippling, in which an image is produced using only dots. Starting with a subject and minimizing elements while maintaining the original idea is always one of my goals.
When I have an idea for a piece, I try to research whether or not something like it has been done before. If the challenge I’ve identified seems like it might have already been addressed, I will wait until I have fully realized it before researching its originality, since the fun part is figuring it out.
After studying graphic design and working in this field for several years, my desire to take the principles of design and apply them to media other than paper or the screen has steadily increased.
A work of art has been fully realized when an artist actively participates in the actions of creating, shaping, and framing. When only one or two of these actions have been done, this only means that the artwork is not as fully realized as it could be. The explanation of these components is intended to be general enough to work with any art form.
I try and make an effort to think about these aspects as much as I can during the art making process.
Creating is the idea behind the piece. Purely, there is no physical component, although realistically there might be a medium required to transmit the idea, such as a recipe or musical score.
When time allows, I use the Osborn Parnes method of Creative Problem Solving to come up with the best solution. This helps to narrow down ideas and gives me a bunch of leftovers for later use.
Shaping is the rearrangement of raw materials to produce organization. A piece of charcoal is a burnt stick becomes a drawing – the same material reorganized.
I try to choose the medium and techniques used for each piece relevant to the idea.
The spatial and temporal boundaries which contain the work establish a frame for an artwork’s beginning and ending – where the art is and where the rest of the world is. This might be the edge of the stage or the resting of a conductor’s baton.
Artwork placement in a space is a form of framing, as the architecture or surroundings themselves might be boundaries.
(I like to make my own actual frames as well.)
Two of the most inspiring artists to me are Donald Judd and M.C. Escher. It would be a stretch to draw parallels between the work of these two, but one thing they do have in common is the effect they have on me. There is something that “rings true” when I look at their work, as if the work answers a question I didn’t realize I had. The work of both artists provides an endless supply of layers and relationships so that viewing never gets old.
A line broken by an inner negative space rotates to form an unknot, a continuous line with no overlapping. The component used as the basis for this piece, as well as the resulting structure, can be unraveled to form a loop.
Tri Unknot was on display at the McGhee Tyson Airport (TYS) in 2023.
A unicursal line is inlaid in copper on the face of a hollow truncated tetrahedron. The knot creates a cognitive visual illusion of three shapes versus two.
The illusion that there are sort of 3 shapes but only really 1 is something I like about this one.
A group of three similar polygons rotate to form a knot. Multiple glass layers show colors, the knot itself, and a faint outline.
As a test, I made a small version with some glass squares I had left over from another test piece. The goal was to create different layers for each of the colors, the black line, and a frosted outline of the whole thing.
As a test, I made a small version with some glass squares I had left over from another test piece. The goal was to create different layers for each of the colors, the black line, and a frosted outline of the whole thing.
The repeated component in this piece had specific colors with it. The idea was to have overlapping layers of transparency in the color. Part of this piece was experimenting with knot lines on an axonometric grid.
The maquette was made with inkjet printing on acetate. It doesn’t look terrible, but I wanted the whole piece to be much larger. This was more for concept than anything, just to see if it might look cool. And it does! But it’s just a stack of loose glass.
The final piece has a frame, I used paint for the colors, vinyl for the black, and etched the glass for the…etched glass part. It ended up being the white background plus 3 layers of glass which fit in slots in the frame.
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 self-intersecting line rotates to create a hexagonal shape. The line can be followed from any point all the way around to loop back to its beginning.
This was another experiment involving cut paper on charred wood. My first attempt at realizing this design was with wire in encaustic medium, shown here.
Even though it was tedious, things were going great until I sealed the next layer. The heat required to fuse the wax (which isn’t much) was enough to bend the very thin brass wire out of whack. After that, I just threw in another experiment, using crumpled tissue paper as a layer in the encaustic.
It was interesting, but I wanted something a little neater. And less disaster-prone. So, I went with cut paper. On charred wood. What could go wrong with knives and fire?
