This is a mobius strip made from sterling silver. Dartmouth College has the truly wonderful Donald Claflin jewelry studio where I've had the pleasure of learning many techniques in my time there. This simple piece was made by twisting a piece of wire and soldering it together. The mobius band is a well known counterintuitive object: it only has one side despite the strip it's made from being two sided. It is a very approachable way to introduce some mathematics, as anyone can make one by taping a twisted strip of paper together. Several topological questions can be asked, for example how many sides does a similar band have with two, three, four, or more twists? And one of my favorites, what happens when you cut along the center? In a normal piece of paper, this cuts it into two pieces, but you might be surprised what happens here! These below three pieces are made with gemstone beads. Several combinatorial questions are naturally suggested when beading. For example, how many necklaces of length n can you make with k beads? Additionally, each piece is made with the same 'type' of gemstone. Most minerals receive their color from impurities, and as such they can appear in many colors despite having the same structure (as compared to gems whose color comes from structural elements, such as peridot, so they only appear in one shade). The gemological terms for this are "allochromatic" and "idiochromatic" respectively. For example, the necklace and longer bracelet are made from varieties of quartz: amethyst (purple), citrine (yellow), rose quartz (pink) and smoky quartz (grey). Quartz, chemically silicon dioxide, is the most common mineral on earth, and have a variety of gem names for historical reasons. Quartz can also be translucent, in which case it is called chalcedony, or opaque, in which case it is called jasper (which also comes in many location-specific forms). The smaller bracelet is made with three varieties of beryl: aquamarine (blue), morganite (pink) and emerald (green). Jade, calcite, and common opal are other minerals that come in nearly every color. 
Here, the longer bracelet encodes a superpermutation. A superpermutation is a sequence of numbers from the set 1, 2, ..., n whose subsequences of length n give every possible permutation of this set. One could simply listing all of these, so the challenge is to find the shortest list that does so. This problem was referenced in a anime, where a character wonders what is the quickest way to watch all 14 episodes of a show in every order. Following this an anonymous 4chan user proved a novel lower bound on the length of the shortest superpermutation. This example reminds us that math is all around is if we look, and great insight can come from anywhere. Similarly, a DeBrujin sequence ennumerates all strings of some length on a finite alphabet. The necklace contains all strings of length four in the alphabet a, b, c, represented by the pink, yellow, and purple beads. These objects can be constructed in a number of fun ways, including finding an Eulerian path in an associated graph. They have applications to allow a sensor to determine it's position in space (it can read a few bits at its location, then since each string only appears once, this determines it's location), card tricks, and procedural generation in video games. As a technical note, DeBrujin sequences are cylic, so it's important the associated necklace does not have a clasp to disrupt the pattern. The bracelet is a similar constrained ennumeration. Here it gives all permutations of three elements where adjacent permutations differ only by swapping two elements. This can also be viewed as trying to find an Eulerian Path in an associated graph, the permutahedron, and is related to the problem of change ringing. An added neat fact is that this constraint means we can represent the sequence are a braid by drawing three dots for each permutation, and connecting them for each swap. The resulting braid, if connected back to its start, makes the Borromean rings! These pieces of artwork can be made interactive by playing a game: pick a random permutation of four elements, for the first, or string of length four on a three letter alphabet, and try to find it in the piece! 