One of my specialties is inlay, here's a possiblility from that POV. Although you could do this all by hand, a small router or Dremel tool with a flat base will make the task faster and cleaner.
Begin with a piece of hardwood ply for your board. Use the router to cut out the infinity pattern then set in a piece of contrasting veneer that fits and will end up flush when done.
Option one, same color for entire infinity pattern. First cut the infinity pattern from some thick veneer. You can use a coping saw for this, cut outside your line and sand the contour smooth. Securely tape this to your board and scribe a line around the contour. Remove the pattern and route the board between the scribed lines to a depth just shy of that of the infinity inlay. Some trial and error will be needed for a good fit. Glue the inlay into the routed slot. Have some sawdust from the inlay veneer material handy that you can rub into the still damp glue at the edges (after scraping off most excess). Overlay with waxed paper, a flat board and either clamp or weight down to bed the inlay in its recess. (If you use black veneer for your inlay material, glue it with black superglue, no sawdust needed. This glue is also less likely to expand over time. http://www.stewmac.com/Materials_and_Supplies/Glues_and_Adhesives/Glues/StewMac_Super_Glues/StewMac_Tinted_Super_Glue.html)
To define squares, you could woodburn across the inlay. Or, with a tiny Dremel bit, and a straight-edge guide, route across the inlay and inlay strips of another color wood or soft metal. Sand the entire surface flush when done.
To keep the underlying board clean, seal it with a wash coat before any routing to keep sawdust form the inlay from dirtying it up. A closed-grain wood such as birch is less likely to be hampered by this.
Option two. Make a pattern of your infinity symbol and use it to scribe the base as outlined above. Route out the pattern in the base and inlay individual squares of alternating colors.
Investment in a Dremel is highly worthwhile if you do any handywork around the house or shop. I recommend a cordless version, batteries are long lasting, much more convenient and you need not compensate for the pull of a cord on an otherwise light weight tool.
You can inlay many other materials besides wood veneer. The image below shows one of my guitar headstocks inlaid with abalone, mother-of-pearl and "stars" which are pieces of gold wire set into drilled holes; all sanded flush with the surface. In the lower image, a box for my niece, the ebony oval is inlaid into the surrounding wood. Sivler, gold and MOP have been inlaid into the ebony. If you use these kinds of materials, you can engrave or scratch (scrimshaw) lines into the inlay.
Here's some resources.
Thick dyed veneers: http://www.dyed-veneer.com/dyed-veneers---thick.aspx
[Dremel base, cheap.] https://www.google.com/express/u/0/product/8389998418404119239_15480024933055777178_8613692?mall=Arizona&directCheckout=1&utm_source=google_shopping&utm_medium=product_ads
Dremel base, nice: http://www.stewmac.com/Luthier_Tools/Tools_by_Job/Tools_for_Inlay_and_Pearl_Cutting/Precision_Router_Base/Precision_Router_Base.html?utm_source=google&utm_medium=shopping&utm_campaign=2017-03-gp&gclid=Cj0KEQjwzd3GBRDks7SYuNHi3JEBEiQAIm6EIwlduJOXoiVw42Eazi4aiQy_wclBhtWWJmvVW-XTKNcaAjiC8P8HAQ
For another option with veneers, search "marquetry".