A parallelogram and a triangular line rotate to create an overall triangular shape. Encaustic medium covers gilded ink.
My first step was putting the main design down in ink. (I phrased the media used in this piece as “Patty Ink” because it once belonged to Pat Lauderdale, someone who always encouraged my artistic endeavors. She died in 2016 and I was given some of her art supplies, so I like to make reference whenever I use them.)
Next, I put gold leaf along the lines, scraping off some of the leaf to reveal the color underneath.
Finally, I added a few layers of encaustic wax to give it a little more depth. The frame is charred wood.
This piece was on display in the Schwarzbart Gallery in 2022.
One component of an interlocking tile extends its connectors and uses color variations to imply depth and an isometric perspective. Stained veneer is inlaid in a wood background.
This piece involves a bit of experimentation with an isometric knot. The interlocking square shapes require more than one color to give the effect that they are 3d-ish.
This piece is a single tile with its connectors extended. A repetition of the tile looks more like this:
Keeping track of the colors in just one tile was involved enough for me:
This piece was on display in the Schwarzbart Gallery and at the City-County Building in Knoxville in 2022.
A single shape tiles the faces of all three equilateral triangle-faced regular polyhedra with knots. Only the bend angle of the shape changes between the solids. The knot also tiles the plane.
Notice that the knot on the tetrahedron appears to be interlocked triangles; squares for the octahedron; and pentagons for the dodecahedron. Hexagons are reserved for the 2-dimensional version of this knot.
This piece shows that a single triangular tile can tile a flat plane as well as the Platonic deltahedra.
Exhibition history for this piece
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.
A shape with interlocking edges tiles a plane.
The edges of these planes bend to form a hexahedron.
A cube or rectangular solid of any size can be created from any multiple of this component. This one uses 24 components and makes a cube.
Twenty-three identical scales are tied together in a continuous loop representing a non-orientable surface. A pattern of accented hexagons covers interlocking segments which twist and bend in an infinite knot.
This band is smaller than others I have done so far, so it is self-supporting and can be displayed resting on a surface or suspended from above.
The component illustration highlights the hexagonal counter-spaces which were painted gold.
The final piece is approximately 18″ in diameter, but let me know if there is a better way to notate the dimensions of something like this….
This piece was accepted as a part of the 2022 Bridges Conference in Aalto, Finland.
It is now part of the permanent collection of the Experience Workshop’s traveling collection.
While working on the artwork, I listened to the album many, many times. I experimented with several techniques, but a semi-controlled corrosion process on metal was the final medium. What is semi-controlled, you ask? Unless you go out of your way to stop rust, it just keeps happening. These image comparisons show the cover artwork before, after, and way after blasting the meticulously etched, pristine lines with flames and chemicals. The stage of deterioration photographed for the album artwork shows some of the original purity of the lines.
Those lines mimic guitar stings. Their arrangement into a loop makes reference to the moon, which is what the phrase “dark star” made me think of. The corrosion relates to gritty, complex sounds from the pedals. Originating from precise metal wires, signals decay and transform into something more chaotic and more complex than the original.
The reverse of the album cover is a silhouette-like image highlighting a contemplative face. I used similar techniques for the corrosion along with masking. I would show the reference photo too but I promised never to show the mystery person’s face.
Variations on the knot design were used as the center labels for the actual LP.
This piece was displayed at the City County building in Knoxville during 2024.
This collection of artworks further explores the use of repeated, simple shapes and lines translated or rotated to create complexity in a knot design. Each piece is based on a component, shown on the respective artwork posts.
A variety of techniques and media were used to bring the designs to life. The shapes and symbols they create are purely geometry, with no hidden meaning beyond the beauty of intricacy.
All work in this collection was created for and displayed at the Schwarzbart Gallery during January of 2022.
A line broken by an inner negative space rotates to form an unknot, a continuous line with no overlapping. The component used as the basis for this piece, as well as the resulting structure, can be unraveled to form a loop. Tri Unknot was on display at the McGhee Tyson Airport (TYS) in 2023.
