Christmas is a fantastic opportunity for me to share some mathematics with loved ones. Among my preferred ways to do this is by stitching geometrical designs on cards. The magic is how the straight lines produce best curves.
Here’s how it’s done. Draw 2 straight lines that intersect. Draw points along each of those lines at equivalent ranges. When you join the dots from one line to the other, as in the star and tree above and in the 4 images listed below, you get a parabola. Strictly speaking, the curve is the envelope of the family of straight lines.
Let’s begin basic: here are 2 parabolas in a square:
Twitter Pinterest The angle between the intersecting lines makes no distinction-you constantly get a parabola. Here are 3 within a triangle.< div class="block-share block-share-- short article hide-on-mobile"data-link-name="block share"> Facebook Twitter< path d=" M16.363 8C12.133 8 10 11.13 10 13.74 c0 1.582.58 2.988 1.823 3.512.204.086.387.003.446 -.23.04 -.16.137
a leading tip). The initial step is to draw the style on a paper the same size as the card– this conserves time in the future, and conserves messing up the cards themselves if you need to redesign the design. Then put a folded tea towel on a flat table (the tea towel safeguards the table), put the card face-up over it, then hold the design firmly in location and utilize a pin to make a hole for each point. Remember to
make the holes on the middle panel of the card! You can begin sewing. I discover that a single strand of stranded cotton works well – stranded cotton is reasonably cheap, and can be found in great deals of colours (including sparkly ones, perfect for Christmas). Use a length around 50cm. If you utilize longer then you’ll get tangled up very rapidly. To start and finish, just use sticky tape on the back to secure the thread neatly. This all gets covered at the end, when you can use double-sided tape to glue one panel over the back. I need to focus very tough at this moment to make certain I glue the right side down, otherwise the card opens in reverse! My last preferred household of styles is likewise based upon a circle of dots, but this time I discover that 72 dots works best. Imagine the points as being numbered consecutively from 1 to 72. Sign up with each point to its double (so 1 to 2, 2 to 4, 3 to 6, 4 to 8, and so on). If you do this with a ruler, it becomes rather simple to discover a pattern without needing the dots to be numbered, merely move one end of the ruler along one dot and the other end 2 dots each time. This is what you get: < a class="rounded-icon block-share __ product block-share __ item-- facebook js-blockshare-link" href="https://www.facebook.com/dialog/share?app_id=180444840287&href=https%3A%2F%2Fwww.theguardian.com%2Fscience%2Falexs-adventures-in-numberland%2F2015%2Fnov%2F26%2Fsolving-for-xmas-how-to-make-mathematical-christmas-cards%3FCMP%3Dshare_btn_fb%26page%3Dwith%3Aimg-10%23img-10&picture=https%3A%2F%2Fmedia.guim.co.uk%2F9492316b1cc21fa6e791e326c31e962e6613acbd%2F0_0_1121_1141%2F1121.jpg" target =" _ blank" data-link-name="social facebook" > Facebook< a class="rounded-icon block-share __ product block-share __ item-- twitter js-blockshare-link" href="https://twitter.com/intent/tweet?text=Solving%20for%20Xmas%3A%20how%20to%20make%20mathematical%20Christmas%20cards&url=https%3A%2F%2Fwww.theguardian.com%2Fscience%2Falexs-adventures-in-numberland%2F2015%2Fnov%2F26%2Fsolving-for-xmas-how-to-make-mathematical-christmas-cards%3FCMP%3Dshare_btn_tw%26page%3Dwith%3Aimg-10%23img-10" target =" _ blank" data-link-name="social twitter" > < path d="M21.3 10.5 v. 5c0 4.7-3.5 10.1-9.9 10.1-2 0-3.8 -.6 -5.3 -1.6.3 0.6.1.8.1 1.6 0 3.1 -.6 4.3-1.5 -1.5 0-2.8 -1 -3.3 -2.4.2 0.4.1.7.1 l. 9 -.1 c-1.6 -.3 -2.8 -1.8 -2.8 -3.5.5.3 1.4 1.6.4 -.9 -.6 -1.6 -1.7 -1.6 -2.9 0 -.6.2 -1.3.5 -1.8 1.7 2.1 4.3 3.6 7.2 3.7 -.1 -.3 -.1 -.5 -.1 -.8 0-2 1.6-3.5 3.5-3.5 1 0 1.9.4 2.5 1.1.8 -.1 1.5 -.4 2.2 -.8 -.3.8 -.8 1.5-1.5 1.9.7 -.1 1.4 -.3 2 -.5 -.4.4 -1 1-1.7 1.5 z" > Twitter Pinterest The envelope of these lines is called a cardioid, due to the fact that of its similarity to a heart.
