### Hyperbolas from Light Fixtures

When you see light from a fixture shining against a wall, you often observe some very clear, sharp curves as the boundaries between light and dark areas. Those curves are generally hyperbolas. Why? The light from the source is generally blocked off to produce a cone of light, and then that cone is intersected with the wall, creating a conic section. For typical arrangements of light fixtures, that section is generally a hyperbola.

### Tours are Modular

Most math tours will consist of a series of “stops” or vignettes, in which something observed in the world around us spurs discussion of a particular mathematical topic, idea, or result. Often these different vignettes can be mixed and matched and regrouped to create new tours in new places or even new tours in places you’ve toured before, but with a different theme. So in constructing tours, it’s extremely useful to have a large “bag of tricks” — a big collection of different things that can come up in the natural world or the built environment, and bits of interesting math that you can hang off of them. So, the idea for this blog in posts going forward is to share such a collection. We’ll post different topics that have come up in different MoMath tours, starting with a rapidly-designed tour that took place at the ASTC 2015 convention. And please feel free to comment, including describing similar activities you’ve done in math tours, or totally new ideas. If you have an interesting one in a comment, we’ll invite you to do a guest post in this blog.

So: go out there and create some great tours and let us know about them.

### Dream a Theme

• ✹ You want the tour to gel into an experience for the participants.
• ✹ Distills and reinforces a “take-home message”.
• ✹ Helps to filter the myriad of ideas that you will encounter once you’re looking at the world through mathematical lenses.
• ✹ Conversely, also helps you to generate fresh ideas, as you flesh out your theme.

### Use the Built Environment

• ✷ Shows math is useful and that people use it to make our world a better place.
• ✷ Mathematical ideas are used so often for and in decoration and adornment. Shows math off as a creative and aesthetic endeavor.
• ✷ These items will be the mainstay of your tour construction.
• ✷ They provide a source of reliable “modules” that can be inserted into virtually any tour.

### Use the Natural Environment

• ❅ Math is intrinsic to our world
• ❅ That’s shown best by “finding” the math in nature
• ❅ Great potential for the “Aha!” effect
• ❅ Shows math as relevant and connected

### The Value of Props

• ✪ Three-dimensional, tactile props for your talk spark engagement and curiosity.
• ✪ Nothing creates a “magic moment” like pulling out the unusual or offbeat, yet highly appropriate, prop from your bag.
• ✪ Props can be something you use, or that the participants use, or be for “show and tell.”
• ✪ Don’t be afraid to use items you encounter, perhaps in surprising ways. A “found prop” is magical, too.

Method A: walk a mile in their shoes

• ✦ The tour can start anywhere and end anywhere
• ✦ Practice being an observant walker
• ✦ Look up, look down, look left and right
• ✦ Invest the time needed
• ✦ Take careful notes

Method B: go with what you know

• ✦ Where do you spend a lot of time? Where are you intimately familiar with?
• ✦ Accumulate items of interest
• ✦ Keep a “math diary”
• ✦ Great for a last-minute/spur of the moment demo

### Finding the Hidden Math

This blog of the National Museum of Mathematics not only records highlights of many of the math tours given by the Museum over the years, but but includes ideas and processes used to create those tours, as a resource so that you can create tours, too. So the first idea to get across is that you indeed can create tours that will help other people perceive the mathematics that infuses the world around and that is nearly ubiquitous. All it takes is a genuine love of mathematics, and opening your own eyes to see the mathematical connections and ramifications that surround us every day.

The first several posts go through a number of the key elements and aspects that underlie essentially any tour. You can create a tour in almost any setting, indoors our outdoors, suitable for rain or shine, and with topics suitable for any target age range/mathematical background. Once you dive into this, you will find that the question becomes not “oh my gosh, how will I find anything to say?” but rather “Wow, how can I ever pare this down to a one-hour tour?” A few key principles lay the groundwork for a successful tour, and we will highlight those first, in the next several postings. With some care and a bit of practice, you will be able to keep tour participants rapt with the wonders of mathematics.