A unicursal line is inlaid in copper on the face of a hollow truncated tetrahedron. The knot creates a cognitive visual illusion of three shapes versus two. The illusion that there are sort of 3 shapes but only really 1 is something I like about this one.
A group of three similar polygons rotate to form a knot. Multiple glass layers show colors, the knot itself, and a faint outline.
Four lines make a reversed knot pattern covering all faces of a hexahedron, regardless of rotation (this same pattern can tile a plane).
A self-intersecting line rotates to create a hexagonal shape. The line can be followed from any point all the way around to loop back to its beginning. This was another experiment involving cut paper on charred wood. My first attempt at realizing this design was with wire in encaustic medium, shown here. Even though it…
A parallelogram and a triangular line rotate to create an overall triangular shape.
One component of an interlocking tile extends its connectors and uses color variations to imply depth and an isometric perspective. Stained veneer is inlaid in a wood background. This piece involves a bit of experimentation with an isometric knot. The interlocking square shapes require more than one color to give the effect that they are…
A single shape tiles the faces of all three equilateral triangle-faced regular polyhedra with knots.
A shape with interlocking edges tiles a plane. The edges of these planes bend to form a hexahedron.
Twenty-three identical scales are tied together in a continuous loop representing a non-orientable surface. A pattern of accented hexagons covers interlocking segments which twist and bend in an infinite knot.
The design is an experiment in using spirals in combination with radial symmetry to produce a unicursal knot. Usually I base knot designs like this off of circles and polygons, so this was a new challenge.
I originally used this design for some cut paper stationery, shown here. Wanting to take things up a notch, and wanting to practice on my new scroll saw, the natural thing to do was scale it up about 600% and use several processes I’d never tried before.
The first thing that needed to be done was choose an exact size based on available materials. Smoky Mountain Vintage Lumber had some really cool reclaimed chestnut for the background and ipe for the frame…the giant, giant circular frame. I’ll come back to that.
The background was constructed by first gluing the chestnut pieces to a piece of 1/8″ MDF. Plywood would have been sturdier but weighed much more. Next I used my super-sophisticated circle cutting jig which consists of 2 pieces of wood and a nail atop my table saw. This method only works when you want to cut a solid circle and don’t mind a tiny hole in the center of the back.
Calculating but ultimately guestimating the frame width I could get from the ipe, I cut the wood into 16 pieces with 11.25° ends. Then, a couple of pieces at a time, I glued the frame segments together knowing it would be a Hanukkah Miracle if they lined up in the end. That’s why I left the final segment out of the gluing process and cut a larger piece to fit the gap at the last minute.
Just as I put off dealing with the frame while working on this, I will put off explaining the process one more time. I was ready to do the fun stuff anyway, cutting out all those little knot segments. So many segments…
I had some mahogany just about the right size for gluing printouts of these templates to. Several days and 33 segments later, I printed out a full size template on letter size paper (tiled and taped together) and determined it was instead well worth $10 to get it printed at FedEx on one big piece of paper.
Well, it would have been if they had followed my instructions to print it at 100% of the file size I gave them even if it cut off some of the margins. Bitter was my disappointment when, after cutting the segments out of the paper, I found that the template had been printed at ~99% size, which made it impossible to use as a placement guide. They were nice enough to print another one for “free” which makes me wonder how much anything really costs anyway….
Just gluing everything in place would have been quicker, but it was so much more amusing to make a stop motion animation of it happening.
Back to the frame. I had only ever made one circular frame before. It was several years ago, my dad helped me, and neither of us could recall exactly how we did it. Coincidentally the cheap mirror in my studio bathroom fell apart (actually fell off the wall) so I used this as an opportunity to practice.
For anyone wondering how to do this starting with a solid circular piece of wood: align your work piece so that a circular dado cut by a plunge bit will be centered in the circle. Make sure your dado is deep enough for a trim bit bearing. Use a jigsaw to cut reasonably close to the inside edge. Clean things up with a trim bit.