A cardioid also develops as the curve traced out by a point on one circle that rolls around another circle of the exact same size. Take 2 1p coins. Stick one to the table. Mark a point on the rim of the other. Roll the moving coin around the edge of the fixed one, and the marked point will trace out a cardioid. Cool, isn’t it ?! Doing something similar however joining each indicate three times itself (so 1 to 3, 2 to 6, 3 to 9, 4 to 12, and so on) results in a curve called a nephroid, due to the fact that of its resemblence to a kidney. This is also the shape that light makes at the bottom of a mug. If you shine parallel light rays at a surface area that’s curved like a circle, the reflections form a nephroid. I’ll leave you to explore what takes place if you join each indicate * 4 * times itself. I’ll just say that it deserves drawing! < div class="block-share block-share-- post hide-on-mobile" data-link-name="block share" > < a class="rounded-icon block-share __ product block-share __ product-- facebook js-blockshare-link" href="https://www.facebook.com/dialog/share?app_id=180444840287&href=https%3A%2F%2Fwww.theguardian.com%2Fscience%2Falexs-adventures-in-numberland%2F2015%2Fnov%2F26%2Fsolving-for-xmas-how-to-make-mathematical-christmas-cards%3FCMP%3Dshare_btn_fb%26page%3Dwith%3Aimg-12%23img-12&picture=https%3A%2F%2Fmedia.guim.co.uk%2F1c13445f86c82c632010cb39edc3c634ba605adb%2F0_152_2048_1438%2F2048.jpg" target =" _ blank" data-link-name="social facebook" > < svg width="32" height="32" viewbox =" -2 -2 32 32" class="inline-share-facebook __ svg inline-icon __ svg" > < course d="M17.9 14h-3v8H12v-8h-2v-2.9 h2V8.7 C12 6.8 13.1 5 16 5c1.2 0 2.1 2.1v3h-1.8 c-1 0-1.2.5 -1.2 1.3 v1.8 h3l -.1 2.8 z" > Facebook < a class="rounded-icon block-share __ product block-share __ product-- twitter js-blockshare-link" href="https://twitter.com/intent/tweet?text=Solving%20for%20Xmas%3A%20how%20to%20make%20mathematical%20Christmas%20cards&url=https%3A%2F%2Fwww.theguardian.com%2Fscience%2Falexs-adventures-in-numberland%2F2015%2Fnov%2F26%2Fsolving-for-xmas-how-to-make-mathematical-christmas-cards%3FCMP%3Dshare_btn_tw%26page%3Dwith%3Aimg-12%23img-12" target =" _ blank" data-link-name="social twitter" > < svg width="32" height="32" viewbox =" -2 -2 32 32" class="inline-share-twitter __ svg inline-icon __ svg" > < path d="M21.3 10.5 v. 5c0 4.7-3.5 10.1-9.9 10.1-2 0-3.8 -.6 -5.3 -1.6.3 0.6.1.8.1 1.6 0 3.1 -.6 4.3-1.5 -1.5 0-2.8 -1 -3.3 -2.4.2 0.4.1.7.1 l. 9 -.1 c-1.6 -.3 -2.8 -1.8 -2.8 -3.5.5.3 1.4 1.6.4 -.9 -.6 -1.6 -1.7 -1.6 -2.9 0 -.6.2 -1.3.5 -1.8 1.7 2.1 4.3 3.6 7.2 3.7 -.1 -.3 -.1 -.5 -.1 -.8 0-2 1.6-3.5 3.5-3.5 1 0 1.9.4 2.5 1.1.8 -.1 1.5 -.4 2.2 -.8 -.3.8 -.8 1.5-1.5 1.9.7 -.1 1.4 -.3 2 -.5 -.4.4 -1 1-1.7 1.5 z" > Twitter < a class="rounded-icon block-share __ product block-share __ product-- pinterest js-blockshare-link" href="http://www.pinterest.com/pin/create/button/?description=Solving%20for%20Xmas%3A%20how%20to%20make%20mathematical%20Christmas%20cards&url=https%3A%2F%2Fwww.theguardian.com%2Fscience%2Falexs-adventures-in-numberland%2F2015%2Fnov%2F26%2Fsolving-for-xmas-how-to-make-mathematical-christmas-cards%3Fpage%3Dwith%3Aimg-12%23img-12&media=https%3A%2F%2Fmedia.guim.co.uk%2F1c13445f86c82c632010cb39edc3c634ba605adb%2F0_152_2048_1438%2F2048.jpg" target =" _ blank" data-link-name="social pinterest" > < svg viewbox="0 0 32 32" width="32" height="32" class="inline-share-pinterest __ svg inline-icon __ svg" > < course d="M16.363 8C12.133 8 10 11.13 10 13.74 c0 1.582.58 2.988 1.823 3.512.204.086.387.003.446 -.23.04 -.16.137 -.568.18 -.737.06 -.23.037 -.312 -.127 -.513 -.36 -.436 -.588 -1 -.588 -1.802 0-2.322 1.684-4.402 4.384-4.402 2.39 0 3.703 1.508 3.703 3.522 0 2.65-1.136 4.887-2.822 4.887 -.93 0-1.628 -.795 -1.405 -1.77.268 -1.165.786 -2.42.786 -3.262 0 -.752 -.39 -1.38 -1.2 -1.38 -.952 0-1.716 1.017-1.716 2.38 0.867.284 1.454.284 1.454l-1.146 5.006 c -.34 1.487 -.05 3.31 -.026 3.493.014.108.15.134.21.05.09 -.117 1.223-1.562 1.61-3.006.108 -.41.625 -2.526.625 -2.526.31.61 1.215 1.145 2.176 1.145 2.862 0 4.804-2.693 4.804-6.298 C22 10.54 19.763 8 16.363 8" > Pinterest Vicky Neale is the Whitehead Speaker at the Mathematical Institute and Balliol College, University of Oxford. This is a visitor post on Alex Bellos’s maths blog. Alex’s newest book Snowflake Seashell Star: A Colouring Adventure in Numberland, consists of images to colour in like the ones above. The book is out today in the United States titled Patterns of deep space.
