Anyway, this is what you need - and it's not exclusive to the 'Mini Club' of course.
Fabric: There are of course many types of fabric you can stitch on, but a good place to start is quilting cotton. You can get it in lots of solid colours and you can also stitch on patterned quilting cotton of course. I normally stitch my pattern samples on white fabric, but you can pick any colour you fancy. Kona Cotton comes in over 300 colours, for example!
Thread/floss: I pretty much always use six stranded cotton embroidery floss from DMC, but there are many other brands available. And different types of thread, perle cotton for example. I will be using two strands of floss when stitching the samples for the club, but do use more if that's what you like. Experimenting is good.
Needle: You will need a crewel/embroidery needle for this type of embroidery. It has a sharp point that can make a temporary hole in the fabric to let you pull the thread through. The size of needle you should use is a bit up to your preference, but sizes between 6-8 should work. You can get packs of needles with different sizes so you can see which size you prefer.
Scissors: You can use fabric or craft scissors to cut the thread, but I really recommend using embroidery scissors. The sharp point makes it easier to cut only what you're supposed to cut and not accidentally snip through something you already stitched.
Transfer method: I use a water soluble pen (from the Hemline brand), but there are lots of other options. You can use a pencil, a Frixion pen, iron on pencil, transfer paper etc etc. I like the water soluble pen because it's pretty reliable and it's easy to get hold of.
Hoop: You don't have to use a hoop, but I find that it makes my stitching a lot neater when I do. It just makes it so much easier when the hoop is holding the fabric taut for you. All the motifs in the club will fit in a 4" hoop.
Speaking of the size of the motifs. Some of them will fit in a 1" hexagon. Others can be easily modified to fit it too. Because the motifs are mostly abstract, organic shapes and elements, you can remove some of these elements to fit the motif in a smaller space.
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