Allow me to interrupt myself and tell you about the side project I squeezed in here: I needed a new studio bathroom mirror for the one that fell of the wall, so I used what I had learned so far about making giant circles.
This was starting to become more about the frame, but I’d rather spend the time to make something like this than go out an buy it. Also, you can’t go out and buy it. Not the big one anyway.
This jig was slightly more sophisticated, but the main difference is that this one is actually clamped to the fence. This way you can use the same side-to side movement to sneak up on cleaning things up as well as switching from the outside cut to the inside cut.
And then I was finally done with the most worrisome parts. The mahogany pieces were finished with boiled linseed oil before placement, and the same was used on the finished frame after some sanding. The chestnut was left alone. I fashioned some aluminum clips to hold the piece in the frame and use a cleat across the back to hang it on the wall.
tri spiral unicursal knot 39"ø, mahogany on chestnut in ipe frame 2021
Here I am on a podcast talking about art and music. It’s proof that I can ramble on for at least an hour. Thanks to Thomas Zachary for asking me to do this.
The artwork I’m talking the most about can be seen on the Knot Theories page.
This work was on display during March, 2021 at The Emporium Gallery in downtown Knoxville.
Knots are continuous lines twisted in such a way to give the illusion of lines overlapping in the same plane. Using a formalized system of knot creation, the artwork here represents knots in multiple “in-between” dimensions in three categories, each based on a partial dimension:
Each piece is based around a simple line or shape translated or rotated to create more complexity. Light and shadow are an important element, so artworks appear different in their environment depending on the time of day. The shapes and symbols they create are purely geometry, with no hidden meaning beyond the beauty of intricacy.
Listen to an interview featuring this show on KAANP.
the quintessential knot
three braided knots as a listing band
stacked slices of an image
deconstructed listing band segment
how to create a piece of art that you can see from all angles?
isolated polyhedra faces
Three lines make a reversed knot pattern covering all faces of a tetrahedron, regardless of rotation (this same pattern can tile a plane).
copper knot inlaid in wood
Three triangular twists are abound together by an encompassing shape echoing the group’s perimeter. The way the paper is cut and overlapped gives the illusion that each of the four main components are made of single pieces of paper. You have to cheat a little to give the illusion of self-intersecting lines made out of a…
A hexagonal ring bound to another triply-twisted ring is suspended vertically.
This is a double-lined, triple-component knot based on an isometric grid. The rules of knot making are slightly bent with the double line, but half the fun of making your own system is breaking its rules. It is “projected” is because its construction uses small wooden dowels to lift the individual pieces above the surface.…
Follow the (implied) line with your eye. The line is unicursal, meaning there’s only one line. At first you might be tempted to assume there are five sections, and of course there are, but the radial symmetry is just a property of this particular line. These are probably the most difficult knots to design because…
knots entangled with smaller versions of themselves
repeated, interlocking shapes forming a Listing band
I chose this knot as one of the representative symbols for the gallery window as it clearly shows the basic rules of knot creation: 1) an alternating over / under pattern 2) with no angle corners intersecting lines and 3) radial symmetry.
Here is the same knot, just for fun, graffitied on metal.
The final piece was done on charred wood with a technique which makes the knot visible only at certain angles and certain lighting conditions (chatoyancy). In direct sunlight, the knot is barely visible until a shadow falls on it.
The position of this piece in the gallery actually lines up with the window at a certain angle from outside.
A knot can be infinite in terms of translation if its left edge lines up with its right edge. That means that a knot can surface a cylinder. But, what about a non-orientable surface?
If a knot (segment) is created so that its upper right lines up with its lower left, and the segment is repeated an odd number of times, the whole unit can surface a one-sided surface.
Here is a maquette of a solid (flattened) version of a knot as a Listing band.
This one shows the 3 individual knots more distinctly:
Once again, look at those shadows! In this example, I could have chopped up the knot to give the illusion of lines going over and under other lines, but the end goal was to make something solid, where this one was simply printed on acetate. I actually started wondering if I could somehow encase a material in something clear so I could make a much heftier version and still have the over-under implication, but I ended up going with something much more down to earth. Metal!
The pieces were designed with three holes at the end of each segment so I could pop rivet everything together and everything would self-align.
I liked the colors I picked for the maquette, but I wasn’t about to paint anything when metal can look so amazing and varied just by using harsh chemicals. After degreasing and shining everything up, I applied a clear sealant for the blue line, gun bluing for the black, and rust juice for the red. Note the lack of photos of bluing the metal because I was scared to death I would poison myself.
I waited to do the final assembly until just before I was ready to display it to avoid the possibility of damage. It’s very delicate for something made out of braided metal.
Perfectly symmetrical listing bands are difficult to display because of one of those things that bring us all down, like gravity. But, if you could slice one (or really anything that’s hard to balance) up, you could stack the slices and create the illusion of a dimensional solid.
Being able to view this from multiple angles makes a lot more sense:
The first version, shown in this video, is made up of 40 pieces of glass, each with a cross-section of a simple listing band. The mind does its wonderful job of smushing and the illusion of a thing, as opposed to just a stack of lines, appears.
The final piece was 80 10″ x 10″ pieces of Plexiglas (which I learned allows for more optical transmission than glass) each with a different shape painted on its surface. The frame was made from busted Chinese chestnut, which I think contrasts nicely with the pristine precision of stacks of thin lines.
This piece sits outside the main gallery because I wanted it to be an introduction to the rest of the pieces: its design is much simpler but its implementation invites the viewer to look closely to find out what’s really going on.
One thing I find fascinating about this shape is that it is the same as the symbol on the window. It just looks completely different from every angle.
The two things you don’t want wood to do are bend and twist. The two things listing bands do are bend and twist. Is there maybe a compromise?
If alternating cuts are made in opposite sides of a piece of wood (or any rigid material) the material can more or less bend. Of course, the material itself isn’t bending as much as the overall bend, but it averages out to a much greater bend that would be possible without the cuts.
The same applies to twisting.
The horizontal spaces allow for twisting and bending to happen.
Notice how the top and bottom fit into one another. This maintains the vertical translation during scale chirality alternation.
A wire was to be placed as a connector as shown here.
I thought a nice touch would be the floating circle in the “head.” Because drilling holes that lined up through a piece of wood thin enough to twist-bend would require a Hanukkah miracle, another solution was to rout out a curved groove half way through mirror images of the scales. Then the wire could be placed in the groove and the wood laminated together.
I CNCed a template which I used to trace on the wood…
…drilled some holes to make life easier…
…and went to the band saw. There were casualties.
Then I used an edge trim bit and the original template to clean things up. I actually designed the whole thing based around the clearance I’d need for the bit to clear the interior zig-zag shapes which was nerve-wracking to say the least.
Again, because of the thickness of the wood, I had to treat it very gingerly. To rout the grooves, I made router passes of 1/256 of an inch at a time. It took a while.
But it was worth it!
When I saw the way the scales looked linked together, I liked it so much that I wanted to show them like that instead of contorted into a band.
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.
Outlined paths of three single twists are chained together.
Even self-intersecting loops (twists) follow the over-under rule of knot creation.
Three triangular twists are abound together by an encompassing shape echoing the group’s perimeter.
The way the paper is cut and overlapped gives the illusion that each of the four main components are made of single pieces of paper.
You have to cheat a little to give the illusion of self-intersecting lines made out of a single piece of paper….
A hexagonal ring bound to another triply-twisted ring is suspended vertically.
Upon closer inspection, you can see that there are nine pieces that make up the two rings. The four supports are placed in a way that supports all the pieces and maintains consistent gaps between them.
This is a double-lined, triple-component knot based on an isometric grid. The rules of knot making are slightly bent with the double line, but half the fun of making your own system is breaking its rules.
