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Gymnocalycium damsii is a very easy to grow and very free-flowering cactus.
 

Description: Gymnocalycium damsii is a very easy to grow and very free-flowering cactus, it may be solitary or freely offsetting.
Stem: Spherical-depressed to short cylindrical, up to 10 cm tall , 8-15 cm in diameter, green, pale-green or brownish-green, and highly decorative,  often with horizontal clear and dark strips.
Ribs: About 10-11, broad to acute, and more or less notched into chin-shape protrusions or warts .
Radial spines: 5-8 up to 12 mm (or more) long, straight or slightly tortuous, short, nearly useless in terms of defence. Young spine are pastel-white or yellowish with a darker tip, later become pale-grey.
Central spines: Usually absent.
Flowers: From the centre, narrow, funnelform, about 6 cm long. External tepals greenish whit reddish margins, internal tepals white to light pinkish or greenish spreading. Scales on the ovary and flower tube small, scattered.
Blooming season: It produces flowers almost continuously during the year.
Fruit: Oblong, 2,5 cm long, 6 mm in diameter, reddish at maturity.

Note: Seems to be a pretty variable plant you might get some other names which are effectively the same plant.... and also not sure all plants identified as this are really this.
 

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Scientific name: Gymnocalycium damsii (K. Schumann) Britton & Rose
The Cactaceae, descriptions and illustrations of plants of the cactus family 3: 163, f. 175, 1922

Origin: Brazil, Bolivia, Paraguay.

Habitat: G. damsii (anisitsii) grows in open areas in semi-shade under the protection of low bushes that have no or very few leaves. The light filters through. The soil is usually  a rich sandy loam. Rains are frequent during the summer. It often form large groups and is extremely variable.

Conservation status: Listed in CITES appendix 2.

Synonyms:

Synonyms:

• Echinocactus damsii K. Schumann
  In: Gesamtb. Kakteen Nachtr. 119. 1903
• Gymnocalycium anisitsii ssp. damsii (Schumann) G. Charles
  In: CSI 20: 17, 2005
• Gymnocalycium anisitsii (K. Schumann) Britton & Rose 1922
• Gymnocalycium damsii var. anisitsii Catalog name in Schneekloth-Liste A
• Gymnocalycium tucavocense Haage SL 1992/93?
• Gymnocalycium damsii var. tucavocense Backeberg nom. inval. (Art. 8.2)
  In: Descriptiones Cactacearum Novarum 3: 6,1963
• Gymnocalycium anisitsii var. tucavocaense
• Gymnocalycium joossensianum
• Gymnocalycium damsii var. multiproliferum P. J. Braun
  Kakteen und Andere Sukkulenten 42 (10): 230, 1991
• Gymnocalycium damsii var. centrispinum Backeberg nom. Inval.
  Descr. Cact. Nov. 3: 6., 1963
• Gymnocalycium damsii var. rotundulum Backeberg nom. Inval.
  Descr. Cact. Nov. 3: 6., 1963
• Gymnocalycium damsii v. robural
• Gymnocalycium damsii v. roseiflorum
• Gymnocalycium damsii v. torulosum
  Descr. Cact. Nov. 3: 6., 1963
• Echinocactus denudatum bruennovianus Haage jr.,
  In: Monatsschr. Kaktee, 8:37. 1898
• Gymnocalycium griseopallidum

Cultivation: It is a summer grower species that offers no cultivation difficulties.  Water regularly in summer (but do not overwater ) Keep this plant almost dry in winter at a minimum temperature of 0°C, prefer relatively rich substrate and  low pH  compost (if possible not limestone) otherwise growth will stop altogether. Feed with a high potassium fertilizer in summer. It is quite frost resistant if kept dry (hardy to -5° C)
Sun Exposure: Light shade. It may tolerate bright situations  but is likely to suffer from sun scorch or stunted growth if over exposed to direct sunlight during the hottest part of the day in summer. This plant needs plenty of space for its roots, repotting should be done every other year or when the it has outgrown its pot. This species is particularly easy and accommodating, seldom suffer of cryptogamic diseases.
Propagation: Direct sow after last frost. (seldom produces offsets)
Seed Collecting: Permit fruit to ripen.  Fruit must be significantly overripe before harvesting seed; clean and dry seeds.

Photo of conspecific taxa, varieties, forms and cultivars
 of plants belonging to the Gymnocalycium anisitii  complex (This Taxon has lots of synonyms ( like many other cacti) with several controversial varieties and subspecies):

• Gymnocalycium damsii
• Gymnocalycium damsii var. rotundulum
• Gymnocalycium damsii var. tucavocense (G. anisitii ssp. damsii)
• Gymnocalycium damsii var. tucavocense "RED FOM"
• Gymnocalycium cv. KIKKO (G. anisitsii forma mostruosa) 
• Gymnocalycium cv. RED BLUSH (hybrid)

All the information and photos in cactus art file are now available also in the new the Enciclopedia of Cacti. We hope you find this new site informative and useful.