It is “projected” is because its construction uses small wooden dowels to lift the individual pieces above the surface. This lets the viewer see the knot at different angles beyond the surface plane just by moving around. Also, the complexity of the pieces is emphasized by the projected shadows.
The first iteration of this design (below) was purchased after being shown by The Arts & Cultural Alliance.
The final piece was constructed from aromatic cedar framed in walnut.
Added bonus: to keep everything organized during assembly, I color-coded an illustration of the piece. The colors were all over the place for easy identification, but they looked so cool, I put them on a shirt.
The finished piece teeters on the edge of simplicity and complexity. I like letting the natural patterns of wood do the work for surface treatments.
Follow the (implied) line with your eye. The line is unicursal, meaning there’s only one line. At first you might be tempted to assume there are five sections, and of course there are, but the radial symmetry is just a property of this particular line.
These are probably the most difficult knots to design because you have to imagine the same numbers of aligning starting and stopping points as you have divisions of a circle. In other words, for a pentagonal rotation like this one, you have to imagine a line which will connect with itself every time you rotate it 72 degrees.
The materials I chose for this are a combination of very fancy and exotic and down-to-earth. My uncle probably had no idea I was going to end up making this when he tore down his old barn. (Thanks for the materials!) The knot itself is padauk, which is so nice I almost hate to use for anything but marimba keys….
This piece was recently on display at RED Gallery as part of an A1 Lab Arts group show.
Self-similarity within structures is always fascinating to me. After creating a knot encompassed in an element of itself (above), I thought it would be nice to also show a progression from the simplest component to a much more complex version of itself which would in turn cycle back to the beginning.
Each of the final pieces changes slightly from the one before it to create the cycle.
You may be wondering, where did that nice oxidization come from? In my studio, I have a clearly labeled bottle of RUST JUICE for just such an occasion. On properly pickled metal, you can actually see the rust form within seconds.
Crest is a music and video collaboration between Adriano Capizzi a.k.a. Metunar and me. We decided that a good starting point might be to take something old and unfinished that seemed like it had potential, but was just sort of stuck.
I dug out something I had started 20 years ago (the first few seconds of the piece) and we started adding on a little every time we sent files back and forth. Using Dropbox and Ableton Live Lite, we each used our preferred methods of music writing to add to or change the other’s ideas. Being across the world from one another, if each of us worked on it a little each day, we could both get up in the morning to find more music added to the project. Communication was done almost entirely through Dropbox Paper, which was much easier than checking emails. It was also nice to use a system integrated with the files themselves.
I used Sibelius to score what I wrote, which made it that much easier to share with Joshua Weinberg (flute) and Tyler Neidermayer (bass clarinet) of Apply Triangle when they graciously agreed to donate their time to playing on this project.
I constructed a single marimba key to play the repeated A♭. It was probably excessive, but now I know why marimbas are so incredibly expensive.
When the music was done, I wanted to make a video in the style of Metunar’s previous videos. He suggested that instead of doing a whole new collaboration on the video, we share the videos and images we had to each create our own. My two older kids helped with making the videos, which were shot in the middle of a field in Knoxville, on a camping trip in Cades Cove, and on a leisurely drive through Gatlinburg. I used Magic to create my visuals and an ancient copy of Premiere for the editing.
Is there a shape, which when repeated, can create a Mobius strip?
Yep, there is. Really all you have to do is chop up a strip into squares, however many pieces you want and there you go, done.
That was easy. Maybe I’m not asking the right question.
Is there an asymmetrical shape which interlocks with itself to create a continuous band on a non-orientable surface?
That one is a lot better, but it seems a little unrealistic that I would have started out wondering that instead of the first question.
To create a simple shape that tiles rectilinearly, you can start with a square, and any change you make to one side, you make the opposite change to the other side. So, if you squish in from the right, you squish out from the left. Continue until you have something interesting.
Listing bands have the additional twist (pun intended) that at some point, the top of one tile (or “scale” which is what I’m calling the individual components) is eventually going to have to fit in with the reverse of the bottom of another. Only bands made up of odd numbers of scales will work. This is probably easiest to understand if you consider that just one scale, twisted into a band would have to fit into itself this way, and one is an odd number. Even numbers simply twist too many times for an asymmetrical scale.
This particular band is made up of nineteen individual scales. They were fabricated from .03″ mild steel and allowed to rust naturally. The scales were designed so that just the right amount of twist and bend could happen with this size and material.
The base was constructed from fir and walnut. Displaying a band this large was a challenge, since it sort of collapses if it’s set on the floor. The base allows easy viewing from multiple angles, which is really necessary to get an idea of how complex the shape is. Fortunately there’s a sweet spot that allows just the right distribution of weight so that it’s balanced and sturdy on the base.
In lieu of a maquette, I did a bunch of calculations, which is usually a recipe for disappointment. Luck was on my side this time.
Is there a shape (solid) which can be repeated 6 times and interlocks to form a hexahedron? I’m still not entirely sure. This may even have search results, but I never looked it up. I wanted to figure it out for myself.
I mean, I still haven’t figured it out, but I got kind of close.
Just thinking about it only got me to the point where I realized I needed something in my hands. My first attempt was with foam core, but as you can see from the painter’s tape, I wasn’t entirely sure this would work. The idea is there, but I think I figured out that 2 of the sides need to be different than the other four.
So, I moved on to balsa wood because it’s very easy to cut and has a thickness you can work with. And it’s wood. More or less.
This is where I got overconfident. I had some leftover poplar from another project so I went straight for the power tools. What you see are the panels that made it. My router ate the rest.
But that was the table router, and I knew that the concept was there if I could just make one and keep all my hand parts. So, I went to the CNC router, using whiteboard because of its very uniform thickness. Also, I wanted a hexahedron I could draw on with dry erase markers for yet another project.
That worked fine, but now I wanted something a little fancier so I returned to poplar. I also wanted to test my band saw skillz [sic] which are lacking yet pretty well concealed here.
The router approach was my favorite, although I had to clean up the interior corners. On the final box, I finished the corners with a roundover bit which I think does a better job of bringing the eye around the corners and makes the box more of an object of interest.
So, if you know the answer to my original question, I guess you can just go ahead and tell me now. I feel like I’ve done enough to not feel lazy.
P.S. Yes I know you can just bevel the edges, but the panels have to interlock and all be identical.
The title of this piece refers to the five groups of repeated motives. The groups are six, twelve, fifteen, twenty-four, and thirty measures long. Each group is made up of a trio of the same instruments (for a total of fifteen instruments). The groups of repeated segments begin at the same time and repeat until their ends line up.
The entire piece is 360 measures long. Each movement is 120 measures long, with a brief pause between movements, and new motives for each player. Around these pauses, the music appears to deconstruct, like the tiles of an offset brick wall without an edge.
Every one in a while, two or more groups share the beginning of a repetition which is accented in the music. Besides these accents and the deconstructions, ten instruments are always playing. The piece is scored for three each of clarinet, organ, vibraphone, piano, and marimba.
The artwork is a visual representation of the piece. The five columns are the five instrument groups and the overlaid rectangles are the length of their respective tessarae.
Knot: Legend III is the third in a series of lengthy, multi-movement instrumental works for chamber ensemble. The movements were written in the order they were with the intention of being able to start from any point and loop your way to where you started, as if following the path of a unicursal knot.
I studied graphic design in college. When it came time to fill a gallery with my best work, I used it as a chance to create all new sculptural works. I’m not sure why this was allowed, other than nobody had tried it before.
Even at the time, I didn’t think of these pieces as sculpture. I thought of them as ideas that had 3-dimensional instances. Proportion, spacing, symmetry, rhythm, symmetry: those are all design things.
Decades after the original show, only a few pieces of concrete and wood